###### Abstract

In epidemics many interesting quantities, like the reproduction number, depend
on the incubation period (time from infection to symptom onset) and/or the
generation time (time until a new person is infected from another infected
person). Therefore, estimation of the distribution of these two quantities is
of distinct interest. However, this is a challenging problem since it is
normally not possible to obtain precise observations of these two variables.
Instead, in the beginning of a pandemic, it is possible to observe for
infection pairs the time of symptom onset for both people as well as a window
for infection of the first person (e.g. because of travel to a risk area). In
this paper we suggest a simple semi-parametric sieve-estimation method based
on Laguerre-Polynomials for estimation of these distributions. We provide
detailed theory for consistency and illustrate the finite sample performance
for small datasets via a simulation study.

Keywords: Laguerre-Polynomials, Semi-Parametric Estimation, Sieve-Estimation,
Epidemics

##  1 Introduction

A dominating question in the public evaluation of the COVID-19 situation
during the 2020 pandemic is the estimation of the basic reproduction number
$R_{0}$, the number of new infections which are caused (on average) from a
single infected individual. While this number is prominently discussed in the
news about COVID-19, it is an important variable for disease transmission in
general (Wallinga and Lipsitch (2007); Leavitt et al. (2020)). In order to
estimate the reproduction number of a disease the so-called generation time
$G$ plays an important role (see Euler-Lotka-Equation e.g. in Britton and
Scalia Tomba (2019)). The generation time $G$ is defined as the time
difference between the infection time of a randomly chosen infected individual
and his or her infector. Let $\varphi_{G}$ denote the density of the
generation time $G$. Then $R_{0}^{-1}=F_{R}(G)$, where (this is the Euler-
Lotka-Equation, see e.g. Britton and Scalia Tomba (2019); Wallinga and
Lipsitch (2007))

$F_{R}(G):=\int_{0}^{\infty}e^{-rt}\varphi_{G}(t)dt$ (1.1)

and where $r$ is the exponential growth rate of the expected incidence. A
straight forward way of estimating $F_{R}(G)$ (and hence $R_{0}$) would thus
be to replace $\varphi_{G}$ by $\hat{\varphi}_{G,n}$ in (1.1), where
$\hat{\varphi}_{G,n}$ is a non-parametric estimator of $\varphi_{G}$. This
strategy is often called _plug-in estimate_ and is of course not limited to
$F_{R}$ but it can be applied to any function $F$ of the density
$\varphi_{G}$, for instance mean value or quantiles. In the following we will
call such functions $F$ _features of $\varphi_{G}$_. In practice it is not
always (or rarely) possible to make observations of the generation time
directly because the infection times cannot be observed (cf. Nishiura et al.
(2009)). However, the urgency of the situation requires estimation with
imperfect data (Ferretti et al. (2020)). Sometimes there is additional
information about the infection time available (e.g. if a person was only
during a certain window $[0,W]$ in a region where the virus is circulating,
see Lauer et al. (2020); Bi et al (2020) for examples related to travel to and
from Wuhan in the early days of the pandemic). This leads to observations of
the infection times with measurement error from which we have to infer
$\varphi_{G}$. Other methods for estimation of $R_{0}$ make sometimes implicit
assumptions about the generation time (cf. Wallinga and Lipsitch (2007)).
Testing for such assumptions can be realized within the same framework if the
test statistic can be written as a function of an estimate
$\hat{\varphi}_{G,n}$, e.g., similarly to Härdle and Mammen (1993): In this
case the feature $F$ of $\varphi_{G}$ of interest would be the test-statistic.
In this paper we aim to do the first step to tackle this type of problems:
Consistent, non-parametric estimation of the generation time density
$\varphi_{G}$. Based on such an estimator the theory for semi-parametric
inference methods about features of $\varphi_{G}$ like $R_{0}$ or test
statistics can be developed. We mentioned above that this is a measurement
error problem. In order to illustrate that the issue cannot be simply
resolved, we need to be more specific about the involved quantities.

We study observations of transmission pairs as shown in Figure 1. The first
person gets infected at time point $T_{1}$ which is unobserved. However, it is
known to lie in the interval $[0,W]$. The exact conditional distribution of
$T_{1}$ in $[0,W]$ is allowed to also depend on a random variable $C$ which
determines for example the location of the infection. In certain situations,
it will turn out that the likelihood will depend only on $\min(W,S_{1})$,
where $S_{1}\geq T_{1}$ denotes the time at which Person 1 shows symptoms. In
that sense, observation of $W$ is only then required if $W\leq S_{1}$.
Otherwise we do not have to observe $W$ but we have to observe that $W\geq
S_{1}$. The _incubation period_ is defined as $I_{1}:=S_{1}-T_{1}$. In an
analogue fashion, $S_{2},T_{2},I_{2}$ are defined for the second person.
Moreover, Person 1 is known to have infected Person 2. Person 2 shows symptoms
at the observed time $S_{2}$. But, of course, the infection time $T_{2}$ is
unobserved. The generation time is thus $G:=T_{2}-T_{1}$ and it is also
unobserved, while the serial interval $S:=S_{2}-S_{1}$ can be observed. Such
data was for example collected by Ferretti et al. (2020); Bi et al (2020);
Lauer et al. (2020).

Figure 1: Schematic depiction of the important quantities and their relation.

The following relations can be directly read off from the definitions and
Figure 1.

$\displaystyle S_{1}=$ $\displaystyle T_{1}+I_{1},$ (1.2) $\displaystyle
S_{2}=$ $\displaystyle T_{1}+I_{2}+G,$ (1.3) $\displaystyle S=$ $\displaystyle
G+I_{2}-I_{1}.$ (1.4)

It is reasonable to assume that the incubation periods $I_{1},I_{2}$ have the
same density $\varphi_{I}$. Denote by $\varphi_{S}$ the density of $S$. Our
interest lies in estimation of $F(G)$, where $F$ is an arbitrary feature of
$G$. As a first example let $F$ be linear, e.g.
$F(G):=\mathbb{E}(G)=\int_{\mathbb{R}}t\varphi_{G}(t)dt$ be the expectation.
In that case by (1.4)

$F(G)=F(S)+F(I_{1})-F(I_{2})=F(S)=\int_{-\infty}^{\infty}t\varphi_{S}(t)dt$

and $F(G)$ can probably be well (maybe even optimally) estimated by replacing
$\varphi_{S}$ above by $\hat{\varphi}_{S,n}$, where $\hat{\varphi}_{S,n}$ is a
non-parametric estimator of $\varphi_{S}$ based on the observed quantities
$S_{1,1},...,S_{1,n}$ and $S_{2,1},...,S_{2,n}$. When letting
$\hat{\varphi}_{S,n}$ be the standard kernel density estimator with an order
two kernel, this strategy results in the average as estimator for the mean
value. For the case of the basic reproduction number, however,
$F_{R}(G)=\mathbb{E}(\exp(-rG))$ is basically a moment generating function,
i.e., it is non-linear. Other examples of interesting non-linear features of
interest are quantiles, a quadratic test statistic or the probability of pre-
symptomatic infection $\mathbb{P}(G\leq I_{1})$. For these features, we cannot
simply estimate the serial interval but it is necessary to have an estimate of
the density $\varphi_{G}$ available. Moreover, since the interest lies in a
function of $\varphi_{G}$ rather than $\varphi_{G}$ itself, it might be
desirable for researchers to not impose parametric assumptions on
$\varphi_{G}$ but to consider a non-parametric approach instead. Further
arguments why a non-parametric approach might be desirable compared to a
parametric approach can be found in Groeneboom (2021). In this paper, we will
construct a sieve estimator based on Laguerre polynomials to estimate
$\varphi_{G}$ and $\varphi_{I}$.

General background information about data analysis in disease transmission can
e.g. be found in Held et al. (2019); Chowell et al. (2009). In the beginning
of a pandemic it is important to estimate its transmission characteristics (Bi
et al (2020)) and it is common to replace the unobserved generation time $G$
by the serial interval $S$ (cf. Britton and Scalia Tomba (2019); Ferretti et
al. (2020); Wallinga and Lipsitch (2007)), because this can be observed in
clinical studies. However, this practice yields biased estimates (cf. Britton
and Scalia Tomba (2019)). The estimation of $R$ from observations of the
serial interval in particular is discussed in Lipsitch et al. (2003). As an
alternative Ferretti et al. (2020) suggest the following two step procedure:
Firstly, fit a parametric model for the incubation time (cf. Lauer et al.
(2020); Bi et al (2020)) and then, secondly, use this to fit a parametric
model to the generation time. Groeneboom (2021) describes how to do non-
parametric estimation of the incubation time. Problems related to under-
reporting or delays (cf. Azmon et al. (2014)) are not considered here because
we have data in mind which was collected by researchers rather than
observational data from self-reporting. In addition to this type of
experimental data, it is also possible to collect larger sets with covariates
and use prediction techniques to identify transmission pairs (cf. Leavitt et
al. (2020)). Specific parametric results about the above mentioned quantities
for Covid-19 can e.g. be found in Bi et al (2020); Ferretti et al. (2020);
Ganyani et al. (2020); Lauer et al. (2020); Tindale et al. (2020) and in many
other places. The setting described in (1.2)-(1.4) is close to de-convolution
and measurement error settings, classical results about which can be found for
example in Fan (1991); Devroye (1989); Carroll et al. (2006). General
background about sieve estimators and series estimation can be found in Newey
(1997); Chen (2007) and the references therein. Ai and Chen (2003); Newey and
Powell (2003) use sieve estimation based on moment conditions while we start
from a likelihood. Further reading about plug-in sieve estimates can be found
in Shen (1997); Chen and Shen (1998).

The main contribution of this paper is to provide the first step for semi-
parametric estimation of features of the generation time: Consistent density
estimation. To this end we extend the methodology of Ferretti et al. (2020);
Ganyani et al. (2020) in two ways: Firstly, we estimate the densities of the
incubation period and the generation time in one step with the same data and,
secondly, we provide a semi-parametric framework which makes no a-priori
assumptions about the incubation and generation times. We pursue this by
constructing a sieve estimator based on Laguerre polynomials highlighting the
flexibility of Laguerre polynomials as approximating functions. In order to
identify the model we need to specify a model for the relation between the
exposure window $[0,W]$ and the infection time $T_{1}$ but we will not make
parametric assumptions about the distributions of the incubation period and
the generation time. Thus, we will do semi-parametric estimation.

The structure of this paper is as follows: In Section 2 we introduce the exact
modelling framework and provide a sieve estimator for the incubation period
and generation time based on Laguerre polynomials. Its properties will be
discussed in Section 3 and we show that our estimator is asymptotically
consistent. The methodology will be studied in Section 4 by means of
simulations and it will be applied to a real-world dataset of CORONA infection
pairs. The R-code which is used for these computations is available on github
(https://github.com/akreiss/ SemiParametric-Laguerre.git). Finally, we finish
the paper with some concluding remarks in Section 5. Additional simulation
results and some proofs are collected in the Appendix.

##  2 Model and Methodology

Let $(W_{i},T_{1,i},C_{i},S_{1,i},S_{2,i},G_{i})_{i=1,...,n}$ be iid copies of
$(W,T_{1},C,S_{1},S_{2},G)$ as in Section 1. However, we suppose that our
observations are only given by
$(S_{1,i},S_{2,i},\widetilde{W}_{i},C_{i})_{i=1,...,n}$, where
$\widetilde{W}_{i}:=\min(S_{1,i},W_{i})$. Recall that $I_{1}:=S_{1}-T_{1}$ and
$I_{2}:=S_{2}-T_{2}$. Since we only have observations for $(S_{1},S_{2})$, we
can only make inference about their joint distribution and the serial interval
$S:=S_{2}-S_{1}$. Thus we have only indirect information about $I$, $G$ and
$T_{1}$ by means of (1.2)-(1.4). In other words, identification of the joint-
distribution of the three random variables $(I,G,T_{1})$ from a joint
distribution of the two variables $(S_{1},S_{2})$ is impossible without
additional assumptions. Throughout we will assume the following to be true:

Assumption (M): Model
_$I_{1},I_{2}$ and $G$ are non-negative and $C$ takes values in
$\\{1,...,K_{C}\\}$. Their distributions have the following densities (with
respect to the Lebesgue measure) or count measures and relate as follows_

$\begin{array}[]{lll}W&\sim&\varphi_{W}\\\ C&\sim&p_{C},\textrm{ independent
of }W\\\ T_{1}&\sim&\varphi_{T_{1}}(\cdot|W,C)\,\textrm{ cond. on }\,(W,C)\\\
I_{1},I_{2}&\overset{iid}{\sim}&\varphi_{I},\textrm{independent of
}(T_{1},W,C)\\\ G&\sim&\varphi_{G},\textrm{ independent of
}(I_{1},I_{2},T_{1},W,C).\end{array}$ (2.1)

_It holds that $\varphi_{T_{1}}(t|w,c)=\mathbbm{1}(t\in[0,w])n(w,c)h(w-t|c)$
where $h:[0,\infty)\to[0,\infty)$ is integrable._

When inspecting the likelihood of Ferretti et al. (2020) and the assumptions
of Ganyani et al. (2020), we see the same independence assumptions for
$I_{1},I_{2},G$ and $T_{1}$. Independence between $I_{1}$ and $T_{1}$ can also
be found in Groeneboom (2021). The variable $C$ can be understood as covariate
and will be interpreted as the location of the infection. It can be used to
capture heterogeneity in the infection dynamics: In a location with an
increasing number of infections, it is more likely that the infection time
$T_{1}$ lies towards the end of the infection window $[0,W]$ conditionally on
$(C,W)$. In contrast, in locations with a low number of infections, $T_{1}$ is
possibly uniformly distributed in $[0,W]$. See Remark 2.1 below for further
details and a motivation for certain choices of $h$. It has been reported (see
e.g. Tindale et al. (2020)) that the transmission dynamics, i.e.,
$\varphi_{I}$ and $\varphi_{G}$, can be different in different scenarios.
Thus, we need to restrict to data coming from the same environment. Moreover,
the above assumption means that we observe true transmission pairs (see e.g.
Tindale et al. (2020) for a parametric model which allows for an unknown
number of intermediate infectors).

######  Remark 2.1.

All discussion below is conditional on $W$. The dataset we imagine is recorded
in the beginning of a pandemic, e.g. a person comes home after travelling to a
region where the virus is circulating. Suppose that the time points, at which
the person of interest has an infectious contact (i.e. a contact which
definitely leads to an infection), are the jump points of a counting process
$N$ with intensity function $\lambda:[0,W]\to[0,\infty)$. It is plausible that
the intensity of infection is proportional to the number of infections in the
population. Since (in most cases) the infection numbers behave exponentially,
we also consider $\lambda(r)=\lambda_{0}\exp(-\alpha(W-r))\sim\exp(\alpha r)$,
where $\alpha\in\mathbb{R}$ describes the exponential growth (or decay) of the
number of infections. In this notation, $T_{1}$, the time of infection, is the
first jump of $N$. In an early stage of the pandemic it is plausible that
there is only one infectious contact. Then, using that counting processes have
independent increments and that the number of jumps in a certain interval is
Poisson distributed with parameter equal to the integral of the intensity over
the respective interval, we obtain

$\displaystyle\mathbb{P}(T_{1}\leq
s|N(t)=1)=\frac{\mathbb{P}(N([0,s])=1,\,N((s,t])=0)}{\mathbb{P}(N([0,t])=1)}=\frac{\int_{0}^{s}\lambda(r)dr}{\int_{0}^{t}\lambda(r)dr}=\frac{e^{-\alpha(W-s)}-e^{-\alpha
W}}{e^{-\alpha(W-t)}-e^{-\alpha W}}.$

Deriving the distribution function above yields that the density of $T_{1}$ is
proportional to $\exp(-\alpha(W-s))$. Thus, we will later consider primarily
the situation in which $h(u|c)=\exp(-r(C)u)$, where $r(C)$ is the (known)
exponential growth of the infection numbers in location $C$. Note that taking
$\alpha\to 0$, i.e., the case of a constant intensity function, yields
$\mathbb{P}(T_{1}\leq s|N(t)=1)\to s/t$ (the distribution function of a
uniform distribution on $[0,t]$).

The discussion in Remark 2.1 is then plausible, if it is plausible to assume
that the person of interest has infectious contacts with a rate which is so
low/the time $W$ is so short that there is only one infectious contact in
$[0,W]$. In particular in the beginning of a pandemic this seems reasonable.
Note that for the subsequent infection (within the observed infection pair)
this assumption is no longer valid: It is, intuitively speaking, more likely
that infections, e.g., in a household happen directly after the infected
person has returned, rather than days after.

Note finally that in the special case $r(c_{0})=0$ for some location $c_{0}$,
i.e., if the infection level remains constant, we get $h(\cdot|c_{0})=1$ (then
$n(w,c_{0})=1/w$), i.e. $T_{1}\sim\mathcal{U}([0,W])$ (uniform distribution).
From a mathematical standpoint, this case leads to another very intuitive
result: The conditional distribution of $S_{1}$ given $W$ and $C=c_{0}$ is
given by the convolution through (1.2)

$\displaystyle\mathbb{P}(S_{1}\leq x|W,C=c_{0})=\mathbb{P}(T_{1}+I_{1}\leq
x|W,C=c_{0})$ $\displaystyle=$
$\displaystyle\frac{1}{W}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\mathbbm{1}(t\in[0,W])\mathbbm{1}(t+s\leq
x)\varphi_{I}(s)dsdt=\frac{1}{W}\int_{0}^{W}\int_{-\infty}^{x-t}\varphi_{I}(s)dsdt.$

The conditional density of $S_{1}$ can then be obtained through
differentiation of the above with respect to $x$ and is given by

$\frac{1}{W}\int_{0}^{W}\varphi_{I}(x-t)dt=\frac{1}{W}\int_{x-W}^{x}\varphi_{I}(u)du.$

Hence, the likelihood of observing $S_{1}=x$ is given by the probability of
observing an incubation period between $x-W$ and $x$, the natural lower and
upper bounds. Thus we can identify $\varphi_{I}$ from observations of $S_{1}$.
Then, in a second step, we can use (1.4) to identify $\varphi_{G}$ from
observing $S$. Hence, under the assumption that $T_{1}\sim\mathcal{U}([0,W])$
conditionally on $W$, we can identify the distributions of $I$ and $G$ from
the distribution of $(S_{1},S_{2})$. We make this mathematically precise also
in the general case in Corollary 3.4 below.

Before turning to the estimators we make a remark about asymptomatic patients.

######  Remark 2.2.

Asymptomatic Patients: It is unclear how asymptomatic patients can be handled
in this framework. However, it should be emphasized that asymptomatic means
here infected people who infect others but who never show symptoms. Pre-
symptomatic infections are well allowed, i.e., Person 1 is allowed to infect
Person 2 before showing symptoms. However, we assume here that Person 1 and
Person 2 both will eventually show symptoms.

After this preliminary discussion, we introduce our estimator by specifying
approximations to $\varphi_{I}$ and $\varphi_{G}$. Consider the following
class of densities on the non-negative real line for
$m\in\mathbb{N}_{0}:=\\{0,1,2,...\\}$

$\mathcal{G}_{m}:=\left\\{\varphi_{\theta}:\quad\theta\in\mathbb{R}^{m+1},\|\theta\|_{2}=1\right\\},$

where $\|\cdot\|_{2}$ is the Euclidean norm and

$\varphi_{\theta}(x)=\mathbbm{1}(x\geq
0)e^{-x}\left(\sum_{k=0}^{m}\theta^{(k)}L_{k}(x)\right)^{2}$ (2.2)

with $\theta=(\theta^{(0)},...,\theta^{(m)})$

$L_{k}(x):=\sum_{i=0}^{k}\binom{k}{i}\frac{(-x)^{i}}{i!}$

being the $k$-th Laguerre polynomial. Since the Laguerre polynomials form an
orthonormal system of functions on $[0,\infty)$ with respect to the weight
function $e^{-x}$ it is simple to show that the above construction yields a
density under the simple condition $\|\theta\|_{2}=1$, cf. Lemma 2.3.
Moreover, it is well known that a large class of densities can be approximated
by Laguerre polynomials. Define to this end the Hellinger distance of two
distributions on $\mathcal{X}$ which have densities $\varphi_{1},\varphi_{2}$
with respect to a measure $\mu$ via

$\rho_{H}(\varphi_{1},\varphi_{2}):=\left(\int_{\mathcal{X}}\left(\varphi_{1}(x)^{\frac{1}{2}}-\varphi_{2}(x)^{\frac{1}{2}}\right)^{2}d\mu(x)\right)^{\frac{1}{2}}.$

The following lemma is essentially a rephrasing of Theorem 1 in Chapter II.8
of Nikiforov and Uvarov (1988) which we state here for the convenience of the
reader.

######  Lemma 2.3.

For any $m\in\mathbb{N}_{0}$ and any $\theta\in\mathbb{R}^{m+1}$ with
$\|\theta\|_{2}=1$ we have that $\varphi_{\theta}$ as defined in (2.2) is a
density function on $[0,\infty)$. Suppose moreover that $\varphi$ is an
arbitrary density on $[0,\infty)$ such that $p(x):=\sqrt{e^{x}\varphi(x)}$ is
continuous on $[0,\infty)$ and has a piecewise continuous derivative
$p^{\prime}(x)$. Suppose that

$\int_{0}^{\infty}p^{\prime}(x)^{2}xe^{-x}dx<+\infty.$ (2.3)

Then, for any sequence $m_{n}\to\infty$ there are
$\theta_{n}\in\mathbb{R}^{m_{n}+1}$ with $\|\theta_{n}\|=1$ such that for
$n\to\infty$

$\rho_{H}(\varphi_{\theta_{n}},\varphi)^{2}=\int_{0}^{\infty}\left(\varphi_{\theta_{n}}(x)^{1/2}-\varphi(x)^{1/2}\right)^{2}dx\to
0.$

Moreover, $\varphi_{\theta_{n}}\to\varphi$ locally uniformly, i.e.,
$\sup_{x\in K}|\varphi_{\theta_{n}}(x)-\varphi(x)|\to 0$ for any compact set
$K\subseteq(0,\infty)$.

###### Proof.

Let $p_{m_{n}}(x;\theta_{n}):=\sum_{k=0}^{m_{n}}\theta_{n}^{(k)}L_{k}(x)$,
where $\theta_{n}^{(k)}$ denotes the $k$-th entry of $\theta_{n}$. Then
$\varphi_{\theta_{n}}(x)=\mathbbm{1}(x\geq
0)e^{-x}p_{m_{n}}(x;\theta_{n})^{2}$. The first part of the lemma is a simple
calculation using the orthonormality of the Laguerre polynomials, i.e., use
that

$\int_{0}^{\infty}e^{-x}L_{k}(x)L_{l}(x)dx=\mathbbm{1}(k=l).$

For the second statement we note

$\int_{0}^{\infty}e^{-x}p(x)^{2}dx=\int_{0}^{\infty}\varphi(x)dx=1<\infty.$

This and (2.3) are exactly the conditions of Theorem 1 in Chapter II.8 of
Nikiforov and Uvarov (1988) which we stated in the Appendix as Theorem 6.2 for
the convenience of the reader. The theorem states that the sequence
$\widetilde{p}_{n}(x):=\sum_{k=0}^{m_{n}}c_{k}L_{k}(x)$ with
$c_{k}=\int_{0}^{\infty}p(x)L_{k}(x)e^{-x}dx$ converges locally uniformly,
i.e., uniformly on compact sets, to $p$. Let

$\mathcal{C}_{n}:=\int_{0}^{\infty}e^{-x}\widetilde{p}_{n}(x)^{2}dx=\sum_{k=0}^{m_{n}}c_{k}^{2}$

and $\theta_{n}^{(k)}:=c_{k}/\mathcal{C}_{n}^{1/2}$ for $k=0,...,m_{n}$.
Clearly, $\|\theta_{n}\|_{2}=1$. By Parseval’s Identity
$\mathcal{C}_{n}\to\sum_{k=0}^{\infty}c_{k}^{2}=\int_{0}^{\infty}e^{-x}p(x)^{2}dx=1$.
Hence, we can conclude that also

$p_{m_{n}}(x;\theta_{n})=\sum_{k=0}^{m_{n}}\theta_{n}^{(k)}L_{k}(x)=\mathcal{C}_{n}^{-\frac{1}{2}}\sum_{k=0}^{m_{n}}c_{k}L_{k}(x)=\mathcal{C}_{n}^{-\frac{1}{2}}\widetilde{p}_{n}(x)$

converges locally uniformly to $p$ because $p$ is bounded on compact sets
$K\subseteq(0,\infty)$. From this we conclude also the locally uniform
convergence of $\varphi_{\theta_{n}}$ to $\varphi$. Moreover, Theorem 6.2
states that

$\int_{0}^{\infty}e^{-x}(\widetilde{p}_{n}(x)-p(x))^{2}dx\to 0.$

The proof of the Lemma is complete since the above implies

$\displaystyle\int_{0}^{\infty}\left(\varphi_{\theta_{n}}^{1/2}-\varphi^{1/2}\right)^{2}dx=\int_{0}^{\infty}e^{-x}\left(p(x)-p_{m_{n}}(x;\theta_{n})\right)^{2}dx$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}e^{-x}\left(p(x)-\widetilde{p}_{n}(x)+\widetilde{p}_{n}(x)\left(1-\mathcal{C}_{n}^{-\frac{1}{2}}\right)\right)^{2}dx$
$\displaystyle\leq$ $\displaystyle
2\int_{0}^{\infty}e^{-x}\left(p(x)-\widetilde{p}_{n}(x)\right)^{2}dx+2\int_{0}^{\infty}e^{-x}\widetilde{p}_{n}(x)^{2}dx\left(1-\mathcal{C}_{n}^{-\frac{1}{2}}\right)^{2}\to
0.$

∎

The conditions of Lemma 2.3 cover a wide class of piecewise continuously
differentiable densities like any sub-Gaussian density, densities with compact
support and sub-exponential densities. We consider condition (2.3) therefore
as not-restrictive. Before we can write down the likelihood, we have to find
the conditional density of $(S_{1},S_{2},W)$ given $C$.

######  Lemma 2.4.

Suppose that the random variables $W,C,T_{1},I_{1},I_{2},G$ are related as in
Assumption (M). The conditional joint density of $(S_{1},S_{2},W)$ given $C$,
where $S_{1}:=I_{1}+T_{1}$ and $S_{2}:=T_{1}+I_{2}+G$, is given by

$f(x_{1},x_{2},\omega|c)=n(\omega,c)\varphi_{W}(\omega)\int_{0}^{x_{2}}\varphi_{G}(y)\int_{0}^{\min(x_{1},\omega)}h(\omega-t|c)\varphi_{I}(x_{1}-t)\varphi_{I}(x_{2}-t-y)dtdy.$

for $x_{1},x_{2},\omega\geq 0$ and $0$ otherwise.

###### Proof.

Let $x_{1},x_{2},\omega\geq 0$ be arbitrary. Note firstly that, by (M), the
conditional density of $(W,T_{1})$ given $C$ is given by

$\varphi_{T_{1}}(t|w,c)\varphi_{W}(w)=\mathbbm{1}(t\in[0,\omega])n(w,c)h(w-t|c)\varphi_{W}(w).$

Below integrals of the type $\int_{0}^{a}$ are to be understood over the set
$(\min(0,a),\max(0,a))$. Since all integrands are supported in $[0,\infty)$,
integrals are zero when $a<0$. We have by the independence assumptions in (M)

$\displaystyle\mathbb{P}(S_{1}\leq x_{1},S_{2}\leq
x_{2},W\leq\omega|C)=\mathbb{P}(T_{1}+I_{1}\leq x_{1},\,T_{1}+G+I_{2}\leq
x_{2},W\leq\omega|C)$ $\displaystyle=$
$\displaystyle\mathbb{E}\left(\mathbbm{1}(W\leq\omega)\mathbb{P}(T_{1}+I_{1}\leq
x_{1},\,T_{1}+G+I_{2}\leq x_{2}|T_{1},W,C)|C\right)$ $\displaystyle=$
$\displaystyle\mathbb{E}\left(\mathbbm{1}(W\leq\omega)\mathbb{P}(I_{1}\leq
x_{1}-T_{1}|T_{1},C)\cdot\mathbb{P}(G+I_{2}\leq x_{2}-T_{1}|T_{1},C)|C\right)$
$\displaystyle=$
$\displaystyle\mathbb{E}\left(\mathbbm{1}(W\leq\omega)\int_{0}^{x_{1}-T_{1}}\varphi_{I}(a)da\cdot\int_{0}^{x_{2}-T_{1}}\int_{0}^{\infty}\varphi_{G}(y)\varphi_{I}(b-y)dydb\Big{|}C\right)$
$\displaystyle=$
$\displaystyle\int_{0}^{\omega}\int_{0}^{v}n(v,c)h(v-t|c)\varphi_{W}(v)\int_{0}^{x_{1}-t}\varphi_{I}(a)da\cdot\int_{0}^{x_{2}-t}\int_{0}^{\infty}\varphi_{G}(y)\varphi_{I}(b-y)dydbdtdv.$

We obtain furthermore by differentiating under the integral and using that
$\varphi_{I}$ and $\varphi_{G}$ are supported on the non-negative real line

$\displaystyle
f(x_{1},x_{2},\omega|c)=\frac{d^{3}}{dx_{1}dx_{2}d\omega}\mathbb{P}(S_{1}\leq
x_{1},S_{2}\leq x_{2},W\leq\omega|C=c)$ $\displaystyle=$
$\displaystyle\frac{d^{2}}{dx_{1}dx_{2}}n(\omega,c)\varphi_{W}(\omega)\int_{0}^{\omega}h(\omega-t|c)\int_{0}^{x_{1}-t}\varphi_{I}(a)da\cdot\int_{0}^{x_{2}-t}\int_{0}^{\infty}\varphi_{G}(y)\varphi_{I}(b-y)dydbdt$
$\displaystyle=$ $\displaystyle
n(\omega,c)\varphi_{W}(\omega)\int_{0}^{\omega}h(\omega-t|c)\varphi_{I}(x_{1}-t)\cdot\int_{0}^{\infty}\varphi_{G}(y)\varphi_{I}(x_{2}-t-y)dydt$
$\displaystyle=$ $\displaystyle
n(\omega,c)\varphi_{W}(\omega)\int_{0}^{\infty}\varphi_{G}(y)\int_{0}^{\omega}h(\omega-t|c)\varphi_{I}(x_{1}-t)\varphi_{I}(x_{2}-t-y)dtdy$
$\displaystyle=$ $\displaystyle
n(\omega,c)\varphi_{W}(\omega)\int_{0}^{x_{2}}\varphi_{G}(y)\int_{0}^{\min(x_{1},\omega)}h(\omega-t|c)\varphi_{I}(x_{1}-t)\varphi_{I}(x_{2}-t-y)dtdy.$

In the last line above we can replace the upper integration bound of the outer
integral by $x_{2}$ because for $y>x_{2}$, we have $x_{2}-t-y<0$ and hence
$\varphi_{I}(x_{2}-t-y)=0$. ∎

By using the density from above, we obtain an expression for the likelihood
for estimation of $\varphi_{I}$ and $\varphi_{G}$.

######  Lemma 2.5.

Let the observations $(S_{1,i},S_{2,i},W_{i},C_{i})_{i=1,...,n}$ be iid copies
of $(S_{1},S_{2},W,C)$ which are related as in Assumption (M). Let
$m_{1},m_{2}\in\mathbb{N}$ and $\theta_{1}\in\mathbb{R}^{m_{1}+1}$,
$\theta_{2}\in\mathbb{R}^{m_{2}+1}$. Consider the candidate densities
$\varphi_{I,\theta_{1}}\in\mathcal{G}_{m_{1}}$ and
$\varphi_{G,\theta_{2}}\in\mathcal{G}_{m_{2}}$. The likelihood for this
configuration is given by, denote $\widetilde{W}_{i}:=\min(W_{i},S_{1,i})$

$\displaystyle\mathcal{L}_{n}(\varphi_{I,\theta_{1}},\varphi_{G,\theta_{2}})$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\log\left(\int_{0}^{S_{2,i}}\varphi_{G,\theta_{2}}(y)\int_{0}^{\widetilde{W}_{i}}h(W_{i}-t|C_{i})\varphi_{I,\theta_{1}}(S_{1,i}-t)\varphi_{I,\theta_{1}}(S_{2,i}-t-y)dtdy\right)$
$\displaystyle\quad\quad\quad+C_{n},$ (2.4)

where $C_{n}$ is a random constant which does not depend on
$\varphi_{I,\theta_{1}}$ and $\varphi_{G,\theta_{2}}$. In the specific case of
$h(u|c)=\exp(-r(c)u)$ for some exponential parameter $r(c)\in\mathbb{R}$ we
even get that

$\displaystyle\mathcal{L}_{n}=$
$\displaystyle\sum_{i=1}^{n}\log\left(\int_{0}^{S_{2,i}}\varphi_{G,\theta_{2}}(y)\int_{0}^{\widetilde{W}_{i}}\exp\left(r(C_{i})t\right)\varphi_{I,\theta_{1}}(S_{1,i}-t)\varphi_{I,\theta_{1}}(S_{2,i}-t-y)dtdy\right)$
$\displaystyle\quad\quad\quad\quad+\widetilde{C}_{n}$ (2.5)

depends on $W_{i}$ only through $\widetilde{W}_{i}$. Above $\widetilde{C}_{n}$
is a random constant which does not depend on $\varphi_{I,\theta_{1}}$ and
$\varphi_{G,\theta_{2}}$.

###### Proof.

Let $f_{\varphi_{I,\theta_{1}},\varphi_{G,\theta_{2}}}(x_{1},x_{2},\omega|c)$
denote the conditional density of $(S_{1},S_{2},W)$ given $C$ as defined in
Lemma 2.4 when $I$ has density $\varphi_{I,\theta_{1}}$ and $G$ has density
$\varphi_{G,\theta_{2}}$. By independence, we have

$\displaystyle\mathcal{L}_{n}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\log\left(f_{\varphi_{I,\theta_{1}},\varphi_{G,\theta_{2}}}(S_{1,i},S_{2,i},W_{i}|C_{i})p_{C}(C_{i})\right)$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\log\left(\int_{0}^{S_{2,i}}\varphi_{G,\theta_{2}}(y)\int_{0}^{\min(S_{1,i},W_{i})}h(W_{i}-t|C_{i})\varphi_{I,\theta_{1}}(S_{1,i}-t)\varphi_{I,\theta_{1}}(S_{2,i}-t-y)dtdy\right)$
$\displaystyle\quad\quad\quad\quad+\sum_{i=1}^{n}\log\left(n(W_{i},C_{i})\varphi_{W}(W_{i})p_{C}(C_{i})\right),$

where the latter term is independent of $\varphi_{I,\theta_{1}}$ and
$\varphi_{G,\theta_{2}}$. This finishes the proof of (2.4). For the specific
choice $h(u|c)=\exp(-r(c)u)$ we get

$\displaystyle\mathcal{L}_{n}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{n}\log\left(\int_{0}^{S_{2,i}}\varphi_{G,\theta_{2}}(y)\int_{0}^{\min(S_{1,i},W_{i})}\exp\left(r(C_{i})t\right)\varphi_{I,\theta_{1}}(S_{1,i}-t)\varphi_{I,\theta_{1}}(S_{2,i}-t-y)dtdy\right)$
$\displaystyle\quad\quad\quad\quad+\sum_{i=1}^{n}\left(\log\left(n(W_{i},C_{i})\varphi_{W}(W_{i})p_{C}(C_{i})\right)-r(C_{i})W_{i}\right)$

which finishes the proof of (2.5) since the second line does not depend on
$\varphi_{I,\theta_{1}}$ or $\varphi_{G,\theta_{2}}$. ∎

Note that Lemma 2.5 implies that for given $\varphi_{W}$ and $p_{C}$, the
likelihood to be optimized is, conveniently, independent of $\varphi_{W}$ and
$p_{C}$. Thus, we can treat $\varphi_{W}$ and $p_{C}$ as known without loosing
any practicality and, therefore, we may define the following estimators for
$\varphi_{I}$ and $\varphi_{G}$: For given sequences
$(m_{1,n})_{n\in\mathbb{N}},(m_{2,n})_{n\in\mathbb{N}}\subseteq\mathbb{N}_{0}$,
we study

$\displaystyle\left(\hat{\varphi}_{I,n},\hat{\varphi}_{G,n}\right):=$
$\displaystyle\underset{\varphi_{I,\theta_{1}}\in\mathcal{G}_{m_{1}},\varphi_{G,\theta_{2}}\in\mathcal{G}_{m_{2}}}{\operatorname{argmax}}\sum_{i=1}^{n}\log\Bigg{(}\int_{0}^{S_{2,i}}\varphi_{G,\theta_{2}}(y)\int_{0}^{\widetilde{W}_{i}}h(W_{i}-t|C_{i})$
$\displaystyle\quad\quad\quad\quad\times\varphi_{I,\theta_{1}}(S_{1,i}-t)\varphi_{I,\theta_{1}}(S_{2,i}-t-y)dtdy\Bigg{)}.$

At this point it is not clear that we can identify the parameters
$\theta_{1}$,$\theta_{2}$. Later, in Corollary 3.4, we will see that
consistent estimation of the distribution functions of $\varphi_{I}$ and
$\varphi_{G}$ is possible if a mild assumption on the characteristic functions
holds. Under the same assumption, a similar proof-technique can be applied to
show that two different sets of parameters lead to different likelihoods.

##  3 Theory

In this section we will prove and discuss a consistency result. For
$\varphi_{I}$ and $\varphi_{G}$ being arbitrary densities of $I_{1},I_{2}$ and
$G$, respectively, we denote by $f_{\varphi_{I},\varphi_{G}}$ the conditional
density of $(S_{1},S_{2},W)$ given $C$ as defined in Lemma 2.4. Then,
$f_{\varphi_{I},\varphi_{G}}p_{C}$ denotes the joint density of
$(S_{1},S_{2},W,C)$. When $\varphi_{I}$ and $\varphi_{G}$ are chosen from the
approximating spaces $\mathcal{G}_{m_{1}}$ and $\mathcal{G}_{m_{2}}$, the
following set denotes the set of all possible approximations to the joint
density of $(S_{1},S_{2},W,C)$

$\mathcal{F}_{m_{1},m_{2}}:=\left\\{f_{\varphi_{I},\varphi_{G}}p_{C}:\,\varphi_{I}\in\mathcal{G}_{m_{1}},\,\varphi_{G}\in\mathcal{G}_{m_{2}}\right\\}.$

In the following we will always assume that we are in the setting presented in
Section 2.

######  Theorem 3.1.

Suppose (M) holds true and let $m_{1,n},m_{2,n}=n^{\beta}$ and
$\varepsilon_{n}=n^{-\gamma}$ for some $\beta,\gamma>0$ such that $\gamma<1/2$
and $\beta\leq 1-2\gamma$. Suppose that the true densities $\varphi_{I}$ and
$\varphi_{G}$ fulfil the assumptions of Lemma 2.3 and let $\varphi_{I,n}$ and
$\varphi_{G,n}$ be the sequences from Lemma 2.3 such that
$\rho_{H}(\varphi_{I},\varphi_{I,n})\to 0$ and
$\rho_{H}(\varphi_{G},\varphi_{G,n})\to 0$. Suppose that there is
$\alpha\in(0,1/2)$ such that for some $N_{0}\in\mathbb{N}_{0}$

$\sup_{n\geq
N_{0}}C_{\alpha}(f_{\varphi_{I},\varphi_{G}},f_{\varphi_{I,n},\varphi_{G,n}})<\infty,$
(3.1)

where

$C_{\alpha}(f_{1},f_{2}):=\left(\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{3}}\left(\frac{f_{1}(x_{1},x_{2},\omega|c)}{f_{2}(x_{1},x_{2},\omega|c)}\right)^{\frac{\alpha}{1-\alpha}}f_{1}(x_{1},x_{2},\omega|c)p_{C}(c)d(x_{1},x_{2},\omega)\right)^{1-\alpha}.$
(3.2)

Then, we have that

$\rho_{H}(f_{\varphi_{I},\varphi_{G}}p_{C},f_{\hat{\varphi}_{I,n},\hat{\varphi}_{G,n}}p_{C})=O_{P}\left(\max\left(\varepsilon_{n},\max\left(\rho_{H}(\varphi_{I},\varphi_{I,n}),\rho_{H}(\varphi_{G},\varphi_{G,n})\right)^{\alpha}\right)\right)=o_{P}(1).$

The proof of this result will be given later in Section 3.2. We will begin
with a discussion of the result and its assumptions in the next subsection.

###  3.1 Discussion of Theorem 3.1

Before turning to the assumptions of Theorem 3.1 we make a remark about the
convergence rate.

######  Remark 3.2.

Note that the requirements on $\beta$ and $\gamma$ in Theorem 3.1 allow for
the choice $\beta=1/5$ and $\gamma=2/5<1/2$. This yields the _classical_
convergence rate $n^{-2/5}$ for series estimation with polynomials if the true
density is sufficiently smooth (cf. Newey (1997) and Proposition 3.6 in Chen
(2007)).

We continue with a discussion of the condition (3.1) above. This can be
understood as a tail condition: For any compact set $K\subseteq(0,\infty)^{3}$
we have by Lemma 2.3 that $f_{\varphi_{I,n},\varphi_{G,n}}$ converges
uniformly to $f_{\varphi_{I},\varphi_{G}}$. Thus, if we restrict the integral
in the definition of
$C_{\alpha}(f_{\varphi_{I},\varphi_{G}},f_{\varphi_{I,n},\varphi_{G,n}})$ to
$K$, the restricted integral remains bounded. Note also that by choosing
$\alpha>0$ small, possible singularities of $f_{2}$ are integrable. Hence,
(3.1) is only restrictive for the integral over $K^{c}$. Thus, if
$f_{\varphi_{I},\varphi_{G}}$ is compactly supported (3.1) follows. Note
furthermore that $f_{\varphi_{I},\varphi_{G}}$ is compactly supported if
$\varphi_{I},\varphi_{G}$ and $\varphi_{W}$ are compactly supported. In our
specific epidemics setting for COVID-19, this is a highly plausible assumption
because people stop being infectious at some point. However, for other
diseases people can stay infectious for an extended period of time, e.g. for
Hepatitis C, the mean generation time is about $20$ years (cf. Wallinga and
Lipsitch (2007)).

In those cases of non-compactly supported distributions, (3.1) is a
restriction: It is required that the approximations do not decrease much
faster to zero than the actual density. The meaning of _much faster_ is to be
understood relative to $f_{\varphi_{I},\varphi_{G}}$ and it can be adjusted by
choosing $\alpha>0$ small. Any polynomial difference can therefore be handled.

Theorem 3.1 above shows, strictly speaking, that we can consistently estimate
the joint distribution of $(S_{1},S_{2},W,C)$. However, from a practical point
of view, the estimation of $\varphi_{I}$ and $\varphi_{G}$ is of interest. In
order to ensure identifiability of $\varphi_{I}$ and $\varphi_{G}$, we need
the following additional assumption:

Assumption (C): Characteristic Functions:
_Let $\Phi_{I}$ denote the characteristic functions of $I$. There are
functions $z_{1}:\mathbb{R}\to\mathbb{R}$ and $z_{2}:\mathbb{R}\to\mathbb{R}$
such that for almost all $x\in\mathbb{R}$_

$\Phi_{I}(x)\neq 0\quad\textrm{ and
}\quad\mathbb{E}\left(e^{iz_{1}(x)W+iz_{2}(x)C}\mathbb{E}\left(e^{ixT_{1}}\big{|}W,C\right)\right)\neq
0.$

In particular the second requirement above looks cryptic in full generality.
In the follow remark we show that in specific scenarios it simplifies to
simpler conditions which are more interpretable.

######  Remark 3.3.

The condition

$\mathbb{E}\left(e^{iz_{1}(x)W+iz_{2}(x)C}\mathbb{E}\left(e^{ixT_{1}}\big{|}W,C\right)\right)\neq
0$ (3.3)

above can be rewritten in more specific scenarios.

  1. 1.

If $\Phi_{T_{1}}(x):=\mathbb{E}(\exp(ixT_{1}))\neq 0$ almost everywhere, (3.3)
holds because we may simply choose $z_{1}\equiv z_{2}\equiv 0$.

  2. 2.

If $C\equiv 1$ and $T_{1}$ is uniformly distributed in $[0,W]$, i.e.,
$h(w-t|c)\equiv 1$ and $n(w,c)=1/w$, we obtain with $z_{2}\equiv 0$ and
$z_{1}(x)=-x/2$ that (note that $\sin(x)/x\to 1$ for $x\to 0$ such that the
below inequality chain can be used also for $x=0$)

$\displaystyle\mathbb{E}\left(e^{iz_{1}(x)W+iz_{2}(x)C}\mathbb{E}\left(e^{ixT_{1}}\big{|}W,C\right)\right)=\mathbb{E}\left(e^{-i\frac{x}{2}W}\frac{1}{W}\int_{0}^{W}e^{ixt}dt\right)$
$\displaystyle=$
$\displaystyle\mathbb{E}\left(\frac{1}{ixW}\left(e^{i\frac{x}{2}W}-e^{-i\frac{x}{2}W}\right)\right)=\mathbb{E}\left(\frac{1}{\frac{x}{2}W}\sin\left(\frac{x}{2}W\right)\right)=\mathbb{E}\left(\int_{0}^{1}\cos\left(\frac{x}{2}Wt\right)dt\right)$
$\displaystyle=$
$\displaystyle\int_{0}^{1}\textrm{Re}\left(\mathbb{E}\left(e^{i\frac{x}{2}Wt}\right)\right)dt=\int_{0}^{1}\textrm{Re}\left(\Phi_{W}\left(\frac{x}{2}t\right)\right)dt,$

where $\Phi_{W}$ is the characteristic function of $W$. The above is non-zero
for almost all $x\in\mathbb{R}$ (implying (3.3)) for many standard
distributions like the exponential distribution which characteristic function
has a strictly positive real part.

Our setting is very similar to the de-convolution set-up: In (1.2) the signal
$I_{1}$ is perturbed by the noise $T_{1}$ (the distribution of which is
determined by $(W,C)$) and in (1.3) the signal $G$ is perturbed by the noise
$I_{2}+T_{1}$. Thus it is clear that assumptions on $W,C$ and $I_{1}$ similar
to those for the de-convolution set-up are necessary for identification. The
assumption of almost everywhere non-vanishing characteristic functions has
already been mentioned in Devroye (1989) (see the Remark below Theorem 1
therein) to be necessary to guarantee consistent estimation.

Let $F_{I}$ and $F_{G}$ denote the distribution functions corresponding to the
densities $\varphi_{I}$ and $\varphi_{G}$, respectively. Similarly, define
$\hat{F}_{I,n}$ and $\hat{F}_{G,n}$ to be the distribution functions of
$\hat{\varphi}_{I,n}$ and $\hat{\varphi}_{G,n}$, respectively. The following
corollary ensures under Assumption (C) that $\hat{F}_{I,n}$ and
$\hat{F}_{G,n}$ are consistent estimators for $F_{I}$ and $F_{G}$,
respectively.

######  Corollary 3.4.

Suppose that, in addition to the assumptions of Theorem 3.1, Assumption (C)
holds true. We then have

$\left\|\hat{F}_{I,n}-F_{I}\right\|_{\infty}=o_{P}(1)\textrm{ and
}\left\|\hat{F}_{G,n}-F_{G}\right\|_{\infty}=o_{P}(1).$

The detailed proof is given in Section 6.3 in the Appendix.

We finally come back to estimating the basic reproduction number $R_{0}$. The
above framework provides us with a methodology to consistently estimate the
reproduction number without making parametric statements about the incubation
period or the generation time. From (1.1) it is evident that we require next
to an estimator for $\varphi_{G}$ also an estimate $\hat{r}_{n}$ of the growth
rate of the expected incidence. In Ferretti et al. (2020) the exponential
growth rate of the reported numbers is estimated to be $r=0.14$. But, strictly
speaking, what we need here is the growth rate of the infection numbers which
is (intuitively speaking) very similar but can be different due to under-
reporting and delays. The estimator is specified in the following corollary,
the proof of which can also be found in Section 6.3 in the Appendix.

######  Corollary 3.5.

Let $\hat{r}_{n}$ denote an estimate of the exponential growth rate of the
expected incidence $r>0$, i.e., let $\hat{r}_{n}-r=o_{P}(1)$. Under the
conditions of Theorem 3.1 and Assumption (C), we have

$\int_{0}^{\infty}e^{-\hat{r}_{n}t}\hat{\varphi}_{G,n}(t)dt\overset{\mathbb{P}}{\to}F_{R}(G).$

###  3.2 Proof of Theorem 3.1

In order to prove Theorem 3.1 we need to introduce the same distance relation
which was used also by Wong and Shen (1995). Define for any two densities
$f_{1},f_{2}$ with respect to a measure $\mu$ on a space $\mathcal{X}$ and
$\alpha\neq 0$ the distance relation

$\rho_{\alpha}(f_{1},f_{2}):=\frac{1}{\alpha}\int_{\mathcal{X}}f_{1}(x)\left(\left(\frac{f_{1}(x)}{f_{2}(x)}\right)^{\alpha}-1\right)d\mu(x).$

Note that

$\displaystyle\rho_{-1/2}(f_{1},f_{2})=-2\int_{\mathcal{X}}\left(\sqrt{f_{1}(x)f_{2}(x)}-f_{1}(x)\right)d\mu(x)=2-2\int_{\mathcal{X}}\sqrt{f_{1}(x)f_{2}(x)}d\mu(x)$
$\displaystyle=$
$\displaystyle\int_{\mathcal{X}}\left(f_{1}(x)+f_{2}(x)-2\sqrt{f_{1}(x)f_{2}(x)}\right)d\mu(x)=\int_{\mathcal{X}}\left(\sqrt{f_{1}(x)}-\sqrt{f_{2}(x)}\right)^{2}d\mu(x)$
$\displaystyle=$ $\displaystyle\rho_{H}(f_{1},f_{2})^{2}.$

A second property of $\rho_{\alpha}$ we mention here, is non-negativity for
$\alpha\geq-1$: Let $g_{\alpha}:(0,\infty)\to\mathbb{R}$ be given by
$g_{\alpha}(x):=\frac{1}{\alpha}\left(x^{-\alpha}-1\right)$. Then
$g_{\alpha}^{\prime\prime}(x)=(\alpha+1)x^{-\alpha-2}\geq 0$ because
$\alpha\geq-1$. Hence, $g_{\alpha}$ is a convex function. We hence obtain by
Jensen’s Inequality (for the measure with density $f_{1}$ with respect to
$\mu$)

$\displaystyle\rho_{\alpha}(f_{1},f_{2})=\int_{\mathcal{X}}g_{\alpha}\left(\frac{f_{2}(x)}{f_{1}(x)}\right)f_{1}(x)d\mu(x)\geq
g_{\alpha}\left(\int_{\mathcal{X}}\frac{f_{2}(x)}{f_{1}(x)}f_{1}(x)\mu(x)\right)=0.$

Lemma 2.3 shows that the sieve spaces $\mathcal{G}_{m}$ lie dense in a large
class of densities with respect to the Hellinger distance. However, for our
later consistency result, we require that the sieve spaces provide also good
approximations with respect to $\rho_{\alpha}$ for $\alpha>0$. The following
lemma provides the main tool.

######  Lemma 3.6.

Let $\varphi_{I,1},\varphi_{I,2}$ be two densities for incubation periods and
let $\varphi_{G,1},\varphi_{G,2}$ be two densities of generation times and
consider $f_{1}(x_{1},x_{2},\omega|c)$ and $f_{2}(x_{1},x_{2},\omega|c)$
defined as in Lemma 2.4 using $(\varphi_{I,1},\varphi_{G,1})$ and
$(\varphi_{I,2},\varphi_{G,2})$, respectively. Define
$C_{\alpha}(f_{1},f_{2})$ as in (3.2). Then, for any $\alpha\in(0,1/2)$,

$\rho_{\alpha}(f_{1}p_{C},f_{2}p_{C})\leq\frac{1}{\alpha}C_{\alpha}(f_{1},f_{2})\left(2\rho_{H}(\varphi_{G,1},\varphi_{G,2})^{2}+8\rho_{H}(\varphi_{I,1},\varphi_{I,2})^{2}\right)^{\alpha}.$
(3.4)

The proof of the above Lemma is presented in Section 6.4 in the Appendix.
Before we can prove the consistency result, we need as a last preparation a
bound on the bracketing entropy of the spaces $\mathcal{F}_{m_{1},m_{2}}$.

######  Definition 3.7.

Let $\varepsilon>0$, $\mathcal{F}_{0}\subseteq\mathcal{F}$ classes of
functions and $\rho:\mathcal{F}\times\mathcal{F}\to[0,\infty)$ a metric on
$\mathcal{F}$ be given. The _bracketing number_
$\mathcal{N}_{[]}(\varepsilon,\mathcal{F}_{0},\rho)$ is defined as the
smallest number of pairs $(l_{i},u_{i})\in\mathcal{F}^{2}$ of functions such
that $\rho(l_{i},u_{i})\leq\varepsilon$ for all pairs and such that for any
$f\in\mathcal{F}_{0}$ there is a pair $(l_{i},u_{i})$ with $l_{i}(x)\leq
f(x)\leq u_{i}(x)$. The pairs $(l_{i},u_{i})$ are called _brackets_.

######  Lemma 3.8.

For $m_{1},m_{2}\geq 1$ and any $\varepsilon>0$ with
$\varepsilon\leq\left(15m_{1}m_{2}^{2}/4\right)^{1/2}$ we have

$\log\mathcal{N}_{[]}(\varepsilon,\mathcal{F}_{m_{1},m_{2}},\rho_{H})\leq(m_{1}+m_{2})\log\left(\frac{\pi\sqrt{15m_{1}m_{2}^{2}}}{\varepsilon}\right)$

This Lemma is also proven in Section 6.4 in the Appendix. We have now all
ingredients together to prove the main result of this paper.

###### Proof of Theorem 3.1.

This theorem is a consequence of Theorem 4 from Wong and Shen (1995). For the
convenience of the reader, we have stated the result in Section 6.1 in the
Appendix as Theorem 6.1. Since we are interested in an asymptotic result we
may assume below that $n\geq N_{0}$.

We apply Theorem 6.1 with $Y_{i}=(S_{1,i},S_{2,i},W_{i},C_{i})$ and
$\mathcal{F}_{n}=\mathcal{F}_{m_{1,n},m_{2,n}}$. We show next that the entropy
condition is fulfilled: We note firstly that
$(15m_{1,n}m_{2,n}^{2})^{\frac{1}{2}}\to\infty$ and $\varepsilon_{n}\to 0$.
Hence, the condition $\varepsilon_{n}\leq(15m_{1,n}m_{2,n}^{2})^{1/2}$ is
eventually fulfilled and we may apply Lemma 3.8. Thus, we obtain for a
suitable $c_{2}<\infty$

$\displaystyle\frac{1}{\sqrt{n}\varepsilon_{n}^{2}}\int_{\frac{\varepsilon_{n}^{2}}{2^{8}}}^{\sqrt{2}\varepsilon_{n}}\sqrt{\log\mathcal{N}_{[]}\left(\frac{u}{c_{1}},\mathcal{F}_{m_{1,n},m_{2,n}},\rho_{H}\right)}du$
$\displaystyle\leq$
$\displaystyle\frac{1}{\sqrt{n}\varepsilon_{n}^{2}}\int_{\frac{\varepsilon_{n}^{2}}{2^{8}}}^{\sqrt{2}\varepsilon_{n}}\sqrt{(m_{1,n}+m_{2,n})\log\left(\frac{c_{1}\pi\sqrt{15m_{1,n}m_{2,n}^{2}}}{u}\right)}du$
$\displaystyle\leq$
$\displaystyle\frac{\sqrt{2}\varepsilon_{n}}{\sqrt{n}\varepsilon_{n}^{2}}\sqrt{(m_{1,n}+m_{2,n})\log\left(\frac{2^{8}c_{1}\pi\sqrt{15m_{1,n}m_{2,n}^{2}}}{\varepsilon_{n}^{2}}\right)}$
$\displaystyle\leq$ $\displaystyle
2n^{-\frac{1}{2}+\gamma+\frac{1}{2}\beta}\sqrt{\log\left(2^{8}c_{1}\pi\sqrt{15}n^{\frac{3}{2}\beta+2\gamma}\right)}\leq
c_{2},$

because $-\frac{1}{2}+\gamma+\frac{1}{2}\beta\leq 0$. Hence, the entropy
condition of Theorem 6.1 is fulfilled. Moreover, by the assumptions and Lemma
3.6 we find that

$\delta_{n}(\alpha)\leq\rho_{\alpha}(f_{\varphi_{I},\varphi_{G}}p_{C},f_{\varphi_{I,n},\varphi_{G,n}}p_{C})\to
0.$

Hence, we eventually have $\delta_{n}(\alpha)<1/\alpha$. Under these
conditions, we have that $\varepsilon_{n}^{*}(\alpha)\to 0$ and
$n\varepsilon_{n}^{*}(\alpha)^{2}\geq
n\varepsilon_{n}^{2}=n^{1-2\gamma}\to\infty$ since $\gamma<1/2$. Hence, we
conclude from Theorem 6.1 that
$\rho_{H}(f_{\varphi_{I},\varphi_{G}}p_{C},f_{\hat{\varphi}_{I,n},\hat{\varphi}_{G,n}}p_{C})=O_{P}(\varepsilon_{n}^{*}(\alpha))$
and the proof of the theorem is complete since

$\varepsilon_{n}^{*}(\alpha)=O(\max(\varepsilon_{n},\max(\rho_{H}(\varphi_{I},\varphi_{I,n}),\rho_{H}(\varphi_{G},\varphi_{G,n}))^{\alpha})$

by Lemma 3.6. ∎

##  4 Empirical Studies

The analytic evaluation of the likelihood presented in Lemma 2.5 is very
tedious. Therefore, the following results were obtained by numerically
approximating the integrals. As illustrated by the simulations below this
approximation does not cause problems in the estimation. However, we believe
that a speed-up of the method is possible if the integrals are analytically
computed. In order to efficiently enforce the constraint that
$\|\theta_{1}\|_{2}=\|\theta_{2}\|_{2}=1$, we optimize the angles of the
polar-coordinates of $\theta_{1}$ and $\theta_{2}$ and fix their radii to $1$.
The angles can then be optimized under the box constraint $[0,\pi]$ (note that
$\theta$ and $-\theta$ yield the same model). Finally, we note that the
likelihood can have singularities because the Laguerre densities (2.2) can be
zero. We overcome the resulting problems in the optimisation, similar to Zhang
and Davidian (2008), by repeating the optimisation of the likelihood with
several randomly chosen starting values. Below we always considered $5$
different, random starting values.

Below we illustrate the performance of our estimator in small samples. As
basis for the simulation we take the dataset from Ferretti et al. (2020) which
is available on the website of the journal (on
https://doi.org/10.1126/science.abb6936 in the supplementary material
section). We begin with an illustration of our methodology with synthetic data
in Section 4.1 followed by a brief analysis of the above dataset in Section
4.2. The R-code which is used for both parts is available on github
(https://github.com/akreiss/SemiParametric-Laguerre.git).

###  4.1 Simulation Study

In this simulation study we use as a data generating process the model as in
(2.1) with the following choices:

$\varphi_{W}$ | exponential with rate $\lambda=0.3820225$
---|---
$p_{C}$ |  $\mathbb{P}(C=0)=0.65$ and $\mathbb{P}(C=1)=0.35$
$\varphi_{T_{1}}$ | $\varphi_{T_{1}}(t|w,c)=\mathbbm{1}(t\in[0,w])\left\\{\begin{array}[]{lll}\frac{re^{-r(w-t)}}{1-e^{-rw}}&,&c=1\\\ \frac{1}{w}&,&c=0\end{array}\right..$
$r$ | $r=\log(2)/5$
$\varphi_{I}$ |  $\varphi_{I,0}=$log-normal distribution: meanlog$=1.644$, standard dev.$=0.363$
$\varphi_{G}$ |  $\varphi_{G,0}=$Weibull distribution: shape$=2.826$, scale$=5.665$

All quantities above are chosen to imitate the data used in Ferretti et al.
(2020): The average window length $\mathbb{E}(W_{i})$ equals roughly the
average length of the observed exposure windows in the dataset of Ferretti et
al. (2020). Likewise, in their dataset the authors distinguish two locations,
those with an exponential growth of cases with growth rate $\log(2)/5$ (in
roughly 35% of the cases) and those with no exponential growth (in roughly 65%
of the cases). Moreover, the density of the incubation period
$\varphi_{I}=\varphi_{I,0}$ is the distribution of incubation times fitted by
Ferretti et al. (2020); Lauer et al. (2020). Lastly, the density of the
generation time is taken from Ferretti et al. (2020). We show firstly in
Figure 2 that these densities can be well approximated through Laguerre type
densities as defined in (2.2). The graphs show those Laguerre densities which
minimize the Hellinger distance to the true densities. Not-surprisingly,
larger choices of the degrees $m_{1}$ and $m_{2}$ lead to better
approximations. However, we also see that, in both cases, already degree two
yields reasonable approximations. Note in particular that the flat beginning
of the density of the incubation period can be well captured by the
approximating densities.

Figure 2: Best approximations of $\varphi_{I}$ (left) and $\varphi_{G}$
(right) through Laguerre densities of the form (2.2) for various choices of
$m_{1}$ and $m_{2}$.

In the dataset in Section 4.2 we have $40$ observations available. In this
simulation study we choose therefore also $n=40$ observations because in
applications this seems to be a reasonable number of transmission pairs for
which the necessary data can be obtained. In order to determine the degree of
the approximating Laguerre polynomials, we use the information criterion BIC.
Since the computations are quite intensive we do the model selection only for
one dataset and choose the selected model for all subsequent repetitions. The
resulting values of the BIC are shown in Table 1 and it can be seen that the
minimal value is obtained for $m_{1}=2$ and $m_{2}=2$. For the remainder of
this Section, we use these degrees.

|  |  | $m_{2}$ |  |
---|---|---|---|---|---
|  | 1 | 2 | 3 | 4
| 1 | 480.54 | 421.08 | 419.51 | 422.26
$m_{1}$ | 2 | 371.39 | 362.87 | 366.01 | 369.63
| 3 | 369.92 | 366.27 | 369.05 | 372.74
| 4 | 373.62 | 369.88 | 372.74 | 376.38
Table 1: BIC values for one synthetic dataset

We simulate now $N=100$ datasets according to the above specified data
generating process and fit our model to it. Figure 3 shows the true densities
in black and the gray lines show the estimates. The dashed lines show in both
cases the Laguerre type densities (2.2) which come closest to the true
densities. It can be seen that the estimation works visibly quite well for the
incubation period. The height of the mode is underestimated but this comes
from the fact that fitting an order two Laguerre density cannot do better. The
estimation for the generation time works a bit less good but still the general
trend is captured well by most estimators if we keep in mind that we have here
only $40$ observations of a heavily convolved variable. In order to assess the
fit of the non-parametric estimator a bit more formally, we compare the
estimators to the true densities in terms of the squared Hellinger-distance
$\rho_{H}^{2}$. The resulting histogram is shown in Figure 4. Note that
squared Hellinger distances are bounded from above by $2$. It can be seen that
the estimation of the Incubation Period works better than the estimation of
the Generation Time. This is not surprising because we observe transmission
pairs each of which contains two independent realisations of incubation
periods but only one generation time. Overall the fit of the generation time
appears to be fairly good ($90\%$ of the distances are smaller than $0.13$ and
$50\%$ of the distances are smaller than $0.04$).

Figure 3: Estimation results (gray lines) for simulated data with $n=40$
observations. True densities are shown in black and the dashed lines show the
Laguerre densities which come closest to the true densities. Figure 4:
Histograms of squared Hellinger distances of estimates to true densities.

In order to illustrate how the feature estimation works we consider estimation
of the basic reproduction number of a fictional pandemic and estimation of
quantiles. In order to estimate the basic reproduction number, we plug the
true and the estimated generation time densities into equation (1.1) and take
the inverse of it (in this fictional pandemic we choose $r=\log(2)/5$ which
means that the case numbers double every five days). The histogram of the
estimates is shown in Figure 5. We emphasize that these are simulated results
and we cannot draw any conclusions about COVID-19. We can see that the
estimation appears to be almost unbiased even in finite samples. Moreover, the
approximation through a normal distribution seems to be reasonably accurate.
This motivates further research in establishing a formal asymptotic normality
result for this type of semi-parametric inference. Figure 6 shows the
histograms of the estimated 30%-quantiles. We can see that the estimates
appear to have a bias in finite samples (the dashed lines indicate the true
quantiles). This bias seems to stem from the fact that the approximation
through low dimensional Laguerre polynomials is not perfect because the
estimates centralize around the quantiles of the best Laguerre approximations
(dotted lines). This effect remains true for other quantiles which are
reported in Appendix 6.2.1.

Figure 5: Histograms of estimated basic reproduction numbers of fictional
pandemic. The dotted line indicates the true $R_{0}$ and the solid line is the
density of a normal distribution with the sample mean and standard deviation.
Figure 6: Histograms of estimated 30%-quantile for the incubation time (left)
and generation time (right). The dashed lines indicate the respective true
values, the dotted lines the quantiles of the best approximating densities and
the solid line is the density of a normal distribution with the corresponding
sample mean and standard deviation.

When inspecting the BIC values in Table 1 we see that the next smallest scores
are obtained for the models $(m_{1}=2,m_{2}=3)$ and $(m_{1}=3,m_{2}=2)$. Even
though the BIC for these two model choices is almost equal, we choose to
inspect the model $(m_{1}=2,m_{2}=3)$ because the additional flexibility in
the generation time seems to be more relevant to us when looking at Figure 2.
The results are similar and are shown in Appendix 6.2.1.

###  4.2 Real Data Application

Finally we apply the methodology to the dataset provided by Ferretti et al.
(2020). This dataset contains 40 transmission pairs for which Ferretti et al.
(2020) were reasonably sure that there was direct transmission within the
pair. In their work the authors’ interest was to estimate the generation time
provided that they already have an estimate for the density of the incubation
period (so the dataset was used in a slightly different context). Thus we can
reasonably illustrate our methodology on this dataset: Semi-parametric
estimation of features of the generation time and the incubation time. The
dataset contains symptom onset dates for all transmission pairs, but the
exposure window is not always reported. In that case we impute the dataset in
the same way as Ferretti et al. (2020): The beginning of the exposure window
is at the earliest 60 days before symptom onset of the infector. The end of
the infection window is at the latest the symptom onset time of any of the two
people in the pair or the end of the exposure window of the second person
which is reported in some cases. We stress that our interest lies in the
theoretical analysis of the methodology and we provide here an illustration
for how our methodology can be used. A complete data analysis would for
example also require a robustness analysis against potential issues like the
question whether the transmission pairs are random samples from the pandemic.

In the following our aim is to use our semi-parametric approach to construct a
test whether the parametric fit suggested by Ferretti et al. (2020) is
appropriate for the data. Let $\varphi_{I,0}$ and $\varphi_{G,0}$ denote the
densities as defined in Section 4.1. More formally, we would like to test the
hypotheses

$H_{0}^{(I)}:\varphi_{I}=\varphi_{I,0}\quad\textrm{and}\quad
H_{0}^{(G)}:\varphi_{G}=\varphi_{G,0}.$

As test statistics we consider
$\rho_{H}(\hat{\varphi}_{I,n},\widetilde{\varphi}_{I,0})^{2}$ and
$\rho_{H}(\hat{\varphi}_{G,n},\widetilde{\varphi}_{G,0})^{2}$, where
$\widetilde{\varphi}_{I,0}$ and $\widetilde{\varphi}_{G,0}$ denote the best
approximations with respect to the Hellinger distance of $\varphi_{I,0}$ and
$\varphi_{G,0}$ through Laguerre polynomials, respectively. In order to choose
the degrees of the approximation we use in the same way as before the BIC. The
resulting values are shown in Table 2 and it can be seen that $m_{1}=2$ and
$m_{2}=1$ yields the smallest BIC. While Figure 2 shows that $m_{1}=2$ yields
a reasonable approximation to $\varphi_{I,0}$, we see that $m_{2}=1$ is a bit
too small to represent $\varphi_{G,0}$ well. Therefore, it is very important
that we compare the estimate with the closest Laguerre-type density
$\widetilde{\varphi}_{G,0}$ rather than with $\varphi_{G,0}$ directly. The
fact that the BIC suggests such a small model indicates that $n=40$ is too
small to properly differentiate the form of $\varphi_{G,0}$ suggested by
Ferretti et al. (2020) from other models. Therefore, the power of our test
would be better if we had more observations.

|  |  | $m_{2}$ |  |
---|---|---|---|---|---
|  | 1 | 2 | 3 | 4
| 1 | 129.96 | 117.17 | 119.80 | 121.26
$m_{1}$ | 2 | 98.43 | 100.55 | 101.82 | 105.14
| 3 | 101.38 | 101.94 | 103.33 | 104.57
| 4 | 99.67 | 100.88 | 106.56 | 110.25
Table 2: BIC values for transmission pair data

In order to assess the distribution of the test statistic under $H_{0}^{(I)}$,
$H_{0}^{(G)}$ or $H_{0}^{(I)}\cap H_{0}^{(G)}$, we suppose that the densities
of the incubation period and generation time are given by $\varphi_{I,0}$ and
$\varphi_{G,0}$, respectively, and simulate data accordingly (note that when
testing for $H_{0}^{(I)}$ or $H_{0}^{G)}$ we suppose that the respective other
density is correctly specified). We do this $N=100$ times and show in Figure 7
the histograms of the squared Hellinger distances between the estimated
densities and those Laguerre densities of type (2.2) (with $m_{1}=2$ and
$m_{2}=1$) which lie closest to the true densities. Note, that in Figure 4 we
compared the estimates with the true densities. This explains the, on first
sight paradox, situation that the histograms for the generation time seems to
show that for the smaller model with $m_{2}=1$ the Hellinger distance is
smaller than for the larger model. The observed squared Hellinger distances
from the dataset are $0.012314$ for the incubation period and $0.000098$ for
the generation time. These values are shown as vertical lines in Figure 7. In
our simulations $63\%$ of the simulated Hellinger distances for the incubation
period are larger than the observed distance. For the generation time this
percentage is $98\%$ and thus even larger large. In $61\%$ of the simulated
cases both distances are simultaneously larger than the observed distances. We
conclude that the dataset does not provide evidence for rejecting the
parametric model.

Figure 7: Histograms of squared Hellinger distances of $N=100$ simulations
from the model under $H_{0}^{(I)}\cap H_{0}^{(G)}$. Distances are computed
between the estimated model and the closest Laguerre type model. The vertical
lines indicate the observed values of the test statistic.

In Table 2 we see that the BIC values for all choices of $m_{1}\geq 2$ and
$m_{2}=2,3$ are roughly equal. Since we see in Figure 2 that for a better
assessment of the generation time we require that $m_{2}\geq 2$, we provide in
Section 6.2.2 in the Appendix a similar analysis for $m_{1}=m_{2}=2$.

We stress that the above analysis should be understood as a recipe for a data
analysis rather than an in-depth analysis of the provided dataset. Other types
of semi-parametric analyses like the ones outlined in Section 4.1 can be
implemented in a similar way.

##  5 Conclusion and Related Questions

We have introduced a semi-parametric estimator for the generation time and
incubation period from observational data. We have shown that both
distributions can be identified from the observations and we presented a
simple, consistent semi-parametric estimator which is based on Laguerre
polynomials. These results are the first steps for more results in the general
realm of semi-parametric inference for epidemics: As specific examples we
mention the reproduction number $R_{0}$ and tests for parametric assumptions.
But also the probability of pre-symptomatic infection $\mathbb{P}(G\leq
I_{1})$ can be of interest. All these quantities are continuous functions of
the densities $\varphi_{I}$ and $\varphi_{G}$ and therefore it can be expected
that asymptotic normality results for estimators based on our estimators
$\hat{\varphi}_{I,n}$ and $\hat{\varphi}_{G,n}$ can be proven. However, it
should be mentioned that such results for $R_{0}$ are possibly more
challenging because they require another estimator $\hat{r}_{n}$.

This work can be extended in several directions. It might for example not be
clear when symptoms exactly start. Therefore, it might be possible that just a
window for symptom onset can be supplied (Lauer et al. (2020)). Moreover, it
can be of interest to include asymptomatic patients by including a certain
_cure-probability_ , i.e. the probability with which patients will never show
symptoms. As an alternative one could also consider follow up studies in which
the symptom onset time of patients is considered as a censored variable (in
which case asymptomatic patients can be interpreted as patients who show
symptoms at $\infty$). Finally, the dataset by Ferretti et al. (2020) includes
also a window for infection of the second person. It would of course be
interesting to include this information in the model. But we would like to
point out that this is not entirely trivial because it is unclear how to model
the time of infection within this window. As we motivated in the beginning of
Section 2 a simple uniformity assumption is possibly difficult to justify.
Therefore, we would suggest to include this distribution in the estimation in
a suitable way. In theoretic terms, the most interesting question would
certainly be to establish asymptotic normality results which allow the
researcher to make quantitative statements. Such statements can for example be
achieved by following the semi-parametric framework as e.g. in Shen (1997).
Another interesting question would be how to adequately incorporate covariates
in the model. It could be the case that e.g. younger and older people have
different incubation periods or generation times.

## Acknowledgements

We are grateful to Niel Hens for useful discussions. Moreover we thank an
unknown associate editor and two anonymous referees for reading and commenting
our paper. Their remarks have lead to a great improvement of the paper.

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##  6 Appendix

###  6.1 A Consistency Result and Approximation Through Laguerre Polynomials

In this section we state two results from the literature which are relevant
for this paper.

The following Theorem is just a re-formulation for a special case of Theorem 4
in Wong and Shen [1995] which is stated here for the convenience of the
reader: In their paper the authors study approximate sieve estimation, i.e.,
they allow that the estimator only approximately maximizes the likelihood. In
their notation this means that we assume in the present paper that
$\eta_{n}=0$, this is already included in the following version of Theorem 4
of Wong and Shen [1995].

######  Theorem 6.1.

Let $Y_{1},...,Y_{n}$ be iid observations which have a density $p_{0}$. Let
moreover $\mathcal{F}_{n}$ be an arbitrary sequence of sieve spaces for
density estimation and let $\hat{p}_{n}$ denote the sieve-MLE. Suppose that
there are constants $c_{1},c_{2}>0$ and a sequence $\varepsilon_{n}>0$ such
that

$\int_{\frac{\varepsilon_{n}^{2}}{2^{8}}}^{\sqrt{2}\varepsilon_{n}}\sqrt{\log\mathcal{N}_{[]}\left(\frac{u}{c_{1}},\mathcal{F}_{n},\rho_{H}\right)}du\leq
c_{2}\sqrt{n}\varepsilon_{n}^{2}.$

Let $\delta_{n}(\alpha):=\inf_{q\in\mathcal{F}_{n}}\rho_{\alpha}(p_{0},q)$ for
$\alpha\in(0,1]$. Suppose that there is $\alpha\in(0,1]$ such that
$\delta_{n}(\alpha)<1/\alpha$. Then, there is a constant $c>0$ (which depends
on the model) such that for

$\varepsilon_{n}^{*}(\alpha):=\max\left(\varepsilon_{n},\sqrt{\frac{4\delta_{n}(\alpha)}{c}}\right)$

we have for another constant $C>0$

$\mathbb{P}\left(\rho_{H}(\hat{p}_{n},p_{0})\geq\varepsilon_{n}^{*}(\alpha)\right)\leq
5\exp\left(-Cn\varepsilon_{n}^{*}(\alpha)^{2}\right)+\exp\left(-\frac{1}{4}n\alpha
c\varepsilon_{n}^{*}(\alpha)^{2}\right).$

The next result is a statement about approximating functions by using Laguerre
polynomials. The following is a combination of Theorem 1 and Remark 2 in
Chapter II.8 of Nikiforov and Uvarov [1988]. We formulate this theorem here in
our setting. The original statement is more general.

######  Theorem 6.2.

Let $f:[0,\infty)\to\mathbb{R}$ be continuous and have a piecewise continuous
derivative $f^{\prime}$. Consider the series

$f_{N}(x):=\sum_{k=0}^{N}c_{k}L_{k}(x),\quad
c_{k}:=\int_{0}^{\infty}f(x)L_{k}(x)e^{-x}dx.$

If the integrals

$\int_{0}^{\infty}f(x)^{2}e^{-x}dx\,\textrm{ and
}\,\int_{0}^{\infty}f^{\prime}(x)^{2}xe^{-x}dx$

converge, we have $f_{N}\to f$ for $N\to\infty$ uniformly over compact sets
$[x_{1},x_{2}]\subseteq(0,\infty)$ and
$\lim_{N\to\infty}\int_{0}^{\infty}\left[f(x)-f_{N}(x)\right]^{2}e^{-x}dx\to
0$.

###  6.2 Further Empirical Results

####  6.2.1 Simulation Study

In this Section we present additional simulation results complementing the
results from Section 4.1. Figures 8-10 show histograms for estimation of the
50%-, 70%- and 90%-quantiles. The results are very similar to the results for
the 30%-quantile which are discussed in Section 4.1.

Figure 8: Histograms of estimated 50%-quantile for the incubation time (left)
and generation time (right). The dashed lines indicate the respective true
values, the dotted lines the quantiles of the best approximating densities and
the solid line is the density of a normal distribution with the corresponding
sample mean and standard deviation. Figure 9: Histograms of estimated
70%-quantile for the incubation time (left) and generation time (right). The
dashed lines indicate the respective true values, the dotted lines the
quantiles of the best approximating densities and the solid line is the
density of a normal distribution with the corresponding sample mean and
standard deviation. Figure 10: Histograms of estimated 90%-quantile for the
incubation time (left) and generation time (right). The dashed lines indicate
the respective true values, the dotted lines the quantiles of the best
approximating densities and the solid line is the density of a normal
distribution with the corresponding sample mean and standard deviation.

Next, we use the same set-up as in Section 4.1 in the main text, however, here
we choose as a model $m_{1}=2$ and $m_{2}=3$, i.e., the model has now more
flexibility for the generation time. The estimation results are visualized in
Figure 11 and the difference in the squared Hellinger distance is shown in
Figure 12. In Figure 11 it appears that for the generation time the mode moves
closer to its true location but it is overestimated (compared to the case
$m_{2}=2$ which was shown in Section 4.1). Moreover, the estimated generation
time densities seem to fluctuate more, i.e., they show a higher variance due
to the higher flexibility. The estimates for the incubation time appear to be
almost identical. In terms of the Hellinger distances, cf. histograms in
Figure 12, the results appear to be very similar to the results obtained in
Section 4.1. In general we see that estimation with $n=40$ observations yields
reasonable results for both model complexities. However, we expect that the
estimation can be improved if more observations are available enabling the
method to choose better approximations.

Figure 11: Estimation results (gray lines) for simulated data with $n=40$
observations. True densities are shown in black and the dashed lines show the
Laguerre densities which come closest to the true densities. Model choices:
$m_{1}=2$ and $m_{2}=3$ Figure 12: Histograms of squared Hellinger distances
of estimates to true densities. Model Choices: $m_{1}=2$ and $m_{2}=3$.

####  6.2.2 Real Data Application

In this section we show an analysis similar to that from Section 4.2 in the
main text, but here we choose $m_{1}=m_{2}=2$. We begin with a simulation: We
generate $N=100$ datasets assuming that the models specified in $H_{0}^{(I)}$
and $H_{0}^{(G)}$ are correct. Figure 13 shows the squared Hellinger distances
of the estimates to the closest Laguerre type densities with $m_{1}=m_{2}=2$.
The vertical lines show the distances which are obtained from the real data
set. In case of the incubation period $60\%$ of the simulated distances are
larger, for the generation time this percentage is lower, $21\%$, but still
high. Finally, both distances are simultaneously larger for both densities in
$14\%$ of the simulated cases. Thus, we conclude that also in this larger
model class there is no evidence in the dataset which would contradict the
parametric models fitted by Ferretti et al. [2020] and Lauer et al. [2020].

Figure 13: Histogram of squared Hellinger distances of $N=100$ simulations
from the model under $H_{0}^{(I)}\cap H_{0}^{(G)}$. Distances are computed
between the estimated model and the closest Laguerre type model (for
$m_{1}=m_{2}=2$). The vertical lines indicate the observed values of the test
statistic.

###  6.3 Proofs of Section 3.1

###### Proof of Corollary 3.4.

For the proof we make the following definitions: Let $\Phi_{I}$, $\Phi_{G}$
and $\Phi_{S_{1},S_{2},W,C}$ be the characteristic functions of the
corresponding random variables, i.e., for real numbers
$x,x_{1},...,x_{4}\in\mathbb{R}$ (note that in the definition of $\varphi_{I}$
it doesn’t matter whether we take $I_{1}$ or $I_{2}$)

$\displaystyle\Phi_{I}(x)$
$\displaystyle:=\mathbb{E}\left(e^{ixI_{1}}\right),\quad\Phi_{G}(x):=\mathbb{E}\left(e^{ixG}\right),$
$\displaystyle\Phi_{S_{1},S_{2},W,C}(x_{1},x_{2},x_{3},x_{4})$
$\displaystyle:=\mathbb{E}\left(e^{i(x_{1}S1+x_{2}S_{2}+x_{3}W+x_{4}C)}\right).$

As a first step, we compute $\Phi_{S_{1},S_{2},W,C}$. We have by definition of
$(S_{1},S_{2},W,C)$ and independence

$\displaystyle\Phi_{S_{1},S_{2},W,C}(x_{1},x_{2},x_{3},x_{4})=\mathbb{E}\left(e^{i(x_{1}+x_{2})T_{1}}e^{ix_{1}I_{1}}e^{ix_{2}I_{2}}e^{ix_{2}G}e^{ix_{3}W}e^{ix_{4}C}\right)$
$\displaystyle=$
$\displaystyle\mathbb{E}\left(e^{ix_{3}W+ix_{4}C}\mathbb{E}\left(e^{i(x_{1}+x_{2})T_{1}}\big{|}W,C\right)\right)\Phi_{I}(x_{1})\Phi_{I}(x_{2})\Phi_{G}(x_{2})$
$\displaystyle=$
$\displaystyle\Psi(x_{1}+x_{2},x_{3},x_{4})\Phi_{I}(x_{1})\Phi_{I}(x_{2})\Phi_{G}(x_{2}),$

where

$\Psi(x,z_{1},z_{2}):=\mathbb{E}\left(e^{iz_{1}W+iz_{2}C}\mathbb{E}\left(e^{ixT_{1}}\big{|}W,C\right)\right).$

Let now $z_{1}(x),z_{2}(x)$ be as in Assumption (C). We obtain the following
two equations

$\displaystyle\Phi_{S_{1},S_{2},W,C}\left(x,0,z_{1}(x),z_{2}(x)\right)$
$\displaystyle=\Psi(x,z_{1}(x),z_{2}(x))\Phi_{I}(x),$
$\displaystyle\Phi_{S_{1},S_{2},W,C}\left(0,x,z_{1}(x),z_{2}(x)\right)$
$\displaystyle=\Psi(x,z_{1}(x),z_{2}(x))\Phi_{I}(x)\Phi_{G}(x).$

Note that Assumption (C) implies that
$\Phi_{S_{1},S_{2},W,C}(x,0,z_{1}(x),z_{2}(x))\neq 0$ almost everywhere. All
of this, in turn, implies the relations (almost everywhere by Assumption (C))

$\displaystyle\Phi_{I}(x)=$
$\displaystyle\Psi(x,z_{1}(x),z_{2}(x))^{-1}\Phi_{S_{1},S_{2},W,C}\left(x,0,z_{1}(x),z_{2}(x)\right),$
(6.1) $\displaystyle\Phi_{G}(x)=$
$\displaystyle\Phi_{S_{1},S_{2},W,C}\left(x,0,z_{1}(x),z_{2}(x)\right)^{-1}\Phi_{S_{1},S_{2},W,C}\left(0,x,z_{1}(x),z_{2}(x)\right).$
(6.2)

Let $\hat{C}_{n},\hat{W}_{n},\hat{T}_{1,n},\hat{I}_{1,n},\hat{I}_{2,n}$ and
$\hat{G}_{n}$ be random variables which have the relation as specified in
(2.1) but where $\varphi_{I}$ is replaced by $\hat{\varphi}_{I,n}$ and
$\varphi_{G}$ is replaced by $\hat{\varphi}_{G,n}$. Let moreover,
$\hat{S}_{1,n}$ and $\hat{S}_{2,n}$ be defined analogously to their
corresponding quantities $S_{1}$ and $S_{2}$ with corresponding characteristic
functions $\Phi_{\hat{S}_{1,n},\hat{S}_{2,n},\hat{W}_{n},\hat{C}_{n}}$,
$\Phi_{\hat{I}_{n}}$ and $\Phi_{\hat{G}_{n}}$.

Since convergence in probability implies the existence of an almost surely
convergent subsequence (cf. Lemma 1.9.2 in van der Vaart and Wellner [1996]),
we find by Theorem 3.1 that there is an event $\Omega$ of probability 1 and a
subsequence which we indicate again by
$f_{\hat{\varphi}_{I,n},\hat{\varphi}_{G,n}}$ such that

$\rho_{H}\left(f_{\varphi_{I},\varphi_{G}}p_{C},f_{\hat{\varphi}_{I,n},\hat{\varphi}_{G,n}}p_{C}\right)\to
0$

for all results in $\Omega$. Theorem 1 in Chapter 3.9 in Shiryaev [2016]
implies that then

$\left|\sum_{c=1}^{K_{C}}\int
r(s_{1},s_{2},w,c)\left(f_{\varphi_{I},\varphi_{G}}(s_{1},s_{2},w|c)-f_{\hat{\varphi}_{I,n},\hat{\varphi}_{G,n}}(s_{1},s_{2},w|c)\right)p_{C}(c)d(s_{1},s_{2},w)\right|\to
0,$

uniformly over all functions
$r:\mathbb{R}^{3}\times\\{1,...,K_{C}\\}\to\mathbbm{C}$ with
$\|r\|_{\infty}\leq 1$. Since the functions
$r(s_{1},s_{2},w,c)=\exp(i(x_{1}s_{1}+x_{2}s_{2}+x_{3}w+x_{4}c))$ are bounded
for all $x_{1},x_{2},x_{3},x_{4}\in\mathbb{R}$, we conclude that
$\Phi_{\hat{S}_{1,n},\hat{S}_{2,n},\hat{W}_{n},\hat{C}_{n}}$ converges
uniformly to $\Phi_{S_{1},S_{2},W,C}$. Hence, for almost any fixed $x$ we
eventually have
$\Phi_{\hat{S}_{1,n},\hat{S}_{2,n},\hat{W}_{n},\hat{C}_{n}}(x,0,z_{1}(x),z_{2}(x))\neq
0$ and thus eventually analogue versions of the relations (6.1) and (6.2) hold
for $\Phi_{\hat{I}_{n}}$ and $\Phi_{\hat{G}_{n}}$. As a consequence, we obtain
the following point-wise almost everywhere convergences

$\displaystyle\Phi_{\hat{I}_{n}}(x)=$
$\displaystyle\Psi(x,z_{1}(x),z_{2}(x))^{-1}\Phi_{\hat{S}_{1,n},\hat{S}_{2,n},\hat{W}_{n},\hat{C}_{n}}\left(x,0,z_{1}(x),z_{2}(x)\right)$
$\displaystyle\quad\quad\to\Psi(x,z_{1}(x),z_{2}(x))^{-1}\Phi_{S_{1},S_{2},W,C}\left(x,0,z_{1}(x),z_{2}(x)\right)=\Phi_{I}(x),$
$\displaystyle\Phi_{\hat{G}_{n}}(x)=$
$\displaystyle\Phi_{\hat{S}_{1,n},\hat{S}_{2,n},\hat{W}_{n},\hat{C}_{n}}\left(x,0,z_{1}(x),z_{2}(x)\right)^{-1}\Phi_{\hat{S}_{1,n},\hat{S}_{2,n},\hat{W}_{n},\hat{C}_{n}}\left(0,x,z_{1}(x),z_{2}(x)\right)$
$\displaystyle\quad\quad\to\Phi_{S_{1},S_{2},W,C}\left(x,0,z_{1}(x),z_{2}(x)\right)^{-1}\Phi_{S_{1},S_{2},W,C}\left(0,x,z_{1}(x),z_{2}(x)\right)=\Phi_{G}(x).$

Now by Levy’s Theorem (cf. Theorem 1.7.6 Billingsley [1968], an inspection of
the proof reveals that in the univariate case convergence almost everywhere is
a sufficient condition), we conclude that $\hat{F}_{I,n}\to F_{I}$ and
$\hat{F}_{G,n}\to F_{G}$ point-wise for all realisations in $\Omega$. Since
the distribution functions are continuous, these convergences are also
uniform. Recall that the above argument holds for a subsequence and all
realisations in $\Omega$. But we can repeat the same argument starting from a
subsequence and showing in that way that every subsequence contains a sub-
subsequence for which the corresponding distribution functions converge almost
surely. This implies convergence in probability by Lemma 1.9.2 in van der
Vaart and Wellner [1996] and the proof is complete. ∎

###### Proof of Corollary 3.5.

We make a similar subsequence of a sub-sequence argument as in Corollary 3.4:
Consider sub-sequences of $\hat{r}_{n}$ and $\hat{F}_{G,n}$ (which we again do
not indicate in the notation) such that $\hat{r}_{n}\to r$ almost surely and
$\left\|\hat{F}_{G,n}-F_{G}\right\|_{\infty}\to 0$ almost surely (the former
is possible by assumption and the latter by Corollary 3.4). The following
considerations are made for each realisation in a set of probability one on
which these two convergences hold. Choose $n$ so large such that
$\hat{r}_{n}\geq r/2$. Choose moreover, $c>0$ such that $t\exp(-tr/2)\leq c$
for all $t\geq 0$. We have

$\displaystyle\int_{0}^{\infty}e^{-\hat{r}_{n}t}\hat{\varphi}_{G,n}dt-\int_{0}^{\infty}e^{-rt}\varphi_{G}(t)dt$
$\displaystyle=$
$\displaystyle\int_{0}^{\infty}\left(e^{-\hat{r}_{n}t}-e^{-rt}\right)\hat{\varphi}_{G,n}dt+\int_{0}^{\infty}e^{-rt}\left(\hat{\varphi}_{G,n}-\varphi_{G}(t)\right)dt.$
(6.3)

By a Taylor expansion we obtain for the first part of the right hand side
above and for some $r^{*}\in[r,\hat{r}_{n}]$ (note that by assumption
$r^{*}\geq r/2$ and hence $t\cdot\exp(-r^{*}t)\leq c$)

$\left|\int_{0}^{\infty}\left(e^{-\hat{r}_{n}t}-e^{-rt}\right)\hat{\varphi}_{G,n}(t)dt\right|\leq\left|\hat{r}_{n}-r\right|\int_{0}^{\infty}te^{-r^{*}t}\hat{\varphi}_{G,n}(t)dt\leq
c\left|\hat{r}_{n}-r\right|\to 0.$

For the second term on the right hand side of (6.3) we obtain via integration
by parts

$\displaystyle\int_{0}^{\infty}e^{-rt}\left(\hat{\varphi}_{G,n}-\varphi_{G}(t)\right)dt$
$\displaystyle=$ $\displaystyle
e^{-rt}\left(\hat{F}_{G,n}(t)-F_{G}(t)\right)\Big{|}_{t=0}^{\infty}+\int_{0}^{\infty}re^{-rt}\left(\hat{F}_{G,n}(t)-F_{G}(t)\right)dt$

which converges to zero since $\hat{F}_{G,n}\to F$ uniformly. Now, a
subsequence of a subsequence argument completes the proof. ∎

###  6.4 Proofs of Section 3.2

###### Proof of Lemma 3.6.

We firstly apply Hoelder’s Inequality with $p=1/(1-\alpha)$ and $q=1/\alpha$
(use that $\alpha\in(0,1)$) to get

$\displaystyle\rho_{\alpha}(f_{1}p_{C},f_{2}p_{C})$ $\displaystyle=$
$\displaystyle\frac{1}{\alpha}\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{3}}f_{1}(x_{1},x_{2},\omega|c)p_{C}(c)\left(\left(\frac{f_{1}(x_{1},x_{2},\omega|c)p_{C}(c)}{f_{2}(x_{1},x_{2},\omega|c)p_{C}(c)}\right)^{\alpha}-1\right)d(x_{1},x_{2},\omega)$
$\displaystyle=$
$\displaystyle\frac{1}{\alpha}\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{3}}\left(\frac{f_{1}(x_{1},x_{2},\omega|c)}{f_{2}(x_{1},x_{2},\omega|c)}\right)^{\alpha}f_{1}(x_{1},x_{2},\omega|c)^{1-\alpha}$
$\displaystyle\quad\quad\quad\quad\times\left(f_{1}(x_{1},x_{2},\omega|c)^{\alpha}-f_{2}(x_{1},x_{2},\omega|c)^{\alpha}\right)p_{C}(c)d(x_{1},x_{2},\omega)$
$\displaystyle\leq$
$\displaystyle\frac{1}{\alpha}\left(\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{3}}\left(\frac{f_{1}(x_{1},x_{2},\omega|c)}{f_{2}(x_{1},x_{2},\omega|c)}\right)^{\frac{\alpha}{1-\alpha}}f_{1}(x_{1},x_{2},\omega|c)p_{C}(c)d(x_{1},x_{2},\omega)\right)^{1-\alpha}$
$\displaystyle\quad\times\left(\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{3}}\left|f_{1}(x_{1},x_{2},\omega|c)^{\alpha}-f_{2}(x_{1},x_{2},\omega|c)^{\alpha}\right|^{\frac{1}{\alpha}}p_{C}(c)d(x_{1},x_{2},\omega)\right)^{\alpha}.$

By using the definition of $C_{\alpha}(f_{1},f_{2})$, we see that (3.4)
follows from the above if we can prove that

$\displaystyle\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{3}}\left|f_{1}(x_{1},x_{2},\omega|c)^{\alpha}-f_{2}(x_{1},x_{2},\omega|c)^{\alpha}\right|^{\frac{1}{\alpha}}p_{C}(c)d(x_{1},x_{2},\omega)$
$\displaystyle\leq$ $\displaystyle
2\rho_{H}(\varphi_{G,1},\varphi_{G,2})^{2}+8\rho_{H}(\varphi_{I,1},\varphi_{I,2})^{2}.$
(6.4)

We begin by applying the reverse triangle inequality for the $L^{1/\alpha}$
norm repeatedly. Define to this end

$\displaystyle B_{1}(t;x_{1},x_{2},c,\omega,y):=$
$\displaystyle\varphi_{G,1}(y)h(\omega-t|c)\varphi_{I,1}(x_{1}-t)\varphi_{I,1}(x_{2}-t-y),$
$\displaystyle B_{2}(t;x_{1},x_{2},c,\omega,y):=$
$\displaystyle\varphi_{G,2}(y)h(\omega-t|c)\varphi_{I,2}(x_{1}-t)\varphi_{I,2}(x_{2}-t-y),$
$\displaystyle A_{1}(y;x_{1},x_{2},c,\omega):=$
$\displaystyle\int_{0}^{\min(x_{1},\omega)}B_{1}(t;x_{1},x_{2},c,y)dt,$
$\displaystyle A_{2}(y;x_{1},x_{2},c,\omega):=$
$\displaystyle\int_{0}^{\min(x_{1},\omega)}B_{2}(t;x_{1},x_{2},c,y)dt.$

We have now by the reverse triangle inequality for $L^{1/\alpha}$ that for any
$x_{1},x_{2},\omega\geq 0$

$\displaystyle\Bigg{|}\left(\int_{0}^{x_{2}}A_{1}(y;x_{1},x_{2},c,\omega)dy\right)^{\alpha}-\left(\int_{0}^{x_{2}}A_{2}(y;x_{1},x_{2},c,\omega)dy\right)^{\alpha}\Bigg{|}^{\frac{1}{\alpha}}$
$\displaystyle=$
$\displaystyle\left|\left\|A_{1}(\cdot;x_{1},x_{2},c,\omega)^{\alpha}\right\|_{L^{1/\alpha}([0,x_{2}])}-\left\|A_{2}(\cdot;x_{1},x_{2},c,\omega)^{\alpha}\right\|_{L^{1/\alpha}([0,x_{2}])}\right|^{\frac{1}{\alpha}}$
$\displaystyle\leq$
$\displaystyle\left\|A_{1}(\cdot;x_{1},x_{2},c,\omega)^{\alpha}-A_{2}(\cdot;x_{1},x_{2},c,\omega)^{\alpha}\right\|_{L^{1/\alpha}([0,x_{2}])}^{\frac{1}{\alpha}}$
$\displaystyle=$
$\displaystyle\int_{0}^{x_{2}}\left|A_{1}(y;x_{1},x_{2},c,\omega)^{\alpha}-A_{2}(y;x_{1},x_{2},c,\omega)^{\alpha}\right|^{\frac{1}{\alpha}}dy$
$\displaystyle=$
$\displaystyle\int_{0}^{x_{2}}\left|\left(\int_{0}^{\min(x_{1},\omega)}B_{1}(t;x_{1},x_{2},c,\omega,y)dt\right)^{\alpha}-\left(\int_{0}^{\min(x_{1},\omega)}B_{2}(t;x_{1},x_{2},c,\omega,y)dt\right)^{\alpha}\right|^{\frac{1}{\alpha}}dy$
$\displaystyle=$
$\displaystyle\int_{0}^{x_{2}}\Big{|}\left\|B_{1}(\cdot;x_{1},x_{2},c,\omega,y)^{\alpha}\right\|_{L^{1/\alpha}([0,\min(x_{1},\omega)])}$
$\displaystyle\quad\quad\quad\quad-\left\|B_{2}(\cdot;x_{1},x_{2},c,\omega,y)^{\alpha}\right\|_{L^{1/\alpha}([0,\min(x_{1},\omega)])}\Big{|}^{\frac{1}{\alpha}}dy$
$\displaystyle\leq$
$\displaystyle\int_{0}^{x_{2}}\left\|B_{1}(\cdot;x_{1},x_{2},c,\omega,y)^{\alpha}-B_{2}(\cdot;x_{1},x_{2},c,\omega,y)^{\alpha}\right\|_{L^{1/\alpha}([0,\min(x_{1},\omega)])}^{\frac{1}{\alpha}}dy$
$\displaystyle=$
$\displaystyle\int_{0}^{x_{2}}\int_{0}^{\min(x_{1},\omega)}\Bigg{|}\left(\varphi_{G,1}(y)h(\omega-t|c)\varphi_{I,1}(x_{1}-t)\varphi_{I,1}(x_{2}-t-y)\right)^{\alpha}$
$\displaystyle\quad\quad\quad\quad-\left(\varphi_{G,2}(y)h(\omega-t|c)\varphi_{I,2}(x_{1}-t)\varphi_{I,2}(x_{2}-t-y)\right)^{\alpha}\Bigg{|}^{\frac{1}{\alpha}}dtdy.$

By using the above inequality chain, we obtain

$\displaystyle\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{3}}\left|f_{1}(x_{1},x_{2},\omega|c)^{\alpha}-f_{2}(x_{1},x_{2},\omega|c)^{\alpha}\right|^{\frac{1}{\alpha}}p_{C}(c)d(x_{1},x_{2},\omega)$
$\displaystyle=$
$\displaystyle\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{3}}\varphi_{W}(\omega)n(\omega,c)p_{C}(c)$
$\displaystyle\times\Bigg{|}\left(\int_{0}^{x_{2}}\varphi_{G,1}(y)\int_{0}^{\min(x_{1},\omega)}h(\omega-t|c)\varphi_{I,1}(x_{1}-t)\varphi_{I,1}(x_{2}-t-y)dtdy\right)^{\alpha}$
$\displaystyle-\left(\int_{0}^{x_{2}}\varphi_{G,2}(y)\int_{0}^{\min(x_{1},\omega)}h(\omega-t|c)\varphi_{I,2}(x_{1}-t)\varphi_{I,2}(x_{2}-t-y)dtdy\right)^{\alpha}\Bigg{|}^{\frac{1}{\alpha}}d(x_{1},x_{2},\omega)$
$\displaystyle=$
$\displaystyle\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{3}}\varphi_{W}(\omega)n(\omega,c)p_{C}(c)$
$\displaystyle\quad\times\Bigg{|}\left(\int_{0}^{x_{2}}A_{1}(y;x_{1},x_{2},c,\omega)dy\right)^{\alpha}-\left(\int_{0}^{x_{2}}A_{2}(y;x_{1},x_{2},c,\omega)dy\right)^{\alpha}\Bigg{|}^{\frac{1}{\alpha}}d(x_{1},x_{2},\omega)$
$\displaystyle\leq$
$\displaystyle\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{3}}\varphi_{W}(\omega)n(\omega,c)p_{C}(c)$
$\displaystyle\quad\quad\times\int_{0}^{x_{2}}\int_{0}^{\min(x_{1},\omega)}\Big{|}\left(\varphi_{G,1}(y)h(\omega-t|c)\varphi_{I,1}(x_{1}-t)\varphi_{I,1}(x_{2}-t-y)\right)^{\alpha}$
$\displaystyle\quad\quad\quad-\left(\varphi_{G,2}(y)h(\omega-t|c)\varphi_{I,2}(x_{1}-t)\varphi_{I,2}(x_{2}-t-y)\right)^{\alpha}\Big{|}^{\frac{1}{\alpha}}dtdyd(x_{1},x_{2},\omega).$
(6.5)

Above we have an iterated integral over a non-negative function, we may thus
re-arrange the order of integration and use substitution. We substitute below
$a=x_{1}-t$ for $x_{1}$ and $b=x_{2}-t-y$ for $x_{2}$. Note that we implicitly
take care of the integration bounds by using the indicator function and the
fact that all densities are zero on the negative real line. Hence, we can
continue the above inequality chain

$\displaystyle\eqref{eq:step1}=$
$\displaystyle\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{5}}\mathbbm{1}(y\leq
x_{2},t\leq\min(x_{1},\omega))\varphi_{W}(\omega)n(\omega,c)p_{C}(c)h(\omega-t|c)$
$\displaystyle\quad\quad\times\Big{|}\left(\varphi_{G,1}(y)\varphi_{I,1}(x_{1}-t)\varphi_{I,1}(x_{2}-t-y)\right)^{\alpha}$
$\displaystyle\quad\quad\quad-\left(\varphi_{G,2}(y)\varphi_{I,2}(x_{1}-t)\varphi_{I,2}(x_{2}-t-y)\right)^{\alpha}\Big{|}^{\frac{1}{\alpha}}dx_{1}\,dx_{2}\,dt\,dy\,d\omega$
$\displaystyle=$
$\displaystyle\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{5}}\mathbbm{1}(y\leq
x_{2},t\leq\min(a+t,\omega))\varphi_{W}(\omega)n(\omega,c)p_{C}(c)h(\omega-t|c)$
$\displaystyle\quad\quad\times\Big{|}\left(\varphi_{G,1}(y)\varphi_{I,1}(a)\varphi_{I,1}(x_{2}-t-y)\right)^{\alpha}$
$\displaystyle\quad\quad\quad-\left(\varphi_{G,2}(y)\varphi_{I,2}(a)\varphi_{I,2}(x_{2}-t-y)\right)^{\alpha}\Big{|}^{\frac{1}{\alpha}}da\,dx_{2}\,dt\,dy\,d\omega$
$\displaystyle=$
$\displaystyle\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{5}}\mathbbm{1}(0\leq
b+t,t\leq\min(a+t,\omega))\varphi_{W}(\omega)n(\omega,c)p_{C}(c)h(\omega-t|c)$
$\displaystyle\quad\quad\times\Big{|}\left(\varphi_{G,1}(y)\varphi_{I,1}(a)\varphi_{I,1}(b)\right)^{\alpha}-\left(\varphi_{G,2}(y)\varphi_{I,2}(a)\varphi_{I,2}(b)\right)^{\alpha}\Big{|}^{\frac{1}{\alpha}}da\,db\,dt\,dy\,d\omega.$
(6.6)

Note next that the indicator equals actually $\mathbbm{1}(t\in[0,\omega])$. We
again interchange the order of integration, to integrate with respect to $t$
first and then with respect to $\omega$. By doing this, recalling the form of
$\varphi_{T_{1}}$ in Assumption (M), keeping in mind that
$|x^{2\alpha}-y^{2\alpha}|\leq|x-y|^{2\alpha}$ for all $x,y\geq 0$ (since
$2\alpha\in(0,1)$) and that $(x-y)^{2}\leq 2x^{2}+2y^{2}$ for all $x,y$, we
continue

$\displaystyle\eqref{eq:step2}=$
$\displaystyle\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{5}}\varphi_{W}(\omega)p_{C}(c)\varphi_{T_{1}}(t|\omega,c)dt\,d\omega$
$\displaystyle\quad\quad\times\Big{|}\left(\varphi_{G,1}(y)\varphi_{I,1}(a)\varphi_{I,1}(b)\right)^{\alpha}-\left(\varphi_{G,2}(y)\varphi_{I,2}(a)\varphi_{I,2}(b)\right)^{\alpha}\Big{|}^{\frac{1}{\alpha}}da\,db\,dy$
$\displaystyle=$
$\displaystyle\int_{[0,\infty)^{3}}\Big{|}\left(\varphi_{G,1}(y)\varphi_{I,1}(a)\varphi_{I,1}(b)\right)^{\alpha}-\left(\varphi_{G,2}(y)\varphi_{I,2}(a)\varphi_{I,2}(b)\right)^{\alpha}\Big{|}^{\frac{1}{\alpha}}da\,db\,dy$
(6.7) $\displaystyle\leq$
$\displaystyle\int_{[0,\infty)^{3}}\Big{|}\left(\varphi_{G,1}(y)\varphi_{I,1}(a)\varphi_{I,1}(b)\right)^{\frac{1}{2}}-\left(\varphi_{G,2}(y)\varphi_{I,2}(a)\varphi_{I,2}(b)\right)^{\frac{1}{2}}\Big{|}^{2}da\,db\,dy$
$\displaystyle\leq$
$\displaystyle\int_{[0,\infty)^{3}}2\left(\varphi_{G,1}(y)^{\frac{1}{2}}-\varphi_{G,2}(y)^{\frac{1}{2}}\right)^{2}\varphi_{I,1}(a)\varphi_{I,1}(b)da\,db\,dy$
$\displaystyle\quad\quad+2\int_{[0,\infty)^{3}}\varphi_{G,2}(y)\left(\varphi_{I,2}(a)^{\frac{1}{2}}\varphi_{I,2}(b)^{\frac{1}{2}}-\varphi_{I,1}(a)^{\frac{1}{2}}\Phi_{I,1}(b)^{\frac{1}{2}}\right)^{2}da\,db\,dy$
$\displaystyle=$ $\displaystyle
2\rho_{H}(\varphi_{G,1},\varphi_{G,2})^{2}+2\int_{[0,\infty)^{2}}\left(\varphi_{I,2}(a)^{\frac{1}{2}}\varphi_{I,2}(b)^{\frac{1}{2}}-\varphi_{I,1}(a)^{\frac{1}{2}}\varphi_{I,1}(b)^{\frac{1}{2}}\right)^{2}da\,db$
$\displaystyle\leq$ $\displaystyle
2\rho_{H}(\varphi_{G,1},\varphi_{G,2})^{2}+4\int_{[0,\infty)^{2}}\left(\varphi_{I,2}(a)^{\frac{1}{2}}-\varphi_{I,1}(a)^{\frac{1}{2}}\right)^{2}\varphi_{I,2}(b)d\,db$
$\displaystyle\quad\quad+4\int_{[0,\infty)^{2}}\varphi_{I,1}(a)\left(\varphi_{I,1}(b)^{\frac{1}{2}}-\varphi_{I,2}(b)^{\frac{1}{2}}\right)^{2}da\,db$
$\displaystyle=$ $\displaystyle
2\rho_{H}(\varphi_{G,1},\varphi_{G,2})^{2}+8\rho_{H}(\varphi_{I,1},\varphi_{I,2})^{2}.$

This is (6.4) and the proof is complete. ∎

###### Proof of Lemma 3.8.

Denote for any $m\in\mathbb{N}_{0}$,
$\Theta_{m}:=\\{\theta\in\mathbb{R}^{m+1}:\,\|\theta\|_{2}=1\\}$. The proof of
this Lemma uses the following strategy which is similar to Lemma 2.1 in
Ossiander [1987]. In the interest of completeness we give the detailed proof:
Let $\delta\in(0,\sqrt{3/2}/2]$ be given. We define for any
$\theta_{1}\in\Theta_{m_{1}}$ and $\theta_{2}\in\Theta_{m_{2}}$ the
$\delta$-ball

$B_{\delta}(\theta_{1},\theta_{2}):=\left\\{\left(\widetilde{\theta}_{1},\widetilde{\theta}_{2}\right)\in\Theta_{m_{1}}\times\Theta_{m_{2}}:\,\|\theta_{1}-\widetilde{\theta}_{1}\|_{2}\leq\delta,\|\theta_{2}-\widetilde{\theta}_{2}\|_{2}\leq\delta\right\\}.$

Find now a set
$\left((\theta_{1,i},\theta_{2,i})\right)_{i=1,...,N(\delta)}\subseteq\Theta_{m_{1}}\times\Theta_{m_{2}}$
such that

$\bigcup_{i=1}^{N(\delta)}B_{\delta}(\theta_{1,i},\theta_{2,i})\supseteq\Theta_{m_{1}}\times\Theta_{m_{2}}.$

In order to bound $N(\delta)$ we construct a specific collection of pairs:
Consider a grid of $[0,\pi]^{m_{1}}$ with side length
$\alpha=2\delta\sqrt{2/3}$ and construct $\theta_{1,i}$ by taking the grid
points as polar coordinates (with radius $1$). Then, it is clear that for any
$\theta\in\Theta_{m_{1}}$, there is a grid point $\theta_{1,i}$ such that the
difference between any two angles of the polar representations of $\theta$ and
$\theta_{1,i}$ is smaller than $\alpha/2=\delta\sqrt{2/3}$. By Lemma 6.3 below
and symmetry of the polar coordinates, we find that
$\|\theta-\theta_{1,i}\|_{2}\leq\sqrt{3/2}\alpha/2=\delta$. The size of this
grid can be bounded by $(\pi/\alpha+1)^{m_{1}}$. We repeat this construction
for $\Theta_{m_{2}}$ and obtain

$N(\delta)\leq\left(\frac{\sqrt{3}\pi}{2\sqrt{2}\delta}+1\right)^{m_{1}+m_{2}}.$
(6.8)

The brackets are now defined as

$\displaystyle l_{i}(x_{1},x_{2},\omega,c):=\inf_{(\theta_{1},\theta_{2})\in
B_{\delta}(\theta_{1,i},\theta_{2,i})}f_{\varphi_{I,\theta_{1}},\varphi_{G,\theta_{2}}}(x_{1},x_{2},\omega|c)p_{C}(c),$
$\displaystyle u_{i}(x_{1},x_{2},\omega,c):=\sup_{(\theta_{1},\theta_{2})\in
B_{\delta}(\theta_{1,i},\theta_{2,i})}f_{\varphi_{I,\theta_{1}},\varphi_{G,\theta_{2}}}(x_{1},x_{2},\omega|c)p_{C}(c).$

For any
$f_{\varphi_{I,\theta_{1}},\varphi_{G,\theta_{2}}}p_{C}\in\mathcal{F}_{m_{1},m_{2}}$
we find thus first a pair $(\theta_{1,i},\theta_{2,i})$ such that
$(\theta_{1},\theta_{2})\in B_{\delta}(\theta_{1,i},\theta_{2,i})$ and thus
also $l_{i}\leq f_{\varphi_{I,\theta_{1}},\varphi_{G,\theta_{2}}}p_{C}\leq
u_{i}$. It remains to compute $\rho_{H}(l_{i},u_{i})$. To this end, we firstly
see that the same arguments which lead to (6.7) (for $\alpha=1/2$) give us
here the following (the $\sup$ refers always to the supremum over all pairs
$(\theta_{1},\theta_{2}),(\widetilde{\theta}_{1},\widetilde{\theta}_{2})\in\Theta_{m_{1}}\times\Theta_{m_{2}}$
such that $\|\theta_{1}-\widetilde{\theta}_{1}\|_{2}\leq\delta$ and
$\|\theta_{2}-\widetilde{\theta}_{2}\|_{2}\leq\delta$)

$\displaystyle\rho_{H}(l_{i},u_{i})^{2}$ $\displaystyle\leq$
$\displaystyle\sum_{c=1}^{K_{C}}\int_{[0,\infty)^{3}}\sup\left(f_{\varphi_{I,\theta_{1}},\varphi_{G,\theta_{2}}}(x_{1},x_{2},\omega)^{\frac{1}{2}}-f_{\varphi_{I,\widetilde{\theta}_{1}},\varphi_{G,\widetilde{\theta}_{2}}}(x_{1},x_{2},\omega)^{\frac{1}{2}}\right)^{2}p_{C}(c)d(x_{1},x_{2},\omega)$
$\displaystyle\leq$
$\displaystyle\int_{[0,\infty)^{3}}\sup\left(\left(\varphi_{G,\theta_{1}}(y)\varphi_{I,\theta_{2}}(a)\varphi_{I,\theta_{2}}(b)\right)^{\frac{1}{2}}-\left(\varphi_{G,\widetilde{\theta}_{1}}(y)\varphi_{I,\widetilde{\theta}_{2}}(a)\varphi_{I,\widetilde{\theta}_{2}}(b)\right)^{\frac{1}{2}}\right)^{2}dydadb$
$\displaystyle\leq$
$\displaystyle\int_{[0,\infty)^{3}}e^{-y-a-b}\sup\Bigg{(}\sum_{k=0}^{m_{1}}\left(\theta_{1,k}-\widetilde{\theta}_{1,k}\right)L_{k}(y)\sum_{k=0}^{m_{2}}\theta_{2,k}L_{k}(a)\sum_{k=0}^{m_{2}}\theta_{2,k}L_{k}(b)$
$\displaystyle-\sum_{k=0}^{m_{1}}\widetilde{\theta}_{1,k}L_{k}(y)\Bigg{(}\sum_{k=0}^{m_{2}}\left(\widetilde{\theta}_{2,k}-\theta_{2,k}\right)L_{k}(a)\sum_{k=0}^{m_{2}}\widetilde{\theta}_{2,k}L_{k}(b)$
$\displaystyle\quad\quad\quad\quad\quad\quad-\sum_{k=0}^{m_{2}}\theta_{2,k}L_{k}(a)\sum_{k=0}^{m_{2}}\left(\theta_{2,k}-\widetilde{\theta}_{2,k}\right)L_{k}(b)\Bigg{)}\Bigg{)}^{2}dydadb$
$\displaystyle\leq$
$\displaystyle\int_{[0,\infty)^{3}}e^{-y-a-b}\Bigg{(}2\delta^{2}\sum_{k=0}^{m_{1}}L_{k}(y)^{2}\sum_{k=0}^{m_{2}}L_{k}(a)^{2}\sum_{k=0}^{m_{2}}L_{k}(b)^{2}$
$\displaystyle+2\sum_{k=0}^{m_{1}}L_{k}(y)^{2}\left(2\delta^{2}\sum_{k=0}^{m_{2}}L_{k}(a)^{2}\sum_{k=0}^{m_{2}}L_{k}(b)^{2}+2\delta^{2}\sum_{k=0}^{m_{2}}L_{k}(a)^{2}\sum_{k=0}^{m_{2}}L_{k}(b)^{2}\right)\Bigg{)}dydadb$
$\displaystyle=$ $\displaystyle 10\delta^{2}m_{1}m_{2}^{2},$

where we used the Cauchy-Schwarz-Inequality in between and the integral
properties of the Laguerre polynomials at the end. Thus, when putting
$\delta=\varepsilon\left(10m_{1}m_{2}^{2}\right)^{-1/2}$ (the condition on
$\delta$ is fulfilled by the assumption on $\varepsilon$) we find together
with (6.8)

$\mathcal{N}_{[]}(\varepsilon,\mathcal{F}_{m_{1},m_{2}},\rho_{H})\leq\left(\frac{\pi\sqrt{15m_{1}m_{2}^{2}}}{2\varepsilon}+1\right)^{m_{1}+m_{2}}\leq\left(\frac{\pi\sqrt{15m_{1}m_{2}^{2}}}{\varepsilon}\right)^{m_{1}+m_{2}}$

and the proof is complete. ∎

######  Lemma 6.3.

Let $n\geq 2$ and $\delta\in[0,1/2]$. Denote by $e_{1}:=(1,0,...,0)^{\prime}$
the first unit vector of $\mathbb{R}^{n}$. Let $x\in\mathbb{R}^{n}$ have
$\|x\|_{2}=1$ and angles of polar coordinates
$\psi_{1},...,\psi_{n-1}\in[0,\delta]$. Then,
$\|e_{1}-x\|_{2}\leq\delta\sqrt{3/2}$.

###### Proof.

By a Taylor expansion, we have for any $\psi\in[0,2\pi]$ (below $\psi^{*}$
denotes different intermediate values between $0$ and $\psi$)

$\displaystyle|\sin(\psi)|=$
$\displaystyle|\cos(\psi^{*})|\cdot|\psi|\leq|\psi|$
$\displaystyle|\cos(\psi)-1|=$
$\displaystyle|\sin(\psi^{*})|\cdot|\psi|\leq|\psi|^{2}.$

By the definition of polar coordinates, we compute (for $n=2$ the sum
disappears)

$\displaystyle\|e_{1}-x\|_{2}^{2}=(1-\cos(\psi_{1}))^{2}+\prod_{k=1}^{n-1}\sin(\psi_{k})^{2}+\sum_{i=2}^{n-1}\prod_{k=1}^{i-1}\sin(\psi_{k})^{2}\cos(\psi_{i})^{2}$
$\displaystyle\leq$
$\displaystyle\psi_{1}^{4}+\prod_{k=1}^{n-1}\psi_{k}^{2}+\sum_{i=2}^{n-1}\prod_{k=1}^{i-1}\psi_{k}^{2}\leq\delta^{4}+\delta^{2(n-1)}+\sum_{i=2}^{n-1}\delta^{2(i-1)}\leq\delta^{2}\frac{\delta^{2}-\delta^{4}+1}{1-\delta^{2}}.$

The statement follows since $(\delta^{2}-\delta^{4}+1)/(1-\delta^{2})\leq 3/2$
for $\delta\in[0,1/2]$. ∎