# Dependent censoring based on copulas

Claudia Czado 111Technical University of Munich, E-mail address:
<EMAIL_ADDRESS>Claudia Czado acknowledges the support of the German Science
Foundation (DFG grant CZ 86/6-1)
Ingrid Van Keilegom 444ORSTAT, KU Leuven, E-mail address:
<EMAIL_ADDRESS>Ingrid Van Keilegom acknowledges the support
of the European Research Council (2016-2021, Horizon 2020 / ERC grant
agreement No. 694409).

(August 27, 2024)

###### Abstract

Consider a survival time $T$ that is subject to random right censoring, and
suppose that $T$ is stochastically dependent on the censoring time $C$. We are
interested in the marginal distribution of $T$. This situation is often
encountered in practice. Consider for instance the case where $T$ is the time
to death of a patient suffering from a certain disease. Then, the censoring
time $C$ is for instance the time until the person leaves the study or the
time until he/she dies from another disease. If the reason for leaving the
study is related to the health condition of the patient or if he/she dies from
a disease that has similar risk factors as the disease of interest, then $T$
and $C$ are likely dependent. In this paper we propose a new model that takes
this dependence into account. The model is based on a parametric copula for
the relationship between $T$ and $C$, and on parametric marginal distributions
for $T$ and $C$. Unlike most other papers in the literature, we do not assume
that the parameter defining the copula function is known. We give sufficient
conditions on these parametric copula and marginals under which the bivariate
distribution of $(T,C)$ is identified. These sufficient conditions are then
checked for a wide range of common copulas and marginal distributions. We also
study the estimation of the model, and carry out extensive simulations and the
analysis of data on pancreas cancer to illustrate the proposed model and
estimation procedure.

Keywords: Copulas; dependent censoring; identifiability, inference.

##  1 Introduction

A very common situation in survival analysis is that duration times are right
censored. This can happen for several reasons. In medical studies it often
happens that patients who are followed over time until they die of a certain
disease are still alive at the end of the study, they leave the study before
the end for various reasons, or they die from another cause. In most papers in
the literature it is assumed that the survival time $T$ is independent of the
censoring time $C$, where independent censoring should here be interpreted as
stochastic independence. However, there are many instances in which this
independence assumption is violated. Consider for instance the case where the
patient leaves the study for reasons related to his/her health, or where
he/she dies of another related disease. In these cases it is important to take
the dependence between $T$ and $C$ into account in the model. However, the
seminal paper by Tsiatis (1975) shows that the bivariate distribution of $T$
and $C$ is not identifiable in a completely nonparametric setting. Some
authors have therefore proposed parametric or semiparametric models for this
bivariate distribution that are identifiable. A popular model in that context
is the copula model, that allows to model the marginal laws of $T$ and $C$
separately from the relation between $T$ and $C$. The first paper in that
context was Zheng and Klein (1995). They worked with a completely known
copula, and proposed a nonparametric estimator of the marginal distribution of
$T$ and $C$, called a copula-graphic estimator, that generalizes the Kaplan
and Meier (1958) estimator to the case of dependent censoring. Later on,
Rivest and Wells (2001) focused on the case of Archimedean copulas and
obtained a closed-form expression for the copula-graphic estimator. However, a
major drawback of this estimator is that it relies on a completely known
copula, for instance a Frank copula with known association parameter. In
practice the association parameter is often not known, and can have a major
influence on the resulting estimator of the marginal distributions. Other
contributions in the same vein (with known copula) are given in Braekers and
Veraverbeke (2005), Huang and Zhang (2008), Chen (2010), de Unã-Álvarez and
Veraverbeke (2017), Sujica and Van Keilegom (2018) and Emura and Chen (2018),
among others, whereas other approaches not based on copulas have been studied
by Nádas (1971), Basu and Ghosh (1978), Basu (1988), Emoto and Matthews
(1990), Scharfstein and Robins (2002), Jackson et al. (2014), Collett (2015),
Hsu et al. (2015), and Deresa and Van Keilegom (2020a, b, 2021), among others.
The latter papers only consider rather specific (semi-)parametric model
assumptions, that can not be easily extended to other contexts.

In this paper we will show that the assumption of a completely known copula is
not a necessary condition to identify the joint distribution of $T$ and $C$.
We will show that if the marginal distributions of $T$ and $C$ and the copula
function are all modelled parametrically, then under certain conditions the
joint model is identifiable. In particular, the association parameter of the
copula function is identifiable, which is an important step forward in the use
of copulas in survival analysis. This might seem surprising, since under right
censoring one either observes $T$ or $C$, but in general not both. We will
show that the identification of the relation between $T$ and $C$ does not
follow using standard lines of reasoning commonly used in parametric models,
but instead it is a delicate exercise that makes use of the available
information in an optimal way. The price to pay for the identifiability of the
association parameter is that the marginals are no longer fully nonparametric.
We believe however that this is an acceptable price, given that one often has
no idea how to choose the association parameter in practice. We will develop
sufficient conditions on the families of marginal distributions and on the
family of copula functions under which the joint model is identifiable. These
sufficient conditions are satisfied for a wide range of common parametric
marginal distributions and copula functions, making the model very useful in
practice.

The paper is organized as follows. In the next section we introduce the copula
model, and some notations, definitions and formulas that are needed later in
the paper. Section 3 develops sufficient conditions under which the model is
identifiable. The estimation of the model is studied in Section 4. Sections 5
and 6 consider the results of extensive simulations and of the analysis of
data on pancreas cancer, respectively. Some ideas for further research are
discussed in Section 7, while the proofs are collected in the Appendix.

##  2 The model, and some notations and definitions

Let $T$ be a survival time, and $C$ be a censoring time. Due to random right
censoring we observe $Y=\min(T,C)$ and $\Delta=I(T\leq C)$. We allow for
stochastic dependence between $T$ and $C$ and will model this dependence with
a copula. Throughout the paper we assume that $T$ and $C$ are non-negative and
that the marginal distributions $F_{T}$ and $F_{C}$ of $T$ and $C$ are
continuous and belong to parametric families:

$\displaystyle
F_{T}\in\\{F_{T,\theta_{T}}:\theta_{T}\in\Theta_{T}\\},\quad\mbox{and}\quad
F_{C}\in\\{F_{C,\theta_{C}}:\theta_{C}\in\Theta_{C}\\},$ (2.1)

for certain parameter spaces $\Theta_{T}$ and $\Theta_{C}$. We denote their
densities by $f_{T}$ and $f_{C}$ (or $f_{T,\theta_{T}}$ and $f_{C,\theta_{C}}$
in the parametric families). We model the bivariate distribution $F_{T,C}$ of
$(T,C)$ by a copula based model. A copula is a bivariate distribution function
${\cal C}:[0,1]\times[0,1]\rightarrow[0,1]$ with uniform margins. Thanks to
Sklar (1959) and the continuity of the marginal distributions $F_{T}$ and
$F_{C}$, we know that there is a unique copula ${\cal C}$ for which

$\displaystyle F_{T,C}(t,c)={\cal C}(F_{T}(t),F_{C}(c))$ (2.2)

holds for any $t,c\geq 0$. We model the copula parametrically:

$\displaystyle{\cal C}\in\\{{\cal C}_{\theta}:\theta\in\Theta\\},$ (2.3)

for some parameter space $\Theta$. Note that instead of (2.2)-(2.3) we could
also assume that $P(T>t,C>c)=\tilde{\cal C}(1-F_{T}(t),1-F_{C}(c))$ with
$\tilde{\cal C}$ belonging to a certain parametric class of survival copulas.
There is no substantial difference between the two ways of modelling the
copula. We will focus on the usual copulas, but survival copulas can be used
as an alternative.

For our approach we need to express conditional distribution functions in
terms of their associated copulas and define

$h_{C|T,\theta}(v|u)=\frac{\partial}{\partial u}{\cal
C}_{\theta}(u,v)\quad\mbox{and}\quad
h_{T|C,\theta}(u|v)=\frac{\partial}{\partial v}{\cal C}_{\theta}(u,v).$

Further we let $F_{Y,\Delta}(y,\delta)=P(Y\leq y,\Delta=\delta)$ and
$f_{Y,\Delta}(y,\delta)=(d/dy)F_{Y,\Delta}(y,\delta)$ for $\delta=0,1$.
Finally, let $F_{Y}(y)=\sum_{\delta=0}^{1}F_{Y,\Delta}(y,\delta)$ and
$f_{Y}(y)=\sum_{\delta=0}^{1}f_{Y,\Delta}(y,\delta)$ be the distribution and
density of the observable random variable $Y$, respectively. Note that we can
express

$\displaystyle F_{T|C}(t|c)=h_{T|C}(F_{T}(t)|F_{C}(c)),\quad\quad\quad
F_{C|T}(c|t)=h_{C|T}(F_{C}(c)|F_{T}(t)),$ $\displaystyle
F_{Y}(y)=F_{T}(y)+F_{C}(y)-{\cal C}(F_{T}(y),F_{C}(y)),$ $\displaystyle
f_{Y,\Delta}(y,1)=f_{T}(y)[1-h_{C|T}(F_{C}(y)|F_{T}(y))],$ (2.4)
$\displaystyle f_{Y,\Delta}(y,0)=f_{C}(y)[1-h_{T|C}(F_{T}(y)|F_{C}(y))].$

For a derivation of the results given in (2.4) we refer to the Appendix. For
all functions introduced above, copula parameters $\theta$ and marginal
parameters $\theta_{T}$ and $\theta_{C}$ can be added to indicate the
parametric versions of these functions, for instance with
$\boldsymbol{\alpha}=(\theta,\theta_{T},\theta_{C})^{\top}$, we denote
$f_{Y,\Delta,\boldsymbol{\alpha}}(y,1)=f_{T,\theta_{T}}(y)[1-h_{C|T,\theta}(F_{C,\theta_{C}}(y)|F_{T,\theta_{T}}(y))]$.

##  3 Identifiability

The study of the identifiability of model (2.1)-(2.3) is the backbone of this
paper, and we investigate it in detail in this section. All proofs are given
in the Appendix.

We start with a general result that gives sufficient conditions under which
the model is identifiable. With identifiability we mean that the parameters
$(\theta,\theta_{T},\theta_{C})\in\Theta\times\Theta_{T}\times\Theta_{C}$
determine in a unique way the density of the observable random variables
$(Y,\Delta)$, i.e. if $f_{Y,\Delta,\boldsymbol{\alpha}_{1}}\equiv
f_{Y,\Delta,\boldsymbol{\alpha}_{2}}$, then
$\boldsymbol{\alpha}_{1}=\boldsymbol{\alpha}_{2}$, where
$\boldsymbol{\alpha}_{j}=(\theta_{j},\theta_{Tj},\theta_{Cj})^{\top}$,
$j=1,2$.

######  Theorem 3.1.

Suppose that

  * (C1)

For $a=0$ and $a=\infty$, and for $\theta_{T1},\theta_{T2}\in\Theta_{T}$ and
$\theta_{C1},\theta_{C2}\in\Theta_{C}$, we have that

$\lim_{t\rightarrow
a}\frac{f_{T,\theta_{T1}}(t)}{f_{T,\theta_{T2}}(t)}=1\Longleftrightarrow\theta_{T1}=\theta_{T2},$

and

$\lim_{t\rightarrow
a}\frac{f_{C,\theta_{C1}}(t)}{f_{C,\theta_{C2}}(t)}=1\Longleftrightarrow\theta_{C1}=\theta_{C2}.$

  * (C2)

The parameter space $\Theta\times\Theta_{T}\times\Theta_{C}$ is such that

$(C2a):\quad\min\Big{\\{}{\max}^{*}\lim_{t\rightarrow
0}h_{T|C,\theta}(u_{t}|v_{t}),{\max}^{*}\lim_{t\rightarrow\infty}h_{T|C,\theta}(u_{t}|v_{t})\Big{\\}}=0,$

and

$(C2b):\quad\min\Big{\\{}{\max}^{*}\lim_{t\rightarrow
0}h_{C|T,\theta}(v_{t}|u_{t}),{\max}^{*}\lim_{t\rightarrow\infty}h_{C|T,\theta}(v_{t}|u_{t})\Big{\\}}=0,$

where $u_{t}=F_{T,\theta_{T}}(t),v_{t}=F_{C,\theta_{C}}(t)$ and ${\max}^{*}$
is the maximum over all
$(\theta,\theta_{T},\theta_{C})\in\Theta\times\Theta_{T}\times\Theta_{C}$.

Then, model (2.1)-(2.3) is identified.

Condition (C1) is valid for a wide range of parametric families for the
marginal densities $f_{T}$ and $f_{C}$, as is shown in the next theorem. For
other families not mentioned below, condition (C1) can be easily checked as
well, but we restrict attention to the most important parametric families in
survival analysis.

######  Theorem 3.2.

Condition (C1) is satisfied for the families of log-normal, log-Student-t,
Weibull, and log-logistic densities.

We will now study two important classes of copulas in more detail, namely
Archimedean copulas and Gaussian copulas. We will verify in which cases
condition (C2) is valid for these classes. Archimedean copulas can be written
as

$\displaystyle{\cal C}(u,v)=\psi^{[-1]}(\psi(u)+\psi(v)),$ (3.1)

where $\psi$ is a generator, i.e. $\psi:[0,1]\rightarrow[0,\infty)$ is a
continuous, strictly decreasing and convex function such that $\psi(1)=0$.
Here, $\psi^{[-1]}$ is the pseudo-inverse of $\psi$, i.e.
$\psi^{[-1]}(t)=\psi^{-1}(t)$ if $0\leq t\leq\psi(0)$ and $\psi^{[-1]}(t)=0$
if $t\geq\psi(0)$. Important families of Archimedean copulas are the Frank
family, corresponding to

$\displaystyle\psi_{\theta}(u)=-\log\Big{(}\frac{e^{-\theta
u}-1}{e^{-\theta}-1}\Big{)},$

with $\theta\in\mathcal{R}\backslash\\{0\\}$, the Clayton family for which
$\psi_{\theta}(u)=\theta^{-1}(u^{-\theta}-1)$, with
$\theta\in[-1,\infty)\backslash\\{0\\}$, and the Gumbel family, defined by
$\psi_{\theta}(u)=(-\log(u))^{\theta}$, with $\theta\in[1,\infty)$.

Differentiation of (3.1) gives

$h_{T|C}(u|v)=\frac{\psi^{\prime}(v)}{\psi^{\prime}\big{(}\psi^{-1}(\psi(u)+\psi(v))\big{)}}$

and

$h_{C|T}(v|u)=\frac{\psi^{\prime}(u)}{\psi^{\prime}\big{(}\psi^{-1}(\psi(u)+\psi(v))\big{)}}$

for $0<u,v<1$, provided the derivatives and inverses in this formula exist.

The following lemma helps in assessing whether condition (C2a) is verified or
not. A similar result exists for condition (C2b).

######  Lemma 3.1.

Suppose the generator $\psi$ is differentiable on $(0,1)$. If
$\lim_{v\rightarrow 1}\psi^{\prime}(v)\in(-\infty,0)$, then
$\lim_{t\rightarrow\infty}h_{T|C,\theta}(F_{T,\theta_{T}}(t)|F_{C,\theta_{C}}(t))=1$.

Lemma 3.1 can be used to assess whether (C2) holds true. Below we do this for
some of the well known parametric Archimedean copula families. We will also
consider the Gaussian copula (which is not an Archimedean copula) defined by

${\cal C}_{\theta}(u,v)=\Phi_{\theta}(\Phi^{-1}(u),\Phi^{-1}(v)),$

where $\Phi$ is the cumulative distribution of a standard normal random
variable, and $\Phi_{\theta}$ is the cumulative distribution of a bivariate
standard normal random vector with correlation $\theta$.

######  Theorem 3.3.

[Frank, Gumbel and Gaussian copula] Condition (C2) is satisfied for

  * (i)

the Frank copula, independently of the marginal distributions and the size of
the parameter space $\Theta\times\Theta_{T}\times\Theta_{C}$.

  * (ii)

the Gumbel copula if $\lim_{t\rightarrow 0}\log F_{T,\theta_{T}}(t)/\log
F_{C,\theta_{C}}(t)\in(0,\infty)$ for all
$(\theta_{T},\theta_{C})\in\Theta_{T}\times\Theta_{C}$.

  * (iii)

for the Gaussian copula if

$\min\Big{\\{}{\max}^{*}\lim_{t\rightarrow
0}A_{\theta,F_{T,\theta_{T}},F_{C,\theta_{C}}}(t),{\max}^{*}\lim_{t\rightarrow\infty}A_{\theta,F_{T,\theta_{T}},F_{C,\theta_{C}}}(t)\Big{\\}}=-\infty$

and

$\min\Big{\\{}{\max}^{*}\lim_{t\rightarrow
0}A_{\theta,F_{C,\theta_{C}},F_{T,\theta_{T}}}(t),{\max}^{*}\lim_{t\rightarrow\infty}A_{\theta,F_{C,\theta_{C}},F_{T,\theta_{T}}}(t)\Big{\\}}=-\infty,$

where
$A_{\theta,F_{1},F_{2}}(t)=\Phi^{-1}(F_{1}(t))-\theta\Phi^{-1}(F_{2}(t))$, and
${\max}^{*}$ is the maximum over all
$(\theta,\theta_{T},\theta_{C})\in\Theta\times\Theta_{T}\times\Theta_{C}$.

Note that it can be easily seen that $\lim_{t\rightarrow 0}\log
F_{T,\theta_{T}}(t)/\log F_{C,\theta_{C}}(t)=\sigma_{C}^{2}/\sigma_{T}^{2}$ if
$\log T\sim N(\mu_{T},\sigma_{T}^{2})$ and $\log C\sim
N(\mu_{C},\sigma_{C}^{2})$, and that this limit equals $\rho_{T}/\rho_{C}$ if
$T\sim$ Weibull$(\lambda_{T},\rho_{T})$ and $C\sim$
Weibull$(\lambda_{C},\rho_{C})$ (we refer to (A.5) for the definition of the
Weibull parameters). Hence, the Gumbel copula can be used for these two
marginal specifications.

Regarding the Gaussian copula, note that for any distribution $F$, the
function $\Phi^{-1}(F(t))$ tends to $-\infty$ and $+\infty$ when $t$ tends to
zero and infinity, respectively. Hence, the limit of
$A_{\theta,F_{1},F_{2}}(t)$ will be determined by which of the functions
$F_{1}$ and $F_{2}$ dominates in the limit. As an example, consider the case
where $\log T\sim N(\mu_{T},\sigma_{T}^{2})$ and $\log C\sim
N(\mu_{C},\sigma_{C}^{2})$. Then,
$\Phi^{-1}(F_{T}(t))=(\log(t)-\mu_{T})/\sigma_{T}$, and hence

$A_{\theta,F_{T},F_{C}}(t)=\big{(}\frac{1}{\sigma_{T}}-\frac{\theta}{\sigma_{C}}\big{)}\log(t)-\big{(}\frac{\mu_{T}}{\sigma_{T}}-\frac{\theta\mu_{C}}{\sigma_{C}}\big{)},$

and this tends to $-\infty$ either when $t$ tends to zero or to infinity
depending on the sign of $\sigma_{C}-\theta\sigma_{T}$. This shows that for
log-normal margins and for a Gaussian copula, condition (C2) is satisfied
locally around the true parameter values. See also Nádas (1971), Basu and
Ghosh (1978) and Deresa and Van Keilegom (2020a), who considered the bivariate
normal case. For other margins the condition is harder to verify, since the
normal quantile function $\Phi^{-1}$ is difficult to handle. Numerical
calculation of the function $A_{\theta,F_{T},F_{C}}(t)$ when $T\sim$
Weibull$(\lambda_{T},\rho_{T})$ and $C\sim$ Weibull$(\lambda_{C},\rho_{C})$
supports however that the limit of this function equals $-\infty$ if
$\rho_{T}<\rho_{C}$ and $t$ tends to $\infty$, or if $\rho_{T}>\rho_{C}$ and
$t$ tends to 0.

Condition (C2) is however a sufficient condition for identification, but not
per se a necessary condition. It only uses the limits when $t$ goes to zero or
infinity, whereas the identification proof can also use all other time points
in between. We will illustrate this for the Clayton copula with $\theta>0$ (in
which case the Clayton copula is strict, i.e. $\psi(0)=\infty$). One can
easily show that for the Clayton copula we have that (see also Aas et al.
(2009))

$\displaystyle
h_{T|C,\theta}(F_{T}(t)|F_{C}(t))=\Big{[}1+\Big{(}\frac{F_{C}(t)}{F_{T}(t)}\Big{)}^{\theta}-F_{C}(t)^{\theta}\Big{]}^{-\frac{\theta+1}{\theta}},$

and

$\displaystyle
h_{C|T,\theta}(F_{C}(t)|F_{T}(t))=\Big{[}1+\Big{(}\frac{F_{T}(t)}{F_{C}(t)}\Big{)}^{\theta}-F_{T}(t)^{\theta}\Big{]}^{-\frac{\theta+1}{\theta}}.$
(3.2)

Hence, it is not possible that both functions tend to zero when $t$ tends to
zero. Since they both converge to 1 when $t$ tends to infinity, we see that
condition (C2) is not satisfied. However, from the fact that
$f_{Y,\Delta}(y,0)=f_{C}(y)\\{1-h_{T|C}(F_{T}(y)|F_{C}(y))\\}$ for all $y$ and
similarly for $f_{Y,\Delta}(y,1)=f_{T}(y)\\{1-h_{C|T}(F_{C}(y)|F_{T}(y))\\}$,
we can identify the parameters $\theta,\theta_{T}$ and $\theta_{C}$, as the
following theorem shows.

######  Theorem 3.4.

[Clayton copula]  Suppose that (C1) holds, that $\Theta_{T}\times\Theta_{C}$
is such that $\lim_{t\rightarrow 0}F_{T,\theta_{T}}(t)/F_{C,\theta_{C}}(t)$ is
either 0 or $+\infty$ for all $\theta_{T}\in\Theta_{T}$ and
$\theta_{C}\in\Theta_{C}$, and that the copula ${\cal C}_{\theta}$ is a
Clayton copula with $\theta>0$. Then, model (2.1)-(2.3) is identified.

Note that the condition that $\lim_{t\rightarrow
0}F_{T,\theta_{T}}(t)/F_{C,\theta_{C}}(t)=0$ or $+\infty$ is satisfied for
many parametric families. Consider e.g. the log-normal family for $T$ and $C$,
i.e. $\log T\sim N(\mu_{T},\sigma_{T}^{2})$ and $\log C\sim
N(\mu_{C},\sigma_{C}^{2})$. Then,

$\displaystyle\lim_{t\rightarrow
0}\frac{F_{T}(t)}{F_{C}(t)}=\lim_{t\rightarrow 0}\frac{f_{T}(t)}{f_{C}(t)}$
$\displaystyle=\frac{\sigma_{C}}{\sigma_{T}}\lim_{t\rightarrow
0}\exp\Big{\\{}-\frac{1}{2}\frac{(\log
t-\mu_{T})^{2}}{\sigma_{T}^{2}}+\frac{1}{2}\frac{(\log
t-\mu_{C})^{2}}{\sigma_{C}^{2}}\Big{\\}}$
$\displaystyle=\frac{\sigma_{C}}{\sigma_{T}}\lim_{t\rightarrow
0}\exp\Big{\\{}-\frac{1}{2}\Big{(}\frac{1}{\sigma_{T}^{2}}-\frac{1}{\sigma_{C}^{2}}\Big{)}(\log
t)^{2}+\Big{(}\frac{\mu_{T}}{\sigma_{T}^{2}}-\frac{\mu_{C}}{\sigma_{C}^{2}}\Big{)}\log
t-\frac{1}{2}\Big{(}\frac{\mu_{T}^{2}}{\sigma_{T}^{2}}-\frac{\mu_{C}^{2}}{\sigma_{C}^{2}}\Big{)}\Big{\\}}.$

This converges to 0 if $\sigma_{T}<\sigma_{C}$ or if $\sigma_{T}=\sigma_{C}$
and $\mu_{T}>\mu_{C}$. Similarly, the expression converges to infinity if
$\sigma_{T}>\sigma_{C}$ or if $\sigma_{T}=\sigma_{C}$ and $\mu_{T}<\mu_{C}$.
Hence, the model is identified locally around the true $(\mu_{T},\sigma_{T})$
and $(\mu_{C},\sigma_{C})$. Note that if $\mu_{T}=\mu_{C}$ and
$\sigma_{T}=\sigma_{C}$, it is easily seen that
$h_{T|C}(F_{T}(t)|F_{C}(t))=(2-F_{C}(t)^{\theta})^{-(\theta+1)/\theta}$ for
the Clayton copula. Hence, $\lim_{t\rightarrow
0}h_{T|C}(F_{T}(t)|F_{C}(t))=2^{-(\theta+1)/\theta}$. It follows that
$\lim_{t\rightarrow
0}f_{C,\mu_{C1},\sigma_{C1}}(t)/f_{C,\mu_{C2},\sigma_{C2}}(t)$
$=(1-2^{-(\theta_{2}+1)/\theta_{2}})/(1-2^{-(\theta_{1}+1)/\theta_{1}})$ for
two sets of parameters $(\theta_{1},\mu_{C1},\sigma_{C1})$ and
$(\theta_{2},\mu_{C2},\sigma_{C2})$. Since the limit $\lim_{t\rightarrow
0}f_{C,\mu_{C1},\sigma_{C1}}(t)/f_{C,\mu_{C2},\sigma_{C2}}(t)$ can only be
equal to 0, 1 or $\infty$ for the log-normal density, it follows that
$\theta_{1}=\theta_{2}$ and hence
$(\mu_{C1},\sigma_{C1})=(\mu_{C2},\sigma_{C2})$. This shows that the model is
also identifiable when $\mu_{T}=\mu_{C}$ and $\sigma_{T}=\sigma_{C}$.

Using similar (but easier) calculations we can show that if $T\sim$
Weibull$(\lambda_{T},\rho_{T})$ and $C\sim$ Weibull$(\lambda_{C},\rho_{C})$,
then $\lim_{t\rightarrow 0}\frac{F_{T}(t)}{F_{C}(t)}=0$ if
$\rho_{T}>\rho_{C}$, it equals $\infty$ if $\rho_{T}<\rho_{C}$ and it equals
$\lambda_{T}/\lambda_{C}$ if $\rho_{T}=\rho_{C}$ (we refer to (A.5) for the
definition of the Weibull parameters).

##  4 Estimation

We now consider parameter estimation of the joint parametric model for the
survival time $T$ and the censoring time $C$ specified in (2.1)-(2.3). For
this we assume that we have an i.i.d. sample ${\cal
D}=\\{(y_{i},\delta_{i}),i=1,\ldots,n\\}$ available. Then the joint log-
likelihood for the parameter vector
$\boldsymbol{\alpha}=(\theta,\theta_{T},\theta_{C})^{\top}$ is given by

$\displaystyle\ell(\boldsymbol{\alpha};{\cal D})$
$\displaystyle=\sum_{i=1}^{n}\log\left(f_{Y,\Delta,\boldsymbol{\alpha}}(y_{i},\delta_{i})\right)$
$\displaystyle=\sum_{i=1,\delta_{i}=1}^{n}\log\left(f_{T,\theta_{T}}(y_{i})\\{1-h_{C|T,\theta}(F_{C,\theta_{C}}(y_{i})|F_{T,\theta_{T}}(y_{i}))\\}\right)$
$\displaystyle\quad+\sum_{i=1,\delta_{i}=0}^{n}\log\left(f_{C,\theta_{C}}(y_{i})\\{1-h_{T|C,\theta}(F_{T,\theta_{T}}(y_{i})|F_{C,\theta_{C}}(y_{i}))\\}\right).$
(4.1)

We follow a maximum likelihood approach by maximizing the log-likelihood
specified in (4.1), i.e. we define parameter estimators by

$\boldsymbol{\hat{\alpha}}=(\hat{\theta},\hat{\theta}_{T},\hat{\theta}_{C})^{\top}=\mbox{argmax}_{\boldsymbol{\alpha}\in
A}\ell(\boldsymbol{\alpha};{\cal D}),$

where $A=\Theta\times\Theta_{T}\times\Theta_{C}$. If we use for instance log-
normal margins for $T$ and $C$ and single-parameter copula families we need to
optimize over five parameters. In the simulation study and the data
application we use unconstrained optimization. For standard one-parameter
copula families such as the Frank, Clayton, Gumbel and Gauss copula there
exist a one-to-one relationship between Kendall’s $\tau$ parameter and the
copula parameter $\theta$. If there is positive dependence we use a logit
transformation for $\tau$ and the Fisher’s z transform of $\tau$ otherwise. So
the Fisher’s z transform of $\tau$ can be used in the Frank and Gauss copula
case. This also facilitates a common interpretation of the dependence strength
measured in terms of Kendall’s $\tau$ over the different copula families.

In order to obtain the asymptotic normality of
$(\hat{\theta},\hat{\theta}_{T},\hat{\theta}_{C})$, we make use of White
(1982), who developed sufficient conditions under which a parameter estimator
defined as the maximizer of a certain criterion function is asymptotically
normal. The results allow for misspecification of the parametric model. Let
$\boldsymbol{\alpha}^{*}=(\theta^{*},\theta_{T}^{*},\theta_{C}^{*})^{\top}$ be
the parameter vector that minimizes the Kullback-Leibler information criterion
$E\big{[}\log f_{Y,\Delta}(Y,\Delta)-\log
f_{Y,\Delta,\boldsymbol{\alpha}}(Y,\Delta)\big{]}$, and let
$d=\mbox{dim}(\Theta)+\mbox{dim}(\Theta_{T})+\mbox{dim}(\Theta_{C})$. Then, we
have the following result.

######  Theorem 4.1.

(i) Under regularity conditions (A1) to (A3) in White (1982),

$(\hat{\theta},\hat{\theta}_{T},\hat{\theta}_{C})\mathrel{\begin{subarray}{c}P\\\
\longrightarrow\\\
n\rightarrow\infty\end{subarray}}(\theta^{*},\theta_{T}^{*},\theta_{C}^{*}).$

(ii) Under regularity conditions (A1) to (A6) in White (1982),

$n^{1/2}\Big{(}(\hat{\theta},\hat{\theta}_{T},\hat{\theta}_{C})-(\theta^{*},\theta_{T}^{*},\theta_{C}^{*})\Big{)}\mathrel{\begin{subarray}{c}d\\\
\longrightarrow\\\ n\rightarrow\infty\end{subarray}}N(0,C),$

where

$C=A(\boldsymbol{\alpha}^{*})^{-1}B(\boldsymbol{\alpha}^{*})A(\boldsymbol{\alpha}^{*})^{-1},$

with

$\displaystyle A(\boldsymbol{\alpha})$
$\displaystyle=\Big{(}E\Big{\\{}\frac{\partial^{2}}{\partial\alpha_{j}\partial\alpha_{k}}\log
f_{Y,\Delta,\boldsymbol{\alpha}}(Y,\Delta)\Big{\\}}\Big{)}_{j,k=1}^{d},$
$\displaystyle B(\boldsymbol{\alpha})$
$\displaystyle=\Big{(}E\Big{\\{}\frac{\partial}{\partial\alpha_{j}}\log
f_{Y,\Delta,\boldsymbol{\alpha}}(Y,\Delta)\frac{\partial}{\partial\alpha_{k}}\log
f_{Y,\Delta,\boldsymbol{\alpha}}(Y,\Delta)\Big{\\}}\Big{)}_{j,k=1}^{d}.$

Note that, if the model is correctly specified, the matrix $C$ appearing in
Theorem 4.1 equals $A(\boldsymbol{\alpha})^{-1}$, the inverse Fisher matrix.

##  5 Simulation study

We study the performance of the maximum likelihood estimators of $\theta$,
$\theta_{T}$ and $\theta_{C}$ for four parametric copula families, namely the
Frank, Clayton, Gumbel and Gauss copula, and for log-normal margins for $T$
and $C$, respectively. In particular the parameters of a log-normal random
variable $X$ are $\mu$ and $\sigma$ and they correspond to the mean of
$\log(X)$ and the standard error of $\log(X)$, respectively. Two different
simulation scenarios are investigated with parameter specification given in
Table 1.

Scenario | $\mu_{T}$ | $\sigma_{T}$ | $\mu_{C}$ | $\sigma_{C}$ | $\tau$
---|---|---|---|---|---
1 | 2.2 | 1.0 | 2.0 | .25 | .2
|  |  |  |  | .5
|  |  |  |  | .8
2 | 2.5 | 1.0 | 2.0 | .50 | .2
|  |  |  |  | .5
|  |  |  |  | .8
Table 1: Parameter specifications of the simulation scenarios with log-normal
margins with parameters $\mu_{T}$ ($\mu_{C}$) and standard deviation
$\sigma_{T}$ ($\sigma_{C}$) for $T$ and $C$, respectively, and dependence
strength measured by Kendall’s $\tau$.

A visualization of the resulting theoretical (sub)densities $f_{Y}$,
$f_{Y,\Delta}(\cdot,0)$ and $f_{Y,\Delta}(\cdot,1)$ from both scenarios based
on the expressions given in (2.4) are given in Figures 1 and 2, respectively.
Especially for Scenario 1 we see that the resulting marginal density of the
observable random variable $Y$ is non standard. This is expected since it is
the sum of two subdensities. Further we see that the strength of the
dependence between $T$ and $C$ influences the skewness of $Y$.

Figure 1: Theoretical (sub)density $f_{Y}$ (left column),
$f_{Y,\Delta}(\cdot,1)$ (middle column) and $f_{Y,\Delta}(\cdot,0)$ (right
column) using a Frank copula (top row), Clayton copula (second row), Gumbel
copula (third row) and Gauss copula (bottom row) for Scenario 1. Figure 2:
Theoretical (sub)density $f_{Y}$ (left column), $f_{Y,\Delta}(\cdot,1)$
(middle column) and $f_{Y,\Delta}(\cdot,0)$ (right column) using a Frank
copula (top row), Clayton copula (second row), Gumbel copula (third row) and
Gauss copula (bottom row) for Scenario 2.

The resulting uncensoring probabilities $P(\Delta=1)$ for all simulation
scenarios are given in Table 2. From this we see that under Scenario 1 the
uncensoring probabilities are quite constant over the studied dependence
strength and similar for all investigated copula families. For Scenario 2 we
observe lower uncensoring probabilities and they decrease as the dependence
strength increases.

Scenario 1: Copula | $\tau=.2$ | $\tau=.5$ | $\tau=.8$
---|---|---|---
Frank | 0.41 | 0.40 | 0.39
Clayton | 0.43 | 0.43 | 0.40
Gumbel | 0.41 | 0.40 | 0.39
Gauss | 0.42 | 0.41 | 0.40
Scenario 2: Copula | $\tau=.2$ | $\tau=.5$ | $\tau=.8$
Frank | 0.28 | 0.24 | 0.16
Clayton | 0.31 | 0.27 | 0.18
Gumbel | 0.30 | 0.24 | 0.18
Gauss | 0.30 | 0.25 | 0.18
Table 2: $P(\Delta=1)$ for all simulation scenarios.

We allow for three different sample sizes $n=200,500$ and $1000$ and repeated
each simulation setting 100 times. We report the average estimate
(average.estimate) together with the average bias (average.bias), the standard
deviation of the average estimate (sd.of.average.estimate) and the empirical
root mean squared error (RMSE) based on the 100 replications.

The results for Scenario 1 are shown in Tables 3, 4, 5 and 6 for the Frank,
Clayton, Gumbel and Gauss copula, respectively. We see satisfactory
performance of the estimation procedure. Only for the Frank copula we see a
considerable bias for the small dependence strength of $\tau=.2$. As we expect
the average RMSE goes down as the sample size increases. The marginal
parameters are well estimated in all cases.

$\tau=.2,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
---|---|---|---|---|---|---
average.estimate | 2.17 | -0.02 | 1.99 | -1.38 | -0.63 | 0.35
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.04 | 0.01
average.bias | -0.02 | -0.02 | -0.01 | 0.01 | 0.76 | 0.15
RMSE | 0.10 | 0.09 | 0.03 | 0.07 | 0.84 | 0.17
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.19 | -0.02 | 1.99 | -1.38 | -0.66 | 0.34
sd.of.average.estimate | 0.00 | 0.01 | 0.00 | 0.00 | 0.02 | 0.01
average.bias | -0.01 | -0.02 | -0.01 | 0.01 | 0.72 | 0.14
RMSE | 0.05 | 0.06 | 0.02 | 0.05 | 0.76 | 0.15
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.18 | -0.01 | 1.99 | -1.38 | -0.73 | 0.33
sd.of.average.estimate | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.00
average.bias | -0.01 | -0.01 | -0.01 | 0.01 | 0.66 | 0.13
RMSE | 0.04 | 0.04 | 0.01 | 0.03 | 0.68 | 0.13
$\tau=.5,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.01 | 2.00 | -1.38 | -0.02 | 0.49
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.05 | 0.01
average.bias | -0.00 | -0.01 | 0.00 | 0.00 | -0.02 | -0.00
RMSE | 0.10 | 0.10 | 0.03 | 0.09 | 0.51 | 0.12
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.01 | 2.00 | -1.38 | -0.02 | 0.49
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.03 | 0.01
average.bias | 0.00 | -0.01 | -0.00 | 0.00 | -0.02 | -0.00
RMSE | 0.06 | 0.07 | 0.02 | 0.05 | 0.34 | 0.08
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | 0.00 | 2.00 | -1.38 | -0.01 | 0.50
sd.of.average.estimate | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.00
average.bias | -0.00 | 0.00 | -0.00 | 0.00 | -0.01 | -0.00
RMSE | 0.05 | 0.04 | 0.01 | 0.04 | 0.22 | 0.05
$\tau=.8,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.21 | 0.00 | 2.00 | -1.40 | 1.39 | 0.80
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.03 | 0.00
average.bias | 0.00 | 0.00 | 0.00 | -0.01 | 0.01 | -0.00
RMSE | 0.10 | 0.08 | 0.02 | 0.09 | 0.29 | 0.04
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.19 | -0.01 | 2.00 | -1.38 | 1.39 | 0.80
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.02 | 0.00
average.bias | -0.01 | -0.01 | -0.00 | 0.00 | 0.00 | -0.00
RMSE | 0.06 | 0.06 | 0.01 | 0.05 | 0.18 | 0.03
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.00 | 2.00 | -1.38 | 1.40 | 0.80
sd.of.average.estimate | 0.00 | 0.00 | 0.00 | 0.00 | 0.01 | 0.00
average.bias | -0.00 | -0.00 | -0.00 | 0.00 | 0.01 | 0.00
RMSE | 0.04 | 0.04 | 0.01 | 0.04 | 0.14 | 0.02
Table 3: Simulation results for the case of a Frank copula under Scenario 1. $\tau=.2,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
---|---|---|---|---|---|---
average.estimate | 2.20 | -0.03 | 1.99 | -1.37 | -2.01 | 0.23
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.22 | 0.02
average.bias | 0.00 | -0.03 | -0.01 | 0.02 | -0.63 | 0.03
RMSE | 0.11 | 0.10 | 0.05 | 0.10 | 2.29 | 0.18
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.19 | -0.01 | 2.00 | -1.38 | -1.89 | 0.21
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.17 | 0.01
average.bias | -0.00 | -0.01 | -0.00 | 0.00 | -0.50 | 0.01
RMSE | 0.07 | 0.06 | 0.04 | 0.08 | 1.81 | 0.14
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | 0.00 | 2.00 | -1.38 | -1.57 | 0.21
sd.of.average.estimate | 0.00 | 0.00 | 0.00 | 0.00 | 0.11 | 0.01
average.bias | 0.00 | 0.00 | -0.00 | 0.00 | -0.18 | 0.01
RMSE | 0.05 | 0.04 | 0.03 | 0.05 | 1.07 | 0.10
$\tau=.5,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.00 | 2.00 | -1.40 | -0.14 | 0.49
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.10 | 0.01
average.bias | 0.00 | -0.00 | 0.00 | -0.01 | -0.14 | -0.01
RMSE | 0.10 | 0.09 | 0.04 | 0.11 | 1.04 | 0.15
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.21 | -0.00 | 2.00 | -1.39 | -0.02 | 0.49
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.04 | 0.01
average.bias | 0.01 | -0.00 | 0.00 | -0.01 | -0.02 | -0.00
RMSE | 0.07 | 0.06 | 0.02 | 0.06 | 0.35 | 0.08
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.01 | 2.00 | -1.38 | 0.02 | 0.51
sd.of.average.estimate | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.00
average.bias | 0.00 | -0.01 | -0.00 | 0.00 | 0.02 | 0.01
RMSE | 0.04 | 0.04 | 0.02 | 0.04 | 0.22 | 0.06
$\tau=.8,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.01 | 2.00 | -1.37 | 1.45 | 0.81
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.03 | 0.00
average.bias | -0.00 | -0.01 | 0.00 | 0.01 | 0.07 | 0.01
RMSE | 0.09 | 0.08 | 0.02 | 0.07 | 0.33 | 0.05
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.01 | 2.00 | -1.39 | 1.41 | 0.80
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.02 | 0.00
average.bias | 0.00 | -0.01 | 0.00 | 0.00 | 0.02 | 0.00
RMSE | 0.07 | 0.06 | 0.01 | 0.05 | 0.20 | 0.03
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.00 | 2.00 | -1.39 | 1.40 | 0.80
sd.of.average.estimate | 0.00 | 0.00 | 0.00 | 0.00 | 0.01 | 0.00
average.bias | -0.00 | -0.00 | 0.00 | -0.00 | 0.01 | 0.00
RMSE | 0.04 | 0.04 | 0.01 | 0.04 | 0.15 | 0.02
Table 4: Simulation results for the case of a Clayton copula under Scenario 1. $\tau=.2,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
---|---|---|---|---|---|---
average.estimate | 2.19 | -0.01 | 2.00 | -1.37 | -1.92 | 0.21
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.19 | 0.01
average.bias | -0.00 | -0.01 | 0.00 | 0.01 | -0.53 | 0.01
RMSE | 0.10 | 0.10 | 0.04 | 0.07 | 2.00 | 0.14
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.00 | 2.00 | -1.39 | -1.58 | 0.21
sd.of.average.estimate | 0.01 | 0.00 | 0.00 | 0.00 | 0.11 | 0.01
average.bias | -0.00 | -0.00 | 0.00 | 0.00 | -0.19 | 0.01
RMSE | 0.06 | 0.05 | 0.02 | 0.04 | 1.14 | 0.09
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.00 | 2.00 | -1.38 | -1.39 | 0.21
sd.of.average.estimate | 0.00 | 0.00 | 0.00 | 0.00 | 0.05 | 0.01
average.bias | 0.00 | -0.00 | -0.00 | 0.00 | -0.00 | 0.01
RMSE | 0.05 | 0.05 | 0.02 | 0.03 | 0.50 | 0.07
$\tau=.5,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.01 | 2.00 | -1.38 | 0.03 | 0.51
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.06 | 0.01
average.bias | -0.00 | -0.01 | -0.00 | 0.01 | 0.03 | 0.01
RMSE | 0.11 | 0.09 | 0.03 | 0.08 | 0.57 | 0.13
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.01 | 2.00 | -1.39 | -0.03 | 0.49
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.04 | 0.01
average.bias | 0.00 | -0.01 | 0.00 | -0.01 | -0.03 | -0.01
RMSE | 0.07 | 0.07 | 0.02 | 0.06 | 0.37 | 0.09
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.21 | 0.00 | 2.00 | -1.38 | 0.01 | 0.50
sd.of.average.estimate | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.01
average.bias | 0.01 | 0.00 | 0.00 | 0.00 | 0.01 | 0.00
RMSE | 0.05 | 0.04 | 0.01 | 0.04 | 0.22 | 0.06
$\tau=.8,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.19 | -0.02 | 2.00 | -1.40 | 1.42 | 0.80
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.03 | 0.00
average.bias | -0.01 | -0.02 | -0.00 | -0.01 | 0.04 | 0.00
RMSE | 0.10 | 0.10 | 0.02 | 0.08 | 0.32 | 0.05
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | 0.00 | 2.00 | -1.39 | 1.40 | 0.80
sd.of.average.estimate | 0.01 | 0.00 | 0.00 | 0.00 | 0.02 | 0.00
average.bias | -0.00 | 0.00 | -0.00 | -0.01 | 0.01 | 0.00
RMSE | 0.06 | 0.05 | 0.02 | 0.05 | 0.19 | 0.03
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | -0.00 | 2.00 | -1.39 | 1.39 | 0.80
sd.of.average.estimate | 0.00 | 0.00 | 0.00 | 0.00 | 0.01 | 0.00
average.bias | 0.00 | -0.00 | 0.00 | 0.00 | 0.00 | -0.00
RMSE | 0.04 | 0.04 | 0.01 | 0.04 | 0.13 | 0.02
Table 5: Simulation results for the case of a Gumbel copula under Scenario 1. $\tau=.2,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
---|---|---|---|---|---|---
average.estimate | 2.46 | -0.05 | 1.98 | -0.68 | -2.08 | 0.24
sd.of.average.estimate | 0.03 | 0.02 | 0.01 | 0.01 | 0.23 | 0.02
average.bias | -0.04 | -0.05 | -0.02 | 0.02 | -0.69 | 0.04
RMSE | 0.31 | 0.17 | 0.13 | 0.11 | 2.37 | 0.20
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.19 | -0.01 | 1.99 | -1.36 | -2.08 | 0.19
sd.of.average.estimate | 0.02 | 0.01 | 0.01 | 0.02 | 0.18 | 0.01
average.bias | -0.01 | -0.01 | -0.01 | 0.03 | -0.70 | -0.01
RMSE | 0.22 | 0.08 | 0.11 | 0.17 | 1.94 | 0.12
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.18 | -0.00 | 1.99 | -1.36 | -1.56 | 0.21
sd.of.average.estimate | 0.02 | 0.01 | 0.01 | 0.02 | 0.11 | 0.01
average.bias | -0.02 | -0.00 | -0.01 | 0.02 | -0.17 | 0.01
RMSE | 0.21 | 0.06 | 0.11 | 0.17 | 1.12 | 0.10
$\tau=.5,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.19 | -0.02 | 1.99 | -1.38 | -0.19 | 0.48
sd.of.average.estimate | 0.02 | 0.01 | 0.01 | 0.02 | 0.11 | 0.01
average.bias | -0.01 | -0.02 | -0.01 | 0.01 | -0.19 | -0.02
RMSE | 0.24 | 0.11 | 0.11 | 0.19 | 1.11 | 0.15
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.18 | -0.01 | 1.99 | -1.38 | -0.14 | 0.48
sd.of.average.estimate | 0.02 | 0.01 | 0.01 | 0.02 | 0.09 | 0.01
average.bias | -0.02 | -0.01 | -0.01 | 0.01 | -0.14 | -0.02
RMSE | 0.21 | 0.07 | 0.11 | 0.18 | 0.94 | 0.10
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.18 | -0.00 | 1.99 | -1.37 | -0.09 | 0.49
sd.of.average.estimate | 0.02 | 0.01 | 0.01 | 0.02 | 0.09 | 0.01
average.bias | -0.02 | -0.00 | -0.01 | 0.02 | -0.09 | -0.01
RMSE | 0.21 | 0.06 | 0.11 | 0.17 | 0.89 | 0.08
$\tau=.8,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.18 | -0.01 | 1.99 | -1.38 | 1.30 | 0.79
sd.of.average.estimate | 0.02 | 0.01 | 0.01 | 0.02 | 0.10 | 0.01
average.bias | -0.02 | -0.01 | -0.01 | 0.00 | -0.09 | -0.01
RMSE | 0.24 | 0.11 | 0.11 | 0.18 | 1.04 | 0.09
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.17 | -0.01 | 1.99 | -1.38 | 1.28 | 0.79
sd.of.average.estimate | 0.02 | 0.01 | 0.01 | 0.02 | 0.10 | 0.01
average.bias | -0.03 | -0.01 | -0.01 | 0.01 | -0.10 | -0.01
RMSE | 0.22 | 0.08 | 0.11 | 0.17 | 1.01 | 0.08
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.18 | -0.00 | 1.99 | -1.38 | 1.29 | 0.79
sd.of.average.estimate | 0.02 | 0.01 | 0.01 | 0.02 | 0.10 | 0.01
average.bias | -0.02 | -0.00 | -0.01 | 0.01 | -0.10 | -0.01
RMSE | 0.21 | 0.06 | 0.11 | 0.17 | 1.00 | 0.08
Table 6: Simulation results for the case of a Gauss copula under Scenario 1.

The results from Scenario 2 are shown in the supplement as Tables 10, 11, 12
and 13 for the Frank, Clayton, Gumbel and Gauss copula, respectively. Compared
to Scenario 1 the positive bias for the Frank copula is even higher under
Scenario 2 for low dependence. In the other copula cases we also see that
there is a slightly higher bias in the low dependence case compared to
Scenario 1. However for stronger dependences the performance is satisfactory
even for sample sizes as low as 200.

##  6 Illustration

We now illustrate the dependent censoring model specified in (2.1)-(2.3) for
pancreas cancer from the Surveillance, Epidemiology, and End Results (SEER)
database (see https://seer.cancer.gov/data-software/). More specifically we
use the monthly survival times of blacks with localized pancreas cancer
between 2000 and 2015. We exclude all patients with 0 survival times. This
leaves us with data on 1549 patients with localized pancreas cancer, of which
777 died of pancreas cancer and 772 patients were still alive or died of other
causes. In our analysis we view the 772 patients as censored observations.
When patients are censored because they died from another disease, their
censoring time is likely related to their (unobserved) survival time, since
many diseases share common risk factors (like stress, eating habits, physical
condition, etc.).

We now fit model (2.1)-(2.3) for the independence, Frank, Clayton, Gumbel and
Gauss copula with log-normal or Weibull margins to this data set. The
resulting parameter estimates are given in Tables 7 and 8, respectively. For
the Weibull margins we have the shape parameter $a>0$ and scale parameter
$b>0$ to estimate, where we use now the parametrization of the Weibull density
used in R (note that the parameters $\lambda$ and $\rho$ used in (A.5) can be
expressed in terms of $a$ and $b$ via the formulas $\lambda=b^{-a}$ and
$\rho=a$). We estimate the unconstrained parameters $\log(a)$ and $\log(b)$.
Similarly the unconstrained parameters $\mu$ and $\log(\sigma)$ for the log-
normal margins are estimated. All parameters are estimated using maximum
likelihood. Standard error estimates are based on using 100 bootstrap samples
with replacement. In view of the asymptotic theory given in Theorem 4.1 we
expect the bootstrap procedure to give valid standard error estimates.

We now compare the different marginal and copula specifications. Since the
model complexities of these single parametric copula models are the same,
considering the maximized log-likelihood is sufficient. From the estimation
results we see that the fitted log likelihoods for the copula models with log-
normal margins are higher than the ones with Weibull margins. The increase
ranges from 13.5 for the Gaussian copula model to 100.8 for the Frank copula
models. These models are non-nested, but all have the same number of
parameters. Therefore there is a preference for the models with log-normal
margins.

With regard to the copula specification the models with the Gumbel copula have
the highest log likelihood. Comparing the dependent censoring models to the
independent censoring one, we see a drop of more than 100 for both marginal
models for the Gumbel specification. Therefore a likelihood ratio test is very
much in favor of the dependent censoring model using the Gumbel copula. This
indicates that upper tail dependence is present between the survival times and
the censoring times. This means that long survival or censoring times occur
much more often together than short times. This is also supported by the worse
performance of the Clayton copula based model, which allows for lower tail
dependence and no upper tail dependence. The symmetric but no tail dependent
Gaussian and Frank copula models perform better than the Clayton copula but
worse than the Gumbel copula. The worse performance of the Frank over the
Gauss copula can be attributed to the fact that the Frank copula has even
lighter joint tails than the Gaussian copula.

The fitted marginal probabilities $P(\Delta=1)=P(T\leq C)$ are given in Table
9. From this table we see that all model specifications give fitted marginal
probabilities that are close to the empirical values.

To further assess the fit we look at the fitted densities of $Y$ given
$\Delta=0$ and $Y$ given $\Delta=1$ for both marginal specifications. These
fitted densities are given in Figure 3. We see that the fitted distribution of
the censored observations have a heavier tail compared to the uncensored
observations for both marginal specifications. We further note that the Gumbel
copula based model is a bit closer to the empirical density for both censored
and uncensored observations.

Finally we give the fitted density of the survival times and censoring times
under all copula and marginal specifications in Figure 4. The effect of the
copula family is visible for both the survival and censoring times. In
particular the copula choice has more influence for the log-normal marginal
specification. Further the Gumbel and Gauss copula give very similar fits
regardless of the studied marginal specification.

Copula | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\theta$ | $\tau$ | log-lik
---|---|---|---|---|---|---|---|---
indep | 3.25 | 0.66 | 3.34 | 0.27 | - | - | - | -7312.37
(stderr) | (0.06) | (0.02) | (0.04) | (0.02) | - | - | - | (50.04)
Frank | 2.66 | 0.41 | 2.68 | 0.17 | 1.36 | 17.81 | 0.80 | -7231.41
(stderr) | (0.04) | (0.02) | (0.04) | (0.02) | (0.17) | (2.90) | (0.03) | (57.56)
Clayton | 3.16 | 0.63 | 3.23 | 0.27 | -2.00 | 0.27 | 0.12 | -7309.74
(stderr) | (0.06) | (0.02) | (0.03) | (0.02) | (0.06) | (0.02) | (0.01) | (43.23)
Gumbel | 2.39 | 0.29 | 2.40 | 0.26 | 3.41 | 31.14 | 0.97 | -7194.61
(stderr) | (0.03) | (0.01) | (0.04) | (0.01) | (0.28) | (9.30) | (0.01) | (48.82)
Gauss | 2.37 | 0.29 | 2.37 | 0.29 | 4.76 | 1.00 | 0.99 | -7237.34
(stderr) | (0.04) | (0.01) | (0.04) | (0.01) | (0.07) | (0.00) | (0.00) | (55.29)
Table 7: Estimated marginal and copula parameters together with the fitted log-likelihood for the SEER data using model (2.1)-(2.3) with the independence, Frank, Clayton, Gumbel and Gauss copula and log-normal margins, respectively (standard errors are based on 100 bootstrap replications). Copula | $\log(a_{T})$ | $\log(b_{T})$ | $\log(a_{C})$ | $\log(b_{C})$ | $\mbox{logit}(\tau)$ | $\theta$ | $\tau$ | log-lik
---|---|---|---|---|---|---|---|---
indep | -0.45 | 3.98 | 0.08 | 3.84 | - | - | - | -7343.55
(stderr) | (0.02) | (0.06) | (0.02) | (0.03) | - | - | - | (37.38)
Frank | -0.40 | 3.77 | 0.08 | 3.63 | -0.83 | 3.00 | 0.30 | -7332.18
(stderr) | (0.02) | (0.10) | (0.04) | (0.08) | (0.51) | (5.35) | (0.10) | (47.49)
Clayton | -0.45 | 3.98 | 0.08 | 3.84 | -5.92 | 0.00 | 0.00 | -7343.66
(stderr) | (0.02) | (0.07) | (0.02) | (0.03) | (0.86) | (0.00) | (0.00) | (47.66)
Gumbel | -0.21 | 3.05 | -0.19 | 3.04 | 3.89 | 50.09 | 0.98 | -7241.48
(stderr) | (0.01) | (0.03) | (0.01) | (0.03) | (0.04) | (1.89) | (0.00) | (47.25)
Gauss | -0.20 | 3.03 | -0.19 | 3.03 | 5.05 | 1.00 | 0.99 | -7250.89
(stderr) | (0.01) | (0.03) | (0.02) | (0.03) | (0.33) | (0.00) | (0.00) | (42.44)
Table 8: Estimated marginal and copula parameters together with the fitted log-likelihood for the SEER data using model (2.1)-(2.3) with the independence, Frank, Clayton, Gumbel and Gauss copula and Weibull margins, respectively (standard errors are based on 100 bootstrap replications). Margin | Empirical | Independence | Frank | Clayton | Gumbel | Gauss
---|---|---|---|---|---|---
log-normal | 0.5016 | 0.5160 | 0.5233 | 0.5180 | 0.5108 | 0.4973
Weibull | 0.5016 | 0.5084 | 0.5022 | 0.5083 | 0.4935 | 0.4932
Table 9: Empirical and fitted probability $P(T\leq C)$ for the SEER data with
the independence, Frank, Clayton, Gumbel and Gauss copula with log-normal or
Weibull margins, respectively.

Figure 3: Fitted density of $Y$ given $\Delta=0$ (left panel) and of $Y$ given
$\Delta=1$ (right panel), respectively, for the SEER data with the
independence, Frank, Clayton, Gumbel and Gauss copula and with log-normal
margins (top row) and Weibull margins (bottom row) together with a histogram
and empirical density of the observed data.

Figure 4: Fitted density of the survival times (top row) and of the censoring
times (bottom row) with log-normal margins (left panel) and Weibull margins
(right panel), respectively, for the SEER data with the independence, Frank,
Clayton, Gumbel and Gauss copula.

##  7 Discussion and future research

In this paper we modelled the bivariate distribution of a survival and a
censoring time by means of a parametric copula model and parametric margins.
We developed sufficient conditions under which this model is identifiable, and
showed that these conditions are satisfied for a wide range of model
specifications. We also studied the estimation of the model and the finite
sample performance of the proposed estimators.

The paper is the first to propose a copula model without assuming that the
association parameter of the copula is fully known. This is the main advantage
of this paper with respect to existing copula models. In order to focus on
this major gain, we kept the setting of this paper rather simple. The model
can however be extended in many ways. First, the model can be made more
flexible by considering semiparametric or nonparametric margins. The question
is then however whether the identifiability of the model can be guaranteed.
Second, covariates can be added to the model. The most simple case is that of
a fully parametric regression model for the survival and censoring time, but
other models like semiparametric Cox proportional hazards or accelerated
failure time models will be worth studying as well. Third, more general
survival models, like competing risks, cure models and truncation in
combination with dependent censoring, will also be very promising. Several of
the above extensions are currently under preparation. Finally, we like to
mention that we focused here on regular copulas, but we could as well consider
survival copulas, or any other rotation of the copula family. This will not
change the essential ideas of the model and its properties.

## Acknowledgments

The authors like to thank Johanna Neslehova for very helpful discussions.

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## Appendix

Derivation of Equations (2.4). Differentiating (2.2) we can express the joint
density of $(T,C)$ as

$\displaystyle f_{T,C}(t,c)=c(F_{T}(t),F_{C}(c))f_{T}(t)f_{C}(c),$ (A.1)

where $c$ denotes the copula density. We are interested in determining the
conditional distribution of $T$ given $C$ and vice versa. From (A.1) it is
straightforward to see that the conditional densities are

$\displaystyle f_{T|C}(t|c)=c(F_{T}(t),F_{C}(c))f_{T}(t)$ (A.2) $\displaystyle
f_{C|T}(c|T)=c(F_{T}(t),F_{C}(c))f_{C}(c)$ (A.3)

and the conditional distribution function of $T$ given $C=c$ can be derived as

$\displaystyle F_{T|C}(t|c)$ $\displaystyle=$
$\displaystyle\int_{0}^{t}c(F_{T}(t^{*}),F_{C}(c))f_{T}(t^{*})dt^{*}$
$\displaystyle=$ $\displaystyle\int_{0}^{t}\frac{\partial^{2}}{\partial
u\partial v}{\cal
C}(u,v)|_{u=F_{T}(t^{*}),v=F_{C}(c)}\frac{dF_{T}(t)}{dt}|_{t=t^{*}}dt^{*}$
$\displaystyle=$ $\displaystyle\frac{\partial}{\partial v}{\cal
C}(u,v)|_{u=F_{T}(t),v=F_{C}(c)}=h_{T|C}(F_{T}(t)|F_{C}(c)).$

Similarly we have $F_{C|T}(c|t)=h_{C|T}(F_{C}(c)|F_{T}(t))$. We now derive the
marginal distribution of $Y=\min(T,C)$:

$\displaystyle F_{Y}(y)$ $\displaystyle=$ $\displaystyle
1-P(Y>y)=1-P(T>y,C>y)$ $\displaystyle=$ $\displaystyle
1-\big{\\{}1-F_{C}(y)-F_{T}(y)+{\cal C}(F_{T}(y),F_{C}(y))\big{\\}}$
$\displaystyle=$ $\displaystyle F_{C}(y)+F_{T}(y)-{\cal
C}(F_{T}(y),F_{C}(y)).$

Finally we derive the expressions for the joint mixed density $f_{Y,\Delta}$
by noting that

$\displaystyle F_{Y,\Delta}(y,1)$ $\displaystyle=P(T\leq y,T\leq
C)=\int_{0}^{y}P(C\geq t|T=t)f_{T}(t)\,dt$
$\displaystyle=\int_{0}^{y}\\{1-h_{C|T}(F_{C}(t)|F_{T}(t))f_{T}(t)\,dt,$

and hence $f_{Y,\Delta}(y,1)=\\{1-h_{C|T}(F_{C}(y)|F_{T}(y))\\}f_{T}(y)$.
Similarly, we get
$f_{Y,\Delta}(y,0)=\\{1-h_{T|C}(F_{T}(y)|F_{C}(y))\\}f_{C}(y)$. $\Box$

Proof of Theorem 3.1. Recall from (2.4) that

$\displaystyle P(Y\leq t,\Delta=1)$ $\displaystyle=P(T\leq t,T\leq
C)=\int_{0}^{t}P(C\geq y|T=y)f_{T}(y)\,dy.$

Hence,

$f_{Y,\Delta}(t,1)=(1-F_{C|T}(t|t))f_{T}(t)=\big{(}1-h_{C|T}(F_{C}(t)|F_{T}(t))\big{)}f_{T}(t).$

From condition (C2b) we know that $\lim_{t\rightarrow
a}h_{C|T}(F_{C}(t)|F_{T}(t))=0$ for $a=0$ or $a=\infty$. Hence,

$\lim_{t\rightarrow a}f_{Y,\Delta}(t,1)=\lim_{t\rightarrow a}f_{T}(t).$

Suppose now that
$f_{Y,\Delta,\boldsymbol{\alpha}_{1}}(t,1)=f_{Y,\Delta,\boldsymbol{\alpha}_{2}}(t,1)$
for all $t$, where
$\boldsymbol{\alpha}_{j}=(\theta_{j},\theta_{Tj},\theta_{Cj})^{\top}$,
$j=1,2$. Then,

$1=\lim_{t\rightarrow
a}\frac{f_{Y,\Delta,\boldsymbol{\alpha}_{1}}(t,1)}{f_{Y,\Delta,\boldsymbol{\alpha}_{2}}(t,1)}=\lim_{t\rightarrow
a}\frac{f_{T,\theta_{T1}}(t)}{f_{T,\theta_{T2}}(t)}.$

It follows from condition (C1) that $\theta_{T1}=\theta_{T2}$. In a similar
way we can show that $\theta_{C1}=\theta_{C2}$. Finally, in order to show that
$\theta_{1}=\theta_{2}$, note that
$F_{Y,\boldsymbol{\alpha}_{j}}(t)=F_{T,\theta_{T1}}(t)+F_{C,\theta_{C1}}(t)-{\cal
C}_{\theta_{j}}(F_{T,\theta_{T1}}(t),F_{C,\theta_{C1}}(t))$ and that
$F_{Y,\boldsymbol{\alpha}_{1}}(t)=F_{Y,\boldsymbol{\alpha}_{2}}(t)$ for all
$t$. Hence, since the copula is unique, it follows that
$\theta_{1}=\theta_{2}$. $\Box$

Proof of Theorem 3.2. Consider first the log-normal density for $T$ depending
on $\boldsymbol{\theta}_{T}=(\mu,\sigma)$. Then,

$\lim_{t\rightarrow
0}\frac{f_{T,\mu_{1},\sigma_{1}}(t)}{f_{T,\mu_{2},\sigma_{2}}(t)}=\lim_{t\rightarrow
0}\frac{\frac{1}{t\sigma_{1}}\phi\Big{(}\frac{\log
t-\mu_{1}}{\sigma_{1}}\Big{)}}{\frac{1}{t\sigma_{2}}\phi\Big{(}\frac{\log
t-\mu_{2}}{\sigma_{2}}\Big{)}}=\lim_{t^{\prime}\rightarrow-\infty}\frac{\sigma_{2}}{\sigma_{1}}\cdot\frac{\phi\Big{(}\frac{t^{\prime}-\mu_{1}}{\sigma_{1}}\Big{)}}{\phi\Big{(}\frac{t^{\prime}-\mu_{2}}{\sigma_{2}}\Big{)}},$

and it is easily seen that this can only be equal to 1 when $\mu_{1}=\mu_{2}$
and $\sigma_{1}=\sigma_{2}$. The same is true when taking the limit for $t$
going to $\infty$.

Similarly, for the log-Student-t, we have

$\displaystyle\lim_{t\rightarrow
0,\infty}\frac{f_{T,\nu_{1},\mu_{1},\sigma_{1}}(t)}{f_{T,\nu_{2},\mu_{2},\sigma_{2}}(t)}=\frac{c_{\nu_{1},\sigma_{1}}}{c_{\nu_{2},\sigma_{2}}}\lim_{t\rightarrow
0,\infty}\frac{\big{(}1+\frac{1}{\nu_{1}}(\frac{\log
t-\mu_{1}}{\sigma_{1}})^{2}\big{)}^{-\frac{\nu_{1}+1}{2}}}{\big{(}1+\frac{1}{\nu_{2}}(\frac{\log
t-\mu_{2}}{\sigma_{2}})^{2}\big{)}^{-\frac{\nu_{2}+1}{2}}},$ (A.4)

where
$c_{\nu,\sigma}=\Gamma(\frac{\nu+1}{2})/\big{[}\sigma\sqrt{\nu\pi}\Gamma(\frac{\nu}{2})]$
and $\Gamma$ is the gamma-function. It is easily seen that the limit in (A.4)
is equal to 1 if and only if
$(\nu_{1},\mu_{1},\sigma_{1})=(\nu_{2},\mu_{2},\sigma_{2})$.

This is also the case for the Weibull density, since

$\displaystyle\lim_{t\rightarrow
0,\infty}\frac{f_{T,\lambda_{1},\rho_{1}}(t)}{f_{T,\lambda_{2},\rho_{2}}(t)}=\frac{\lambda_{1}\rho_{1}}{\lambda_{2}\rho_{2}}\lim_{t\rightarrow
0,\infty}\frac{t^{\rho_{1}-1}\exp(-\lambda_{1}t^{\rho_{1}})}{t^{\rho_{2}-1}\exp(-\lambda_{2}t^{\rho_{2}})},$
(A.5)

and this equals one only if $\rho_{1}=\rho_{2}$ and $\lambda_{1}=\lambda_{2}$.

Finally, for the log-logistic density, the limit equals

$\lim_{t\rightarrow
0,\infty}\frac{f_{T,\lambda_{1},\kappa_{1}}(t)}{f_{T,\lambda_{2},\kappa_{2}}(t)}=\frac{\kappa_{1}\lambda_{1}^{\kappa_{1}}}{\kappa_{2}\lambda_{2}^{\kappa_{2}}}\lim_{t\rightarrow
0,\infty}\frac{t^{\kappa_{1}-1}[1+(\lambda_{2}t)^{\kappa_{2}}]^{2}}{t^{\kappa_{2}-1}[1+(\lambda_{1}t)^{\kappa_{1}}]^{2}},$

and again this can only be equal to one when $\kappa_{1}=\kappa_{2}$ and
$\lambda_{1}=\lambda_{2}$. Hence, condition (C1) is satisfied for each of
these densities. $\Box$

Proof of Lemma 3.1. We will use the abbreviated notation
$u_{t}=F_{T,\theta_{T}}(t)$ and $v_{t}=F_{C,\theta_{C}}(t)$. Note that
$\psi(1)=0$, $\lim_{t\rightarrow 0}\psi^{-1}(t)=1$ and $\lim_{u\rightarrow
1}\psi^{\prime}(u)=c\in(-\infty,0)$. Hence,

$\displaystyle\hskip
76.82234pt\lim_{t\rightarrow\infty}h_{T|C,\theta}(u_{t}|v_{t})$
$\displaystyle=\lim_{t\rightarrow\infty}\frac{\psi^{\prime}(v_{t})}{\psi^{\prime}\big{(}\psi^{-1}(\psi(u_{t})+\psi(v_{t}))\big{)}}=\frac{c}{c}=1.\hskip
59.75095pt\Box$

Proof of Theorem 3.3. We start with the Frank copula,. Straightforward
calculations show that $\lim_{u\rightarrow 1}\psi^{\prime}(u)=\theta
e^{-\theta}/(e^{-\theta}-1)<0$ for $\theta\neq 0$, and hence we know from
Lemma 3.1 that $\lim_{t\rightarrow\infty}h_{T|C}(F_{T}(t)|F_{C}(t))=1$. Hence,
condition (C2a) can only be satisfied if $h_{T|C}(F_{T}(t)|F_{C}(t))$
converges to zero for $t\rightarrow 0$. Some straightforward but tedious
calculations show that

$\displaystyle\lim_{t\rightarrow 0}h_{T|C}(F_{T}(t)|F_{C}(t))$
$\displaystyle=\lim_{t\rightarrow
0}\frac{\psi^{\prime}(F_{C}(t))}{\psi^{\prime}\big{(}\psi^{-1}\big{(}\psi(F_{T}(t))+\psi(F_{C}(t))\big{)}\big{)}}$
$\displaystyle=\lim_{t\rightarrow 0}\frac{e^{-\theta F_{C}(t)}(e^{-\theta
F_{T}(t)}-1)}{(e^{-\theta F_{C}(t)}-1)(e^{-\theta
F_{T}(t)}-1)+e^{-\theta}-1}=0$

for $\theta\neq 0$. Hence, condition (C2a) is satisfied. Similarly it can be
shown that condition (C2b) is satisfied.

For the Gumbel family we have that $\lim_{u\rightarrow 1}\psi^{\prime}(u)=0$,
and hence Lemma 3.1 is not applicable. Hence, we calculate both the limit for
$t$ going to 0 and infinity (see Aas et al. (2009) for the formula of
$h_{T|C}(F_{T}(t)|F_{C}(t))$ for the Gumbel family):

$\displaystyle\lim_{t\rightarrow 0,\infty}h_{T|C}(F_{T}(t)|F_{C}(t))$
$\displaystyle=\lim_{t\rightarrow 0,\infty}\Big{\\{}1+(-\log
F_{C}(t))^{-\theta}(-\log F_{T}(t))^{\theta}\Big{\\}}^{-1+1/\theta}$
$\displaystyle\hskip 14.22636pt\times\lim_{t\rightarrow
0,\infty}\exp\Big{\\{}-\big{[}(-\log F_{T}(t))^{\theta}+(-\log
F_{C}(t))^{\theta}\big{]}^{1/\theta}-\log F_{C}(t)\Big{\\}}.$

Under the assumption that $\log F_{T}(t)/\log F_{C}(t)\rightarrow c$ for some
$0<c<\infty$ when $t$ tends to zero, the exponential factor above tends to 0
when $t$ tends to 0, and tends to 1 when $t$ tends to infinity, whereas the
factor in front of this exponential factor converges to some constant in the
interval $[0,1]$, depending on the limit of $\log F_{T}(t)/\log F_{C}(t)$ (for
$t$ tending to 0 or infinity). This shows that the product of the two limits
equals 0 when $t$ tends to 0, whereas when $t$ tends to infinity the limit can
be zero but it can also be strictly positive depending on the limit of $\log
F_{T}(t)/\log F_{C}(t)$ for $t$ growing to infinity.

Finally, we consider the Gaussian copula. Note that

$\displaystyle P\big{(}\Phi^{-1}(F_{T}(T))\leq t,\Phi^{-1}(F_{C}(C))\leq
c\big{)}$ $\displaystyle=P\big{(}T\leq F_{T}^{-1}(\Phi(t)),C\leq
F_{C}^{-1}(\Phi(c))\big{)}=\Phi_{\theta}(t,c),$

and hence $\Phi^{-1}(F_{T}(T))|\Phi^{-1}(F_{C}(C))\sim
N(\theta\Phi^{-1}(F_{C}(C)),1-\theta^{2})$. It follows that (we omit the
parameters $\theta,\theta_{T}$ and $\theta_{C}$ for simplicity)

$\displaystyle h_{T|C}(F_{T}(t)|F_{C}(t))=F_{T|C}(t|t)$
$\displaystyle=P\Big{(}\Phi^{-1}(F_{T}(T))\leq\Phi^{-1}(F_{T}(t))\,\Big{|}\,\Phi^{-1}(F_{C}(C))=\Phi^{-1}(F_{C}(t))\Big{)}$
$\displaystyle=\Phi\Big{(}\frac{\Phi^{-1}(F_{T}(t))-\theta\Phi^{-1}(F_{C}(t))}{(1-\theta^{2})^{1/2}}\Big{)}$
$\displaystyle=\Phi\Big{(}\frac{A_{\theta,F_{T},F_{C}}(t)}{(1-\theta^{2})^{1/2}}\Big{)}.$

Since $A_{\theta,F_{T},F_{C}}(t)$ tends to $-\infty$ either when $t$ tends to
0 or to $\infty$, it follows that $h_{T|C}(F_{T}(t)|F_{C}(t))$ tends to 0
either when $t$ tends to 0 or to $\infty$. Hence, condition $(C2a)$ is
satisfied. In a similar way we can prove condition $(C2b)$. $\Box$

Proof of Theorem 3.4. Suppose that $\lim_{t\rightarrow
0}F_{T,\theta_{T}}(t)/F_{C,\theta_{C}}(t)=\infty$ (the case where the limit
equals 0 can be handled similarly). Then, it follows from (3.2) that
$\lim_{t\rightarrow
0}h_{C|T,\theta}(F_{C,\theta_{C}}(t)|F_{T,\theta_{T}}(t))=0$. Hence, using
similar arguments as in the proof of Theorem 3.1, it follows from condition
(C1) that $\theta_{T}$ is identifiable. From the formula of
$f_{Y,\Delta}(\cdot,1)$ given in (2.4) it then follows that the function
$t\rightarrow h_{C|T,\theta}(F_{C,\theta_{C}}(t)|F_{T,\theta_{T}}(t))$ is
identifiable.

Next, note that for $t$ large enough,
$F_{T}(t)^{\theta}/F_{C,\theta_{C}}(t)^{\theta}-F_{T}(t)^{\theta}$ is close to
zero (we omit $\theta_{T}$ since it is identifiable), and hence a Taylor
expansion can be used to write

$\displaystyle\log h_{C|T,\theta}(F_{C,\theta_{C}}(t)|F_{T}(t))$
$\displaystyle=-\frac{\theta+1}{\theta}\log\Big{(}1+\frac{F_{T}(t)^{\theta}}{F_{C,\theta_{C}}(t)^{\theta}}-F_{T}(t)^{\theta}\Big{)}$
$\displaystyle=-\frac{\theta+1}{\theta}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\big{\\{}F_{T}(t)^{\theta}(F_{C,\theta_{C}}(t)^{-\theta}-1)\big{\\}}^{k}$

for $t$ large. This is a polynomial in $u_{t}=F_{T}(t)$. Hence, for two sets
of parameters $(\theta,\theta_{C})$ and $(\theta^{*},\theta_{C}^{*})$ we have
that

$\frac{\theta+1}{\theta}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\big{(}F_{C,\theta_{C}}(t)^{-\theta}-1\big{)}^{k}u_{t}^{\theta
k}=\frac{\theta^{*}+1}{\theta^{*}}\sum_{k=1}^{\infty}\frac{(-1)^{k-1}}{k}\big{(}F_{C,\theta_{C}^{*}}(t)^{-\theta^{*}}-1\big{)}^{k}u_{t}^{\theta^{*}k}$

for $t$ large. And this is only possible if $\theta=\theta^{*}$ and
$\theta_{C}=\theta_{C}^{*}$. $\Box$

## Supplement

###  1.1 Results for Scenario 2

The results from Scenario 2 are shown in Tables 10, 11, 12 and 13 for the
Frank, Clayton, Gumbel and Gauss copula, respectively. Compared to Scenario 1
the positive bias for the Frank copula is even higher under Scenario 2 for low
dependence. In the other copula cases we also see that there is a slightly
higher bias in the low dependence case compared to Scenario 1. However for
stronger dependences the performance is satisfactory even for sample sizes as
low as 200.

$\tau=.2,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
---|---|---|---|---|---|---
average.estimate | 2.43 | -0.06 | 1.98 | -0.69 | -0.51 | 0.38
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.01 | 0.06 | 0.01
average.bias | -0.07 | -0.06 | -0.02 | 0.00 | 0.88 | 0.18
RMSE | 0.17 | 0.13 | 0.04 | 0.06 | 1.08 | 0.23
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.44 | -0.04 | 1.99 | -0.69 | -0.59 | 0.36
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.05 | 0.01
average.bias | -0.06 | -0.04 | -0.01 | -0.00 | 0.80 | 0.16
RMSE | 0.13 | 0.10 | 0.03 | 0.04 | 0.95 | 0.19
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.18 | -0.01 | 1.99 | -1.38 | -0.73 | 0.33
sd.of.average.estimate | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.00
average.bias | -0.01 | -0.01 | -0.01 | 0.01 | 0.66 | 0.13
RMSE | 0.04 | 0.04 | 0.01 | 0.03 | 0.68 | 0.13
$\tau=.5,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.52 | -0.00 | 2.00 | -0.69 | -0.02 | 0.49
sd.of.average.estimate | 0.02 | 0.01 | 0.00 | 0.01 | 0.07 | 0.01
average.bias | 0.02 | -0.00 | -0.00 | 0.00 | -0.02 | -0.01
RMSE | 0.22 | 0.14 | 0.04 | 0.06 | 0.66 | 0.15
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.52 | 0.00 | 2.00 | -0.69 | -0.18 | 0.46
sd.of.average.estimate | 0.02 | 0.01 | 0.00 | 0.00 | 0.08 | 0.02
average.bias | 0.02 | 0.00 | 0.00 | 0.00 | -0.18 | -0.04
RMSE | 0.20 | 0.13 | 0.03 | 0.04 | 0.78 | 0.17
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.20 | 0.00 | 2.00 | -1.38 | -0.01 | 0.50
sd.of.average.estimate | 0.00 | 0.00 | 0.00 | 0.00 | 0.02 | 0.00
average.bias | -0.00 | 0.00 | -0.00 | 0.00 | -0.01 | -0.00
RMSE | 0.05 | 0.04 | 0.01 | 0.04 | 0.22 | 0.05
$\tau=.8,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.52 | -0.00 | 2.00 | -0.69 | 1.54 | 0.81
sd.of.average.estimate | 0.02 | 0.01 | 0.00 | 0.01 | 0.06 | 0.01
average.bias | 0.02 | -0.00 | 0.00 | -0.00 | 0.15 | 0.01
RMSE | 0.21 | 0.14 | 0.03 | 0.07 | 0.60 | 0.07
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.52 | 0.00 | 2.00 | -0.70 | 1.40 | 0.80
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.03 | 0.00
average.bias | 0.02 | 0.00 | 0.00 | -0.01 | 0.02 | -0.00
RMSE | 0.14 | 0.09 | 0.03 | 0.04 | 0.30 | 0.05
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.52 | 0.01 | 2.00 | -0.69 | 1.38 | 0.80
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.02 | 0.00
average.bias | 0.02 | 0.01 | 0.00 | -0.00 | -0.01 | -0.00
RMSE | 0.10 | 0.07 | 0.02 | 0.03 | 0.17 | 0.03
Table 10: Simulation results for the case of a Frank copula under Scenario 2. $\tau=.2,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
---|---|---|---|---|---|---
average.estimate | 2.45 | -0.04 | 1.98 | -0.67 | -1.60 | 0.27
sd.of.average.estimate | 0.02 | 0.01 | 0.01 | 0.01 | 0.20 | 0.02
average.bias | -0.05 | -0.04 | -0.02 | 0.02 | -0.21 | 0.07
RMSE | 0.20 | 0.14 | 0.07 | 0.07 | 1.97 | 0.22
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.46 | -0.02 | 1.99 | -0.68 | -1.72 | 0.25
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.19 | 0.02
average.bias | -0.04 | -0.02 | -0.01 | 0.01 | -0.33 | 0.05
RMSE | 0.15 | 0.10 | 0.05 | 0.05 | 1.89 | 0.18
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.48 | -0.01 | 2.00 | -0.68 | -1.50 | 0.22
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.11 | 0.01
average.bias | -0.02 | -0.01 | -0.00 | 0.01 | -0.11 | 0.02
RMSE | 0.11 | 0.07 | 0.04 | 0.04 | 1.13 | 0.12
$\tau=.5,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.52 | 0.01 | 2.00 | -0.69 | -0.10 | 0.49
sd.of.average.estimate | 0.02 | 0.01 | 0.00 | 0.01 | 0.08 | 0.01
average.bias | 0.02 | 0.01 | 0.00 | -0.00 | -0.10 | -0.01
RMSE | 0.20 | 0.14 | 0.05 | 0.07 | 0.85 | 0.15
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.50 | -0.01 | 2.00 | -0.69 | 0.04 | 0.51
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.04 | 0.01
average.bias | -0.00 | -0.01 | 0.00 | -0.00 | 0.04 | 0.01
RMSE | 0.14 | 0.10 | 0.03 | 0.05 | 0.42 | 0.10
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.49 | -0.01 | 2.00 | -0.69 | 0.04 | 0.51
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.03 | 0.01
average.bias | -0.01 | -0.01 | -0.00 | -0.00 | 0.04 | 0.01
RMSE | 0.10 | 0.07 | 0.02 | 0.03 | 0.29 | 0.07
$\tau=.8,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.50 | -0.01 | 2.00 | -0.69 | 1.52 | 0.81
sd.of.average.estimate | 0.02 | 0.01 | 0.00 | 0.01 | 0.05 | 0.01
average.bias | -0.00 | -0.01 | -0.00 | -0.00 | 0.13 | 0.01
RMSE | 0.20 | 0.15 | 0.04 | 0.06 | 0.51 | 0.07
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.50 | -0.02 | 2.00 | -0.70 | 1.43 | 0.80
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.02 | 0.00
average.bias | -0.00 | -0.02 | 0.00 | -0.01 | 0.04 | 0.00
RMSE | 0.11 | 0.08 | 0.02 | 0.04 | 0.23 | 0.04
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.50 | -0.01 | 2.00 | -0.70 | 1.40 | 0.80
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.02 | 0.00
average.bias | -0.00 | -0.01 | -0.00 | -0.01 | 0.01 | 0.00
RMSE | 0.08 | 0.06 | 0.02 | 0.03 | 0.16 | 0.02
Table 11: Simulation results for the case of a Clayton copula under Scenario 2. $\tau=.2,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
---|---|---|---|---|---|---
average.estimate | 2.46 | -0.04 | 1.99 | -0.69 | -1.76 | 0.26
sd.of.average.estimate | 0.02 | 0.02 | 0.00 | 0.01 | 0.24 | 0.02
average.bias | -0.04 | -0.04 | -0.01 | 0.00 | -0.37 | 0.06
RMSE | 0.22 | 0.19 | 0.05 | 0.07 | 2.42 | 0.22
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.48 | -0.02 | 1.99 | -0.69 | -1.45 | 0.23
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.12 | 0.01
average.bias | -0.02 | -0.02 | -0.01 | 0.00 | -0.06 | 0.03
RMSE | 0.14 | 0.10 | 0.03 | 0.03 | 1.20 | 0.11
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.49 | -0.01 | 2.00 | -0.69 | -1.38 | 0.21
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.05 | 0.01
average.bias | -0.01 | -0.01 | -0.00 | 0.00 | 0.01 | 0.01
RMSE | 0.09 | 0.07 | 0.03 | 0.02 | 0.49 | 0.08
$\tau=.5,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.50 | -0.03 | 2.00 | -0.68 | 0.15 | 0.53
sd.of.average.estimate | 0.03 | 0.02 | 0.00 | 0.01 | 0.08 | 0.02
average.bias | 0.00 | -0.03 | 0.00 | 0.01 | 0.15 | 0.03
RMSE | 0.26 | 0.20 | 0.05 | 0.06 | 0.86 | 0.18
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.49 | -0.01 | 2.00 | -0.69 | 0.06 | 0.51
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.04 | 0.01
average.bias | -0.01 | -0.01 | -0.00 | -0.00 | 0.06 | 0.01
RMSE | 0.15 | 0.10 | 0.03 | 0.04 | 0.42 | 0.10
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.50 | -0.00 | 2.00 | -0.69 | 0.03 | 0.51
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.03 | 0.01
64 average.bias | -0.00 | -0.00 | -0.00 | 0.00 | 0.03 | 0.01
RMSE | 0.10 | 0.07 | 0.02 | 0.03 | 0.29 | 0.07
$\tau=.8,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.49 | -0.03 | 2.00 | -0.68 | 1.53 | 0.81
sd.of.average.estimate | 0.02 | 0.02 | 0.00 | 0.01 | 0.06 | 0.01
average.bias | -0.01 | -0.03 | 0.00 | 0.01 | 0.14 | 0.01
RMSE | 0.21 | 0.17 | 0.04 | 0.06 | 0.57 | 0.07
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.51 | -0.01 | 2.00 | -0.70 | 1.44 | 0.80
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.03 | 0.00
average.bias | 0.01 | -0.01 | 0.00 | -0.01 | 0.05 | 0.00
RMSE | 0.15 | 0.11 | 0.02 | 0.04 | 0.28 | 0.04
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.50 | -0.01 | 2.00 | -0.69 | 1.42 | 0.80
sd.of.average.estimate | 0.01 | 0.01 | 0.00 | 0.00 | 0.02 | 0.00
average.bias | -0.00 | -0.01 | 0.00 | 0.00 | 0.03 | 0.00
RMSE | 0.11 | 0.09 | 0.02 | 0.03 | 0.21 | 0.03
Table 12: Simulation results for the case of a Gumbel copula under Scenario 2. $\tau=.2,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
---|---|---|---|---|---|---
average.estimate | 2.46 | -0.05 | 1.98 | -0.68 | -2.08 | 0.24
sd.of.average.estimate | 0.03 | 0.02 | 0.01 | 0.01 | 0.23 | 0.02
average.bias | -0.04 | -0.05 | -0.02 | 0.02 | -0.69 | 0.04
RMSE | 0.31 | 0.17 | 0.13 | 0.11 | 2.37 | 0.20
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.45 | -0.03 | 1.98 | -0.68 | -1.77 | 0.24
sd.of.average.estimate | 0.03 | 0.01 | 0.01 | 0.01 | 0.18 | 0.02
average.bias | -0.05 | -0.03 | -0.02 | 0.01 | -0.38 | 0.04
RMSE | 0.29 | 0.12 | 0.12 | 0.10 | 1.84 | 0.17
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.47 | -0.01 | 1.98 | -0.68 | -1.69 | 0.21
sd.of.average.estimate | 0.03 | 0.01 | 0.01 | 0.01 | 0.14 | 0.01
average.bias | -0.03 | -0.01 | -0.02 | 0.01 | -0.31 | 0.01
RMSE.2 | 0.26 | 0.09 | 0.11 | 0.10 | 1.39 | 0.11
$\tau=.5,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.49 | -0.01 | 1.99 | -0.68 | -0.08 | 0.50
sd.of.average.estimate | 0.04 | 0.02 | 0.01 | 0.01 | 0.12 | 0.02
average.bias | -0.01 | -0.01 | -0.01 | 0.02 | -0.08 | -0.00
RMSE | 0.34 | 0.17 | 0.12 | 0.12 | 1.21 | 0.18
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.48 | -0.01 | 1.99 | -0.68 | -0.05 | 0.50
sd.of.average.estimate | 0.03 | 0.01 | 0.01 | 0.01 | 0.10 | 0.01
average.bias | -0.02 | -0.01 | -0.01 | 0.01 | -0.05 | 0.00
RMSE | 0.29 | 0.12 | 0.11 | 0.10 | 1.00 | 0.13
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.48 | -0.01 | 1.99 | -0.68 | -0.07 | 0.50
sd.of.average.estimate | 0.03 | 0.01 | 0.01 | 0.01 | 0.09 | 0.01
average.bias | -0.02 | -0.01 | -0.01 | 0.02 | -0.07 | -0.00
RMSE | 0.26 | 0.09 | 0.11 | 0.10 | 0.92 | 0.10
$\tau=.8,n=200$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.46 | -0.04 | 1.99 | -0.67 | 1.45 | 0.81
sd.of.average.estimate | 0.03 | 0.02 | 0.01 | 0.01 | 0.11 | 0.01
average.bias | 0.04 | -0.04 | -0.01 | 0.02 | 0.06 | 0.01
RMSE.6 | 0.31 | 0.16 | 0.12 | 0.11 | 1.10 | 0.10
$n=500$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.46 | -0.03 | 1.99 | -0.68 | 1.36 | 0.80
sd.of.average.estimate | 0.03 | 0.01 | 0.01 | 0.01 | 0.10 | 0.01
average.bias | -0.04 | -0.03 | -0.01 | 0.01 | -0.02 | 0.00
RMSE | 0.28 | 0.12 | 0.11 | 0.10 | 1.03 | 0.09
$n=1000$ | $\mu_{T}$ | $\log(\sigma_{T})$ | $\mu_{C}$ | $\log(\sigma_{C})$ | $\mbox{logit}(\tau)$ | $\tau$
average.estimate | 2.49 | 0.00 | 1.99 | -0.69 | 1.27 | 0.79
sd.of.average.estimate | 0.03 | 0.01 | 0.01 | 0.01 | 0.10 | 0.01
average.bias | -0.01 | 0.00 | -0.01 | 0.00 | -0.12 | -0.01
RMSE | 0.26 | 0.09 | 0.11 | 0.10 | 1.01 | 0.09
Table 13: Simulation results for the case of a Gauss copula under Scenario 2.