# Nonparametric inference about increasing odds rate distributions

Tommaso Lando<EMAIL_ADDRESS>Idir Arab  Paulo Eduardo Oliveira

###### Abstract

To improve nonparametric estimates of lifetime distributions, we propose using
the increasing odds rate (IOR) model as an alternative to other popular, but
more restrictive, “adverse ageing” models, such as the increasing hazard rate
one. This extends the scope of applicability of some methods for statistical
inference under order restrictions, since the IOR model is compatible with
heavy-tailed and bathtub distributions. We study a strongly uniformly
consistent estimator of the cumulative distribution function of interest under
the IOR constraint. Numerical evidence shows that this estimator often
outperforms the classic empirical distribution function when the underlying
model does belong to the IOR family. We also study two different tests, aimed
at detecting deviations from the IOR property, and we establish their
consistency. The performance of these tests is also evaluated through
simulations.

###### keywords:

hazard rate, heavy tails , nonparametric test , odds

††journal: Elsevierlabel1label1footnotetext: Department of Economics,
University of Bergamo, Italy; Department of Finance, VŠB-TU Ostrava, Czech
Republiclabel2label2footnotetext: CMUC, Department of Mathematics, University
of Coimbra, Portugal

##  1 Introduction

In statistics, the problem of estimating a cumulative distribution function
(CDF) $F$, often depends on any prior information on $F$. This is, for
example, the idea that gives rise to parametric inference. On the other hand,
no information on $F$ leads to the most basic nonparametric estimator of $F$,
namely, the empirical CDF $\mathbb{F}_{n}$. The approach based on shape
restricted inference represents an intermediate case, in which it is assumed
that $F$ belongs to some important, and as broad as possible, nonparametric
family of distributions, satisfying some shape constraints. It is obvious
that, using a stronger shape constraint, one may expect a larger estimation
improvement upon $\mathbb{F}_{n}$, but, on the other hand, less applicability
and higher risk, if the constraint is false. Thus, these shape restrictions
should be chosen to reasonably conform to the data, and validated through
statistical testing.

In reliability and survival analysis, one is generally interested in random
lifetimes, so it is typically assumed that distributions may exhibit “adverse
ageing”, vaguely meaning that ageing has a negative effect on lifetime
(Marshall and Olkin, 2007). The classic (but not unique) way of translating
this intuitive notion into a mathematical language is assuming that $F$ has an
increasing hazard rate (IHR). For this reason, nonparametric inference for IHR
distributions represents an extremely relevant case, which has been studied by
Grenander (1956), Marshall and Proschan (1965) or Prakasa Rao (1970), among
many other authors. However, the IHR assumption is sometimes considered to be
too strong. For example, it is not compatible with heavy-tailed (as it
requires the existence of all moments) and bathtub distributions, namely,
distributions that exhibit a U-shaped hazard rate (Marshall and Olkin, 2007).
The solution to this problem may be using a weaker shape restriction, which is
still coherent with the “adverse ageing” notion. For example, Wang (1987),
Rojo and Samaniego (1994) or El Barmi et al. (2021) estimate distributions
which exhibit an IHR, but only on average (IHRA). Although the IHRA class
contains the IHR class, this family still requires the existence of all
moments and does not contain any bathtub model. Therefore, the problem of
estimating $F$ within a weaker “adverse ageing” setting, compatible with
heavy-tailed and bathtub distributions, is particularly interesting.

In a recent paper, Lando et al. (2022a) proposed an ageing class that is based
on the monotonicity of the odds rate (OR), instead of the hazard rate. In
fact, ageing classes are typically defined in terms of some suitable
stochastic ordering constraint, taking the exponential distribution as the
classic benchmark for “no ageing”. This is due to the well-known “lack of
memory” property. However, a wider family than the IHR may be constructed by
replacing the exponential with a different reference distribution, providing
an alternative interpretation of “no ageing”. In particular, as discussed in
Lando et al. (2022a), the log-logistic distribution (with shape parameter
equal to 1), hereafter referred to as LL(1), may be a suitable benchmark, as
it satisfies a “multiplicative lack of memory” property (Galambos and Kotz,
2006), and it is the only distribution such that the odds of failure by time
$x$, that is the probability of failure over survival at $x$, has a constant
growth rate with respect to time. Moreover, unlike the exponential
distribution, the LL(1) distribution has an infinite expectation, which is
also more coherent with a literal intuitive translation of the “no ageing”
concept. The family of distributions that are dominated by the LL(1) with
respect to the convex transform order of Van Zwet (1964) is called the
increasing odds rate (IOR) family. Besides having some interesting properties,
the IOR class has a wide range of applicability: it does not require the
existence of moments, it contains all IHR distributions, plus some important
heavy-tailed and bathtub ones (see Section 2.1 of Lando et al. (2022a)).
Accordingly, the objective of this paper is to obtain an estimator of $F$ that
satisfies the IOR assumption, hence improving upon $\mathbb{F}_{n}$ under weak
conditions. In Section 3, we construct such an estimator, denoted as
$\widetilde{\mathbb{F}}_{n}$, using similar methods as those discussed in the
books of Barlow et al. (1971) or Robertson et al. (1988).
$\widetilde{\mathbb{F}}_{n}$ is strongly uniformly consistent, and simulations
show that it is generally more accurate than $\mathbb{F}_{n}$, provided that
the IOR assumption holds. On the other hand, it is clear that the method does
not work properly when the IOR assumption is false, therefore it is important
to decide, based on statistical testing, whether the data support the IOR
assumption or not. In Section 4, we obtain two different nonparametric tests
of the IOR null hypothesis. We study the properties of these tests, establish
their consistency, and compare their performances simulating the values of the
power function under some critical alternatives.

##  2 The IOR property

Let us begin with some preliminary notations. Throughout this paper,
“increasing” signifies “non-decreasing” and “decreasing” signifies “non-
increasing”. We define the greatest convex minorant (GCM) of a function $g$,
denoted as $g_{cx}$, as the largest convex function that does not exceed $g$.
Similarly, the least concave majorant (LCM) of $g$, denoted as $g_{cv}$, is
the smallest concave function which is larger than or equal to $g$. Throughout
this paper, $X$ denotes a random variable with an absolutely continuous CDF
$F$ and probability density function (PDF) $f$. Moreover, for the sake of
simplicity, any map $\phi$ which depends on a CDF $F$, will be denoted as
$\phi_{F}$ or just $\phi$, whenever it is clear from the context.

We will now recall the definition and properties of the IOR family. The reader
is referred to Lando et al. (2022a) for a more detailed discussion.

Let us denote with $L(x)=\frac{x}{1+x}$, $x>0$, the CDF of the LL(1)
distribution. The function

$\Lambda_{F}(x)=L^{-1}\circ F(x)=\frac{F(x)}{1-F(x)}$

defines the odds of failure by time $x$. The function $\Lambda_{F}$ is
obviously increasing, so it is natural to get interested in its growth rate.
In fact, one may expect that $F$ exhibits “no ageing”, in some sense, when the
odds rate of $F$, defined as

${\lambda_{F}(x)=\Lambda_{F}^{\prime}(x)=\frac{f(x)}{(1-F(x))^{2}}},$

is constant. Similarly, an increase of $\lambda$ means that the probability of
failure over survival is accelerating with respect to time, suggesting an
“adverse ageing” scenario. Therefore, we are interested in distributions with
an increasing behaviour of $\lambda$, or, equivalently, with a convex odds
function $\Lambda$. This family, formally
$\mathcal{F}_{IOR}=\\{F:\Lambda_{F}\text{ is convex}\\}$, is referred to as
the IOR class.

Note that the convexity of $\Lambda$ does not imply the existence of a
density. Indeed, this convexity allows for a single point mass of $F$ at the
right endpoint of the support. However, for simplicity of the presentation, we
shall assume the existence of a PDF.

It is easy to see that the IOR class contains the well known IHR class, which
may be characterized as $\\{F:-\ln(1-F)\text{ is convex}\\}$. These two
classes may be equivalently defined in terms of the convex transform order,
requiring convexity of $G^{-1}\circ F$, where $G$ is some suitable (and fixed)
reference CDF. In general, $G$ dominates $F$ in the convex transform order,
denoted as $G\geq_{c}F$, if $G^{-1}\circ F$ is convex (Shaked and
Shanthikumar, 2007). The IHR class is obtained by choosing
$G(x)=\mathcal{E}(x)=1-e^{-x}$, $x>0$, namely, the unit exponential
distribution. The main reason for choosing $\mathcal{E}$ as a benchmark is
that it satisfies the “lack of memory” property. Likewise, for $G={L}$, we
obtain the IOR class, and it is also easy to see that, if $X\sim L$, then the
shifted random variable $X+1$ satisfies the “multiplicative lack of memory”
property, that is, $P(X+1>ab|X+1>a)=P(X+1>b)$, for every $a,b>1$ (Galambos and
Kotz, 2006). Bear in mind that ageing notions are typically location and scale
independent, in particular, $X$ and $\sigma X+\mu$ are equivalent in terms of
the convex transform order. In this regard, we may refer to the LL(1)
location-scale family, determined by the CDF ${L}$ up to location and scale
transformations.

We now discuss the applicability of the IOR model. First, note that the IOR
class may contain bathtub distributions, characterised by a “decreasing then
increasing” behaviour of the HR, and are commonly employed in survival
analysis and reliability (see, for example, Marshall and Olkin (2007), or
Glaser (1980)). For instance, under some conditions on the parameters, the
bathtub distributions defined by Topp and Leone (1955), Hjorth (1980), Schäbe
(1994) or Haupt and Schäbe (1997) are IOR. This class contains also many
heavy-tailed distributions, such as the log-logistic, the Pareto, the Burr
XII, the Fréchet, the Student’s $t$ (with shape parameter(s) larger than or
equal to 1), the lognormal (under some conditions) and the Cauchy
distributions. It is not possible to establish inclusion relations between the
IOR class and other relevant classes which contain the IHR class, such as the
IHRA and the DMRL families (Marshall and Olkin, 2007). However, like the IHR,
these latter classes require the existence of all moments, therefore they
cannot be used as shape constraints to estimate heavy-tailed distributions,
which is one of the main advantages of the IOR model. This discussion
motivates us to study nonparametric inference on lifetime distributions using
the IOR constraint, instead of the IHR one.

##  3 An IOR estimator

Since we are interested in random lifetimes, hereafter we will focus on the
case in which the distribution of interest $F$ has support included in the
interval $[0,\infty)$. Given an ordered sample $X_{(1)},\ldots,X_{(n)}$ from
$F$, the empirical CDF is defined as
$\mathbb{F}_{n}(x)=\frac{1}{n}\sum_{i=1}^{n}{\mathbf{1}_{X_{(i)}\leq x}}$,
where $\mathbf{1}_{A}$ is the indicator of the event $A$. A realisation of the
random process $\mathbb{F}_{n}$ will be denoted with $F_{n}$. The empirical
CDF represents the most natural way of estimating $F$ without any prior
information, as it satisfies several important properties, in particular, the
Glivenko-Cantelli Theorem establishes strong and uniform convergence of
$\mathbb{F}_{n}$ to $F$. Assume that $F\in\mathcal{F}_{IOR}$: we are
interested in determining an estimator, say $\widetilde{\mathbb{F}}_{n}$, such
that $\widetilde{\mathbb{F}}_{n}\in\mathcal{F}_{IOR}$ and
$\widetilde{\mathbb{F}}_{n}$ converges strongly and uniformly to $F$. A first
intuitive solution would be estimating $\Lambda_{F}$ using its empirical
counterpart
$\Lambda_{\mathbb{F}_{n}}=L^{-1}\circ\mathbb{F}_{n}=\frac{\mathbb{F}_{n}}{1-\mathbb{F}_{n}}$
(note that $\Lambda_{\mathbb{F}_{n}}(X_{(i)})=\frac{i}{n-i}$) and then
estimate $F$ using $L\circ(\Lambda_{\mathbb{F}_{n}})_{cx}$. However, the
function $L^{-1}(p)=\frac{p}{1-p}$ is unbounded in $[0,1)$, accordingly, for
every sample size,
$\sup{{}_{x}}|\Lambda_{\mathbb{F}_{n}}{(x)}-\Lambda{(x)}|=\infty$, and the
corresponding IOR estimator defined earlier would not have the desired
property. Therefore, we will consider a somewhat more involved approach.

For some given CDF $F\in\mathcal{F}_{IOR}$, let us define the following
integral transform

$T_{F}^{-1}(p)=Z_{F}\circ F^{-1}(p)=\int_{0}^{F^{-1}(p)}(1-F(x))^{2}\,dx,\quad
p\in[0,1],$

where $Z_{F}(t)=\int_{0}^{t}(1-F(x))^{2}\,dx$, and the corresponding scale-
free version $\overline{T}_{F}^{-1}(p)=\frac{T_{F}^{-1}(p)}{T_{F}^{-1}(1)}$.
We shall focus on distributions such that
$T_{F}^{-1}(1)=\int_{0}^{\infty}(1-F(x))^{2}\,dx<\infty$. In particular,
$T^{-1}(1)<\infty$ if $1-F$ is integrable, that is, if $F$ has finite mean,
because
$\int_{0}^{\infty}(1-F(x))^{2}\,dx\leq\int_{0}^{\infty}(1-F(x))\,dx={\mathbb{E}(X)}$.
However, the existence of the mean is not necessary: for example, for the CDF
$L$ we have $T_{L}^{-1}(1)=1$, although $L$ has infinite mean. The function
$T^{-1}$ belongs to the family of generalized total time on test transforms,
which has been studied by Barlow and Van Zwet (1969). Note also that the
inverse of $T_{F}^{-1}$, namely $T_{F}=F\circ Z^{-1}_{F}$, is a CDF with
support $[0,T_{F}^{-1}(1)]$.

It can be seen that

$(T^{-1})^{\prime}(p)=Z^{\prime}\circ
F^{-1}(p)(F^{-1})^{\prime}(p)=\frac{(1-F\circ F^{-1}(p))^{2}}{f\circ
F^{-1}(p)},$ (1)

so that $T^{-1}$ is concave, or, equivalently, $T$ is convex, if and only if
$\frac{f(x)}{(1-F(x))^{2}}$ is increasing, that is, $F$ is IOR. Note that
$T^{-1}$ is the identity function if and only if $F$ is an LL(1) distribution.

For $k=1,\ldots,n$, the empirical counterpart of $T_{F}^{-1}(\frac{k}{n})$ is

$T^{-1}_{\mathbb{F}_{n}}\left(\frac{k}{n}\right)=\int_{0}^{X_{(k)}}(1-\mathbb{F}_{n}(t))^{2}\,dt=\frac{1}{n^{2}}\left(\sum_{i=1}^{k-1}(2n-2i+1)X_{(i)}+(n-k+1)^{2}X_{(k)}\right),$

where, in particular, $X_{(0)}:=0$, $T_{\mathbb{F}_{n}}^{-1}{(0)}:=0$, and
$T^{-1}_{\mathbb{F}_{n}}(1)=\frac{1}{n^{2}}\sum_{i=1}^{n}(2n-2i+1)X_{(i)}.$
Note that $T^{-1}_{\mathbb{F}_{n}}(\frac{k}{n})$ can also be expressed as

$T^{-1}_{\mathbb{F}_{n}}\left(\frac{k}{n}\right)=\sum_{j=1}^{k}\left(\frac{n-j+1}{n}\right)^{2}(X_{(j)}-X_{(j-1)}).$

For general $p\in[0,1]$, $T_{\mathbb{F}_{n}}^{-1}(p)$ is defined, by linear
interpolation, as the piecewise linear function joining the points
$\left(\frac{k-1}{n},\,T^{-1}_{\mathbb{F}_{n}}\left(\frac{k-1}{n}\right)\right)$
and
$\left(\frac{k}{n},\,T^{-1}_{\mathbb{F}_{n}}\left(\frac{k}{n}\right)\right)$,
$k=1,\ldots,n$. Similarly, the estimator of $\overline{T}^{-1}$ is
$\overline{T}^{-1}_{\mathbb{F}_{n}}=\frac{T^{-1}_{\mathbb{F}_{n}}(p)}{T^{-1}_{\mathbb{F}_{n}}(1)}$.
These estimators are strongly uniformly consistent.

######  Lemma 1.

  * 1.

$T^{-1}_{\mathbb{F}_{n}}$ ($\overline{T}^{-1}_{\mathbb{F}_{n}}$) converges
strongly and uniformly to $T_{F}^{-1}$ ($\overline{T}_{F}^{-1}$) in $[0,1]$;

  * 2.

$T_{\mathbb{F}_{n}}$ ($\overline{T}_{\mathbb{F}_{n}}$) converges strongly and
uniformly to $T_{F}$ ($\overline{T}_{F}$) in $[0,T_{F}^{-1}(1)]$ ($[0,1]$).

###### Proof.

The strong uniform convergence of $T^{-1}_{\mathbb{F}_{n}}$ is a special case
of Theorem 2.1 of Barlow and Van Zwet (1969). Since $T^{-1}_{\mathbb{F}_{n}}$
is bounded and has bounded domain, its inverse function $T_{\mathbb{F}_{n}}$
is also strongly uniformly consistent. Since $T^{-1}_{\mathbb{F}_{n}}(1)$
converges a.s. to $T^{-1}_{F}(1)$, it is clear that the same results hold for
$\overline{T}^{-1}_{\mathbb{F}_{n}}$ and, likewise, for
$\overline{T}_{\mathbb{F}_{n}}$. ∎

Henceforth, we will abbreviate $T^{-1}_{\mathbb{F}_{n}}$ and
$\overline{T}^{-1}_{\mathbb{F}_{n}}$ with $T^{-1}_{n}$ and
$\overline{T}^{-1}_{n}$, respectively, whenever it will be clear from the
context that they correspond to a sample from $\mathbb{F}_{n}$.

Note that under the IOR constraint, $T^{-1}$ is concave. Therefore, an IOR
estimator may be constructed using $(T_{n}^{-1})_{cv}$, namely, the LCM of
$T_{n}^{-1}$. In fact, $({T}_{n}^{-1})_{cv}$ is a concave piecewise linear
function, so it always has a right derivative, denoted as
$\partial_{+}({T}_{n}^{-1})_{cv}$, which, by construction, is a decreasing
right-continuous step function. Accordingly, the relation in (1) may be used
to obtain an increasing estimator of the OR, defined as follows:

$\widetilde{\lambda}_{n}(x)=\frac{1}{\partial_{+}({T}_{n}^{-1})_{cv}\circ\mathbb{F}_{n}(x)},\;\text{
for }x\in(0,X_{(n)}),$

whereas $\widetilde{\lambda}_{n}(x)=0$, for $x\leq 0$, and
$\widetilde{\lambda}_{n}(x)=+\infty$, for $x\geq X_{(n)}$. In other words,
$\widetilde{\lambda}_{n}(x)$ is the reciprocal of the slope (from the right)
of $({T}_{n}^{-1})_{cv}$, evaluated at the point $\mathbb{F}_{n}(x)$.
Accordingly, the odds function is estimated by

$\widetilde{\Lambda}_{n}(x)=\int_{0}^{x}{\widetilde{\lambda}_{n}(t)\,dt}=\int_{0}^{x}\frac{1}{\partial_{+}(T_{n}^{-1})_{cv}\circ\mathbb{F}_{n}(t)}\,dt.$
(2)

Finally, an IOR estimator of $F$ is given by

$\widetilde{\mathbb{F}}_{n}=L\circ\widetilde{\Lambda}_{n}=1-\frac{1}{1+\widetilde{\Lambda}_{n}}.$

$\widetilde{\mathbb{F}}_{n}$ is clearly IOR, and it is absolutely continuous,
except in the right endpoint of its support, $X_{(n)}$, at which it has a
jump. Correspondingly, the PDF $f$ can be estimated, in $(0,X_{(n)})$, as

$\widetilde{f}_{n}=\frac{\widetilde{\lambda}_{n}}{(1+\widetilde{\Lambda}_{n})^{2}}.$
(3)

Using the methods described in the book of Robertson et al. (1988), it is
possible to establish the asymptotic properties of these estimators.

######  Theorem 2.

Assume that $F$ is IOR, then:

  1. 1.

$\widetilde{\lambda}_{n}\longrightarrow\lambda$ almost surely and uniformly in
$[0,x_{0}]$, for every $x_{0}$ such that $\lambda(x_{0})<\infty$;

  2. 2.

$\widetilde{\mathbb{F}}_{n}{\longrightarrow}F$ almost surely and uniformly in
$[0,\infty)$;

  3. 3.

$\widetilde{f}_{n}{\longrightarrow}f$ almost surely and uniformly in
$[0,x_{0}]$, for every $x_{0}$ such that $\lambda(x_{0})<\infty$.

###### Proof.

Following the approach as in Theorem 7.4.1 of Robertson et al. (1988) we prove
that
$\lambda(x_{0}^{-})\leq\liminf_{n\rightarrow\infty}\widetilde{\lambda}_{n}(x_{0})\leq\limsup_{n\rightarrow\infty}\widetilde{\lambda}_{n}(x_{0})\leq\lambda(x_{0}^{+})$.
Indeed, since the LCM $(T_{n}^{-1})_{cv}$ is concave by construction, then,
for some small $\epsilon>0$, we have that

$\frac{(T_{n}^{-1})_{cv}(\mathbb{F}_{n}(x_{0})+\epsilon)-(T_{n}^{-1})_{cv}(\mathbb{F}_{n}(x_{0}))}{\epsilon}\leq{\partial_{+}(T_{n}^{-1})_{cv}\circ\mathbb{F}_{n}(x_{0})}=\frac{1}{\widetilde{\lambda}_{n}(x_{0}^{+})}\\\
\leq\frac{1}{\widetilde{\lambda}_{n}(x_{0}^{-})}={\partial_{-}(T_{n}^{-1})_{cv}\circ\mathbb{F}_{n}(x_{0})}\leq\frac{(T_{n}^{-1})_{cv}(\mathbb{F}_{n}(x_{0}))-(T_{n}^{-1})_{cv}(\mathbb{F}_{n}(x_{0})-\epsilon)}{\epsilon},$

for every $x_{0}>0$ and $\epsilon\in(0,\min(F_{n}(x_{0}),1-F_{n}(x_{0})))$.
$T^{-1}_{n}$ converges uniformly to $T^{-1}$ on $[0,1]$, consequently, the
Marshall’s inequality (Groeneboom and Jongbloed, 2014, Exercise 3.1) gives

$\sup_{p\in[0,1]}|(T^{-1}_{n})_{cv}(p)-T^{-1}(p)|\leq\sup_{p\in[0,1]}|{T}^{-1}_{n}(p)-T^{-1}(p)|\longrightarrow
0,$

with probability 1. Then, letting $n\longrightarrow\infty$ in the above
inequality, we obtain

$\frac{{T}^{-1}(F(x_{0})+\epsilon)-{T}^{-1}(F(x_{0}))}{\epsilon}\leq\limsup_{n\rightarrow\infty}\frac{1}{\widetilde{\lambda}_{n}(x_{0})}\\\
\leq\liminf_{n\rightarrow\infty}\frac{1}{\widetilde{\lambda}_{n}(x_{0})}\leq\frac{{T}^{-1}(F(x_{0}))-{T}^{-1}(F(x_{0})-\epsilon)}{\epsilon},$

which gives the claimed bound allowing $\epsilon\longrightarrow 0$. In
particular, $\widetilde{\lambda}_{n}$ is uniformly strongly consistent at the
continuity points of $\lambda$. Now, if $\lambda$ is increasing and continuous
on some compact set $[0,x]$, $\widetilde{\lambda}_{n}$ converges uniformly to
$\lambda$ on $[0,x]$. Setting $x=x_{0}$, we obtain part 1). Subsequently, the
dominated convergence theorem (note that $\widetilde{\lambda}_{n}$ is bounded
on $[0,x]$) implies
$\lim_{n\rightarrow\infty}\int_{0}^{y}\widetilde{\lambda}_{n}(t)\,dt=\lim_{n\rightarrow\infty}\widetilde{\Lambda}_{n}(y)=\int_{0}^{y}{\lambda}(t)\,dt=\Lambda(y)$,
for every $y\in[0,x]$. Therefore, since the CDF $L$ is uniformly continuous in
$[0,\infty)$, it follows that

$\widetilde{\mathbb{F}}_{n}(x)=L\circ\widetilde{\Lambda}_{n}(x)\longrightarrow
L\circ{\Lambda}(x)=F(x)$

almost surely, for every $x$. Finally, following the same arguments in the
proof of the Glivenko-Cantelli theorem, we can ensure that
$\widetilde{\mathbb{F}}_{n}\longrightarrow F$ almost surely and uniformly in
$[0,\infty)$. Since $f(x)=\lambda(x)(1-F(x))^{2}$ and, writing
$\widetilde{\Lambda}_{n}=\Lambda_{\widetilde{\mathbb{F}}_{n}}$, we may express
$\widetilde{f}_{n}(x)=\widetilde{\lambda}_{n}(x)(1-\widetilde{\mathbb{F}}_{n}(x))^{2}$.
Then, part 1) and part 2) imply that $\widetilde{f}_{n}\longrightarrow f$
strongly and uniformly in $[0,x_{0}]$. ∎

###  3.1 Simulations

We illustrate the numerical performance of $\widetilde{\mathbb{F}}_{n}$ and
$\mathbb{F}_{n}$ in terms of MSE, as is well known,
$\text{MSE}(\mathbb{F}_{n}(x))=\frac{1}{n}F(x)(1-F(x))$. With regard to
$\widetilde{\mathbb{F}}_{n}$, the MSE was simulated using the following
distributions: (i) log-logistic, $F(x)=\frac{1}{1+x^{-a}}$, $a>0$, hereafter
LL($a$), which is IOR for $a>1$ and has a constant OR for $a=1$; (ii) Weibull
$F(x)=1-e^{-x^{a}}$, $a>0$, hereafter W($a$), which is IOR for $a\geq 1$ (and
also log-concave, IHR, IHRA and DMRL); (iii) beta type II,
$F(x)=\beta(\frac{x}{1+x};a,b)$, $a,b>0$, hereafter B2($a,b$), which is IOR
for $a>1,b\geq 1$ ($\beta$ denotes the regularized incomplete beta function);
(iv) the Haupt and Schabe’s distribution (Haupt and Schäbe, 1997)
$F(x)=\sqrt{a^{2}+(2a+1)x}-a$, $a>-1/2$, $x\in[0,1]$, hereafter HS($a$), which
is bathtub for $a\in(\frac{1}{2},1)$ and IOR at the same time. Note that scale
parameters are conveniently not considered in our analysis, since ageing
properties are scale invariant. Figure 1 illustrates the estimators
$\tilde{\mathbb{F}}_{n}$ and $\mathbb{F}_{n}$ in the IOR case, using a small
sample size. Differently, the behaviour of such estimators in the non-IOR case
is discussed in the next section and depicted in Figure 5.

Figure 1: Simulated small sample ($n=20$) from the IOR distribution HS(0.6).
The plot on the left shows the CDFs $F$ (dot-dashed), $\mathbb{F}_{n}$ (solid)
and $\widetilde{\mathbb{F}}_{n}$ (dashed); the plot on the right shows
$\lambda_{F}$ (dot-dashed) and $\widetilde{\lambda}_{n}$ (dashed).

In Figure 2 we plot simulated MSE values at all percentiles, based on 1000
runs, standardised by the corresponding ones for $\mathbb{F}_{n}$, that is,
$\frac{\text{MSE}(\widetilde{\mathbb{F}}_{n}(x))}{\text{MSE}(\mathbb{F}_{n}(x))}$.
The simulation results show that, when $F$ is IOR,
$\widetilde{\mathbb{F}}_{n}$ often outperforms $\mathbb{F}_{n}$, especially
for extremely small deciles. This is particularly visible for small sample
sizes, as clearly when $n$ grows both estimators converge to $F$. According to
our results, the largest improvements often occur in the left tails, around
deciles 0.1–0.3, however, in some cases $\widetilde{\mathbb{F}}_{n}$ performs
worse at very small percentiles (0.01–0.05) for very small sample sizes
($n=10$) and some particular models, as it can be seen in Figure 2. Note that,
among the distributions considered, only the W and the HS distributions have
all moments, whereas the LL(1) has an infinite mean, the LL(2) and the B2 with
$b\leq 2$ have infinite variance, and the B2 with $b\leq 3$ has an infinite
third moment. This represents one of the main advantages of the proposed
method, as the majority of the shape conditions which are commonly used in
order-restricted inference under an “adverse ageing scenario” (log-concavity,
IHR, IHRA, DMRL) require the existence of all moments. Similarly, the HS($a$)
is bathtub for the values of $a$ considered, so it is not compatible with the
log-concavity, IHR, and IHRA assumptions (only the W($a$), for $a\geq 1$,
satisfies all these latter properties). It can also be noted that convergence
is somewhat slower for distributions with infinite variance. The case of the
LL(1) is especially critical, because, besides having an infinite mean, more
importantly, it has a constant OR, so we expect no improvement, or less,
compared to the cases in which the OR is strictly increasing. Surprisingly,
even in this case, $\widetilde{\mathbb{F}}_{n}$ performs slightly better than
$\mathbb{F}_{n}$ for smaller sample sizes, whereas, in general,
$\widetilde{\mathbb{F}}_{n}$ performs better at the tails and $\mathbb{F}_{n}$
performs better around the median.

Figure 2: Standardised MSE for $n=10$ (solid), $n=30$ (dashed), $n=50$ (dot-
dashed), $n=100$ (dotted), evaluated at all percentiles.

###  3.2 Smoothed estimators

One may wonder whether the improvement of $\widetilde{\mathbb{F}}_{n}$ over
${\mathbb{F}}_{n}$ depends on the fact that $\widetilde{\mathbb{F}}_{n}$ is a
smooth function. In response to this, a numerical comparison of
$\widetilde{\mathbb{F}}_{n}$ with kernel estimators of the CDF, may show that
the MSE of $\widetilde{\mathbb{F}}_{n}$ is often smaller even in this case,
suggesting that the improvement of $\widetilde{\mathbb{F}}_{n}$ over
${\mathbb{F}}_{n}$ depends mostly on shape aspects, rather than on smoothness.
Although the objective of this paper is to estimate the CDF, we will now
briefly discuss the possibility of obtaining smooth estimators even for the
PDF. In fact, while $\widetilde{\mathbb{F}}_{n}$ is absolutely continuous in
$[0,X_{(n)})$, the corresponding estimator of the PDF $\widetilde{f}_{n}$,
defined by (3), is typically discontinuous and may exhibit spikes at the
observed points (see Figure 3). Hence, it is generally not a good pointwise
estimator of $f$. For this reason, it may be interesting to obtain a smooth
estimator of the PDF which preserves the OR properties of $\widetilde{f}_{n}$,
which may be achieved by smoothing the step function $\widetilde{\lambda}_{n}$
and then applying the approach discussed above.

Let $k$ be a zero-mean PDF with support $[-1,1]$, for technical convenience.
Given a bandwidth $h>0$ define as usual $k_{h}(x)=\frac{1}{h}k(\frac{x}{h})$,
and denote by $K_{h}$ the corresponding CDF, that is,
$K_{h}(x)=\int_{-\infty}^{x}k_{h}(t)\,dt$. A smooth version of
$\widetilde{\lambda}_{n}$ is given by

$\lambda_{n,h}^{s}(x)=\int
k_{h}(x-t)\widetilde{\lambda}_{n}(t)\,dt=\int(1-K_{h}(x-t))\,d\widetilde{\lambda}_{n}(t){,}$

after integration by parts. Therefore, as $\widetilde{\lambda}_{n}$ is
increasing by construction, it follows immediately that $\lambda^{s}_{n,h}$ is
also increasing. Note that, since $\widetilde{\lambda}_{n}(x)=+\infty$, for
$x\geq X_{(n)}$, then $\lambda_{n,h}^{s}(x)=+\infty$, for $x\geq X_{(n)}-h$,
hence the smoothing has an effect just at points smaller than $X_{(n)}-h$.
Now, a convex estimator of $\Lambda$ is given by
$\Lambda^{s}_{n,h}=\int_{-\infty}^{x}\lambda_{n,h}^{s}(t)\,dt$, where, unlike
(2), here the integration starts at $-\infty$, meaning that it assigns
positive mass to the negative half line (this may be adjusted, if needed).
Correspondingly, an IOR estimator of the CDF is defined as
$\mathbb{F}^{s}_{n,h}=L\circ\Lambda_{n,h}^{s}$, and a smooth estimator of the
PDF is given by
$f^{s}_{n,h}=\frac{\lambda_{n,h}^{s}}{(1+\Lambda_{n}^{s})^{2}}$.

Since $k$ has support $[-1,1]$,
$\lambda_{n,h}^{s}(x)=\int_{x-h}^{x+h}k_{h}(x-t)\widetilde{\lambda}_{n}(t)\,dt$.
Therefore, by monotonicity,
$\widetilde{\lambda}_{n}(t)\in[\widetilde{\lambda}_{n}(x-h),\widetilde{\lambda}_{n}(x+h)]$,
which holds for $t\in[x-h,x+h]$, implies that
$\lambda_{n,h}^{s}(x)\in[\widetilde{\lambda}_{n}(x-h),\widetilde{\lambda}_{n}(x+h)]$.
Then, if $h\longrightarrow 0$ as $n\longrightarrow\infty$, the asymptotic
behaviour of $\widetilde{\lambda}_{n}$, proved in Theorem 2, entails that
$\lambda_{n,h}^{s}\longrightarrow\lambda$ strongly and uniformly in
$[0,x_{0}]$ for every $x_{0}$ such that $\lambda(x_{0})<\infty$. So, using the
same arguments as in the proof of Theorem 2,
$\mathbb{F}_{n}^{s}\longrightarrow F$ uniformly in $[0,\infty)$ and
$f_{n}^{s}\longrightarrow f$ strongly and uniformly in $[0,x_{0}]$.

As usual in kernel estimation problems, the optimal choice of the bandwidth is
the crucial issue. We are not dealing with this problem in the present paper,
as the involved construction of $\widetilde{\lambda}_{n}$ makes it very hard
to derive expressions for the MSE of $\lambda_{n,h}^{s}$. This estimator is
also quite demanding from a computational point of view. As an illustration,
Figure 3 shows $\lambda_{n,h}^{s}$ and $f_{n,h}^{s}$ for a sample of size
$n=20$ from a B2(2,3) distribution, using the Epanechnikov kernel with
bandwidth $h=\frac{1}{4}$, that is,
$k_{1/4}(x)=\frac{3}{4}\left(1-16x^{2}\right)$,
$x\in[-\frac{1}{4},\frac{1}{4}]$.

Figure 3: Simulated small sample ($n=20$) from the IOR distribution B2(2,3).
The plot on the left shows the ORs $\lambda_{F}$ (dot-dashed)
$\widetilde{\lambda}_{n}$ (solid) and $\lambda_{n,h}^{s}$ (dashed), for
$h=\frac{1}{4}$; the plot on the right shows $f$ (dot-dashed),
$\widetilde{f}_{n}$ (solid) and $f_{n,h}^{s}$ (dashed).

##  4 Tests for the IOR null hypothesis

Tests of different ageing properties, or, as discussed above, shape
restrictions, have been discussed extensively in the literature: among many
other authors, Proschan and Pyke (1967), Barlow and Proschan (1969), Bickel
(1969) or Bickel and Doksum (1969) consider the null hypothesis of
exponentiality versus the IHR alternative; Deshpande (1983), Kochar (1985),
Link (1989), Wells and Tiwari (1991), and Ahmad (1994), consider the null
hypothesis of exponentiality versus the IHRA alternative; while Tenga and
Santner (1984), Hall and Van Keilegom (2005), Durot (2008), Groeneboom and
Jongbloed (2012), Gijbels and Heckman (2004) or Lando (2022) consider the IHR
null hypothesis versus the alternative that the HR is non-monotone. This
approach, that is, setting the shape restriction as the null hypothesis, may
be more demanding from a computational point of view, but it has several
advantages in controlling the behaviour of the test. We follow this idea and
study tests of the null hypothesis $\mathcal{H}_{0}:F\in\mathcal{F}_{IOR}$
versus ${\mathcal{H}_{1}}:F\notin\mathcal{F}_{IOR}$. Tests of this kind have
already been studied in Lando et al. (2022a), who focused on a restricted null
hypothesis $\mathcal{H}_{0}^{\nu}:\text{``}\lambda\text{ is increasing in
}S_{\nu}\text{''}$, where $S_{\nu}=\\{x:x\leq F^{-1}(1-\nu)\\}$,
$\nu\in(0,1)$. The domain restriction was necessary to control the behaviour
of the test statistic, given by the maximum distance between the empirical
odds function
$\Lambda_{\mathbb{F}_{n}}=\frac{\mathbb{F}_{n}}{1-\mathbb{F}_{n}}$ and its GCM
$(\Lambda_{\mathbb{F}_{n}})_{cx}$, due to the lack of uniform strong
convergence of $\Lambda_{\mathbb{F}_{n}}$ and
$(\Lambda_{\mathbb{F}_{n}})_{cx}$ to $\Lambda_{F}$ under $\mathcal{H}_{0}$.
Differently, transporting the testing problem to the maximum distance between
the estimators $\overline{T}_{n}$, or ${\mathbb{F}}_{n}$, and their
corresponding constrained versions $(\overline{T}_{n})_{cx}$, or
$\widetilde{\mathbb{F}}_{n}$, respectively, means that, under
$\mathcal{H}_{0}$, we do have uniform strong convergence in the whole domain,
hence restrictions are no longer necessary.

###  4.1 A test based on the convexity of $\overline{T}$

Similarly to Tenga and Santner (1984), who deal with the IHR property, we may
obtain a first test which detects departures from the IOR property by
considering a suitable distance between $\overline{T}_{n}$ and
$(\overline{T}_{n})_{cx}$, or, similarly, $\overline{T}^{-1}_{n}$ and
$(\overline{T}_{n}^{-1})_{cv}$. We will consider the following scale-
independent test statistic

${\rm
KT}(\mathbb{F}_{n})=\sup_{u\in[0,1]}|\overline{T}_{n}(u)-(\overline{T}_{n})_{cx}(u)|=\max_{1\leq
i\leq
n}|\frac{i}{n}-(\overline{T}_{n})_{cx}\circ\overline{T}_{n}^{-1}(\tfrac{i}{n})|.$

It can be seen that, if the jump points of the empirical odds function
$\Lambda_{\mathbb{F}_{n}}$, determined by the coordinates
$(X_{(i)},\frac{i}{n-1})$, $i=1,\ldots,n-1$, lay on a convex curve, then ${\rm
KT}(\mathbb{F}_{n})=0$.

To obtain a conservative test, the determination of the least favourable
distribution of ${\rm KT}$ under the null hypothesis is especially critical.
As established in the following theorem, owing to a stochastic ordering
result, such a distribution coincides with the distribution of ${\rm KT}$ in
the case when the data are randomly sampled from the LL(1). To understand this
behaviour, we need to introduce some additional notations. Let us denote
$\overline{Z}_{F}=\frac{Z_{F}}{T^{-1}_{F}(1)}$, so that
$\overline{T}_{F}=F\circ\overline{Z}_{F}^{-1}$, and define the functional

$\Phi_{p}(F)=(F\circ\overline{Z}_{F}^{-1})_{cx}\circ\overline{T}_{F}^{-1}(p)=(\overline{T}_{F})_{cx}\circ\overline{T}_{F}^{-1}(p).$

The empirical counterpart of $\Phi_{p}(F)$, for $p=\frac{i}{n}$, is

$\Phi_{i/n}(\mathbb{F}_{n})=(\mathbb{F}_{n}\circ\overline{Z}_{\mathbb{F}_{n}}^{-1})_{cx}\circ\overline{T}_{\mathbb{F}_{n}}^{-1}(\tfrac{i}{n})=(\overline{T}_{\mathbb{F}_{n}})_{cx}\circ\overline{T}_{\mathbb{F}_{n}}^{-1}(\tfrac{i}{n}).$

Let us recall that $X$ is larger than $Y$ in the usual stochastic order,
denoted as $X\geq_{st}Y$, if $P(X\geq t)\geq P(Y\geq t)$, for every $t$
(Shaked and Shanthikumar, 2007). We can now establish the following result.

######  Theorem 3.

Let $\mathbb{L}_{n}$ be the empirical CDF corresponding to a random sample
from $L$. Under $\mathcal{H}_{0}$, ${\rm KT}(\mathbb{L}_{n})\geq_{st}{\rm
KT}(\mathbb{F}_{n})$.

###### Proof.

Let $L_{n}$ be a realization of $\mathbb{L}_{n}$, corresponding to the
observed sample $(y_{(1)},\ldots,y_{(n)})$. Since ${L}\geq_{c}F$, Theorem 1 of
Lando et al. (2022b) implies that
$\Phi_{i/n}(\mathbb{F}_{n})\geq_{st}\Phi_{i/n}(\mathbb{L}_{n})$, for
$i=1,\ldots,n$, provided that

$(\overline{T}_{F^{*}_{n}})_{cx}\circ\overline{T}_{F^{*}_{n}}^{-1}(\tfrac{i}{n})=\Phi_{i/n}(F_{n}^{*})\geq\Phi_{i/n}({L}_{n})=(\overline{T}_{L_{n}})_{cx}\circ\overline{T}_{L_{n}}^{-1}(\tfrac{i}{n}),$
(4)

where $F_{n}^{*}=F^{-1}\circ L\circ L_{n}=\tau\circ L_{n}$ is the empirical
CDF corresponding to the values $\tau(y_{(i)})$, which determine an ordered
sample from $F$. Let

$u_{i}\\!=\\!\overline{T}^{-1}_{L_{n}}(\tfrac{i}{n})\\!=\\!\frac{\sum_{j=1}^{i}(\frac{n-j+1}{n})^{2}(y_{(j)}-y_{(j-1)})}{\sum_{j=1}^{n}(\frac{n-j+1}{n})^{2}(y_{(j)}-y_{(j-1)})}\;\mbox{and}\;u^{*}_{i}\\!=\\!\overline{T}_{F^{*}_{n}}^{-1}(\tfrac{i}{n})\\!=\\!\frac{\sum_{j=1}^{i}(\frac{n-j+1}{n})^{2}(\tau(y_{(j)})-\tau(y_{(j-1)}))}{\sum_{j=1}^{n}(\frac{n-j+1}{n})^{2}(\tau(y_{(j)})-\tau(y_{(j-1)}))}.$

The denominators
$\theta=\sum_{j=1}^{n}(\frac{n-j+1}{n})^{2}(y_{(j)}-y_{(j-1)})$ and
$\eta=\sum_{j=1}^{n}(\frac{n-j+1}{n})^{2}(\tau(y_{(j)})-\tau(y_{(j-1)}))$ are
constant because the samples are fixed. We now define the function
$h:[0,1]\rightarrow[0,1]$ by

$h(u_{i})=h\left(\frac{1}{\theta}{\sum_{j=1}^{i}(\frac{n-j+1}{n})^{2}(y_{(j)}-y_{(j-1)})}\right)=\frac{1}{\eta}{\sum_{j=1}^{i}(\frac{n-j+1}{n})^{2}(\tau(y_{(j)})-\tau(y_{(j-1)}))}=u^{*}_{i},$

and by linear interpolation between the $u_{i}$’s values. The increments may
be expressed as
$u_{i}-u_{i-1}=\frac{1}{\theta}(\frac{n-j+1}{n})^{2}(y_{(i)}-y_{(i-1)})$ and
$u^{*}_{i}-u^{*}_{i-1}=\frac{1}{\eta}(\frac{n-j+1}{n})^{2}(\tau(y_{(i)})-\tau(y_{(i-1)}))$,
respectively. To prove (4), note that, under $\mathcal{H}_{0}$, the function
$\tau$ is increasing and concave, hence

$\frac{h(u_{i+1})-h(u_{i})}{u_{i+1}-u_{i}}=\frac{\theta}{\eta}\frac{\tau(y_{i+1})-\tau(y_{i})}{y_{i+1}-y_{i}}\leq\frac{\theta}{\eta}\frac{\tau(y_{i})-\tau(y_{i-1})}{y_{i}-y_{i-1}}=\frac{h(u_{i})-h(u_{i-1})}{u_{i}-u_{i-1}}.$

Therefore the slopes of the piecewise linear function $h$ are decreasing, that
is, $h$ is concave. Now, Theorem 2.2 of Tenga and Santner (1984) yields (4).
Subsequently, as
$\Phi_{i/n}(\mathbb{F}_{n})\geq_{st}\Phi_{i/n}(\mathbb{L}_{n})$, ${\rm
KT}(\mathbb{F}_{n})=\max_{i}(\frac{i}{n}-\Phi_{i/n}(\mathbb{F}_{n}))$ and
similarly ${\rm
KT}(\mathbb{L}_{n})=\max_{i}(\frac{i}{n}-\Phi_{i/n}(\mathbb{L}_{n}))$, we
obtain the desired result. ∎

Theorem 3 states that the random variable ${\rm KT}(\mathbb{L}_{n})$
represents a stochastic (upper) bound for ${\rm KT}(\mathbb{F}_{n})$ under the
null hypothesis. Accordingly, the problem boils down to testing
$\mathcal{H}_{0}^{L}:$ “$F$ is LL(1)” against $\mathcal{H}_{1}$, so we reject
$\mathcal{H}_{0}$ when ${\rm KT}(F_{n})\geq c_{\alpha,n}$, where
$c_{\alpha,n}$ is the solution of $P({\rm KT}(\mathbb{L}_{n})\geq
c_{\alpha,n})=\alpha$. Theorem 3 ensures that the probability of rejecting
$\mathcal{H}_{0}$ when it is true, is at most $\alpha$, that is, the test has
size $\alpha$ under $\mathcal{H}_{0}$. Similarly, if $F$ is DOR, it is easy to
see that $P({\rm KT}(\mathbb{F}_{n})\geq c_{\alpha,n})\geq P({\rm
KT}(\mathbb{L}_{n})\geq c_{\alpha,n})=\alpha$, that is, the test is unbiased
under DOR alternatives. For a given realisation $F_{n}$, the $p$-value of the
test is $p=P({\rm KT}(\mathbb{L}_{n})\geq{\rm KT}(F_{n}))$.

As established by the following theorem, ${\rm KT}$ is also capable of
detecting any deviation from the IOR null hypothesis.

######  Theorem 4.

Under $\mathcal{H}_{1}$, $\lim_{n\rightarrow\infty}P({\rm
KT}(\mathbb{F}_{n})>c_{\alpha,n})=1$.

###### Proof.

If $\mathcal{H}_{0}$ is true, considering the special case when
$\overline{T}_{n}$ is obtained by sampling from the LL(1), $\overline{T}_{n}$
and $(\overline{T}_{n})_{cx}$ converge strongly and uniformly to the identity
function. In particular, Marshall’s inequality gives
$\sup_{u\in[0,1]}|(\overline{T}_{n})_{cx}(u)-u|\leq\sup_{u\in[0,1]}|{\overline{T}}_{n}(u)-u|,$
with probability 1. Then, for every fixed $\alpha\in(0,1)$, there exists some
$n_{0}$ such that, for $n>n_{0}$

$P(\sup_{u}|\overline{T}_{n}(u)-u|\leq\frac{\epsilon}{2}\land\sup_{u}|(\overline{T}_{n})_{cx}(u)-u|\leq\frac{\epsilon}{2})\geq
1-\alpha$

However, the function $u-\frac{\epsilon}{2}$ is convex, therefore, for
$n>n_{0},$
$u-\frac{\epsilon}{2}\leq(\overline{T}_{n})_{cx}(u)\leq\overline{T}_{n}(u)\leq
u+\frac{\epsilon}{2},\forall u$, and by inclusion we obtain

$P(-\epsilon\leq\overline{T}_{n}(u)-(\overline{T}_{n})_{cx}(u)\leq\epsilon,\forall
u)=P(\sup_{u}(\overline{T}_{n}(u)-(\overline{T}_{n})_{cx}(u))\leq\epsilon)\geq
1-\alpha.$

Since $\epsilon$ can be arbitrarily small, $c_{\alpha,n}\longrightarrow 0$ for
$n\longrightarrow\infty$.

Suppose that $\mathcal{H}_{1}$ is true. Then
$d=\sup_{u}(\overline{T}(u)-(\overline{T}_{n})_{cx}(u))>0$. Moreover,
$\overline{T}_{n}$ converges strongly and uniformly to $\overline{T}$, whereas
$(\overline{T}_{n})_{cx}$ converges strongly and uniformly to
$\overline{T}_{cx}$ (this can be seen using the same argument as in Theorem 3
of Lando (2022)). Therefore, given some $\epsilon>0$, there exists some
$n_{0}$ such that, for $n>n_{0}$,
$\sup_{u}|\overline{T}(u)-\overline{T}_{n}(u)|<\frac{\epsilon}{2}$ and
$\sup|(\overline{T}_{n})_{cx}(u)-\overline{T}_{cx}(u)|<\frac{\epsilon}{2}$,
with probability 1. Then, for $n>n_{0}$

$\overline{T}_{n}(u)-(\overline{T}_{n})_{cx}(u)>\overline{T}(u)-\frac{\epsilon}{2}-(\overline{T}_{cx}(u)+\frac{\epsilon}{2})=\overline{T}(u)-\overline{T}_{cx}(u)-\epsilon$

almost surely, for every $u$, which implies

$\sup_{u}(\overline{T}_{n}(u)-(\overline{T}_{n})_{cx}(u))>\sup_{u}(\overline{T}(u)-\overline{T}_{cx}(u)-\epsilon)=d-\epsilon>0.$

Therefore, since $\epsilon$ can be arbitrarily small,
$P(\sup_{u}(\overline{T}_{n}(u)-(\overline{T}_{n})_{cx}(u))\geq
d)\longrightarrow 1$, for $n\longrightarrow\infty$. But since
$c_{\alpha,n}\rightarrow 0$, then $P({\rm
KT}(\mathbb{F}_{n})>c_{\alpha,n})\longrightarrow 1$. ∎

It is important to remark that one may replace the sup-norm with a different
type of distance between $(\overline{T}_{n})_{cx}$ and $\overline{T}_{n}$ and
still obtain a consistent test, since the key steps in the proof of Theorem 4
are the uniform consistency of these two estimators under $\mathcal{H}_{0}$,
while only $\overline{T}_{n}$ converges to $\overline{T}$ under
$\mathcal{H}_{1}$.

###  4.2 A test based on $\widetilde{\mathbb{F}}_{n}$

A second test can be obtained by considering a suitable distance between
$\mathbb{F}_{n}$ and $\widetilde{\mathbb{F}}_{n}$, such as the uniform norm.
Let us define the following Kolmogorov-Smirnov type test statistic

${\rm
KS}({\mathbb{F}}_{n})=\sup_{x>0}|\mathbb{F}_{n}(x)-\widetilde{\mathbb{F}}_{n}(x)|.$

About the IHR property, a Kolmogorov-Smirnov test of this type has been
studied by Lando (2022). Note that ${\rm KS}$ is also scale-independent.
However, if the jump points of $\Lambda_{\mathbb{F}_{n}}$ lay on a convex
curve, then ${\rm KS}(\mathbb{F}_{n})$ does not coincide with 0, differently
from ${\rm KT}(\mathbb{F}_{n})$. To fulfil this property, one should consider
a modified version of ${\rm KS}$, say $\widehat{{\rm
KS}}(\mathbb{F}_{n})=\sup_{x>0}|\widehat{\mathbb{F}}_{n}(x)-\widetilde{\mathbb{F}}_{n}(x)|$,
where $\widehat{\mathbb{F}}_{n}$ has the same construction of
$\widetilde{\mathbb{F}}_{n}$, as described in Section 3, with the exception
that it does not include any shape constraint, namely,

$\widehat{\mathbb{F}}_{n}(x)=L\left(\int_{0}^{x}\frac{1}{\partial_{+}(T_{n}^{-1})\circ\mathbb{F}_{n}(t)}dt\right).$

$\widehat{\mathbb{F}}_{n}$ and $\mathbb{F}_{n}$ are very similar and it is
easy to see that they are asymptotically equivalent, moreover, the computation
of $\widehat{\mathbb{F}}_{n}$ is quite demanding without having an improvement
on the power of the test. Therefore, henceforth we will focus only on ${\rm
KS}$.

As in the previous subsection, the distribution of ${\rm KS}$ may be
determined by simulating from $L$. However, in this case, the stochastic
ordering arguments used in Theorem 3 do not hold, because of the complicated
construction of $\widetilde{\mathbb{F}}_{n}$. Accordingly, we cannot formally
establish a stochastic bound for the size of the test under $\mathcal{H}_{0}$,
so it is more correct to present this as a test for $\mathcal{H}^{L}_{0}$
against $\mathcal{H}_{1}$. We reject the null $\mathcal{H}^{L}_{0}$ when ${\rm
KS}(F_{n})\geq c_{\alpha,n}$, where $c_{\alpha,n}$ is the solution of $P({\rm
KS}(\mathbb{L}_{n})\geq c_{\alpha,n})=\alpha$. However, even in some critical
IOR cases, numerical evidence, reported in the next section, shows that the
type-I error of ${\rm KS}$ is always smaller than $\alpha$. Moreover, the test
is consistent against non-IOR alternatives, as established by the following
theorem. The proof is omitted because it can be obtained using the same
arguments as in the proof of Theorem 3 of Lando (2022).

######  Theorem 5.

Under $\mathcal{H}_{1}$, $\lim_{n\rightarrow\infty}P({\rm
KS}(\mathbb{F}_{n})>c_{\alpha,n})=1$.

Similarly to what has been discussed in the previous subsection, one may
replace the sup-norm with a different type of distance between
$\widetilde{\mathbb{F}}_{n}$ and ${\mathbb{F}}_{n}$, and the corresponding
test would still be consistent.

###  4.3 Simulations

We compare the performance of the two tests proposed simulating from some
popular parametric families of distributions. As these tests are scale
invariant, we set the scale parameters to 1. Some special cases of interest
are (i) IOR models, (ii) decreasing OR (DOR) models, (iii) non-monotone OR
models. As for (i), we consider the LL($a$), with shape parameter $a$ ranging
in the interval $[1,1.2]$. With regard to (ii), we consider the LL($a$) with
$a\in[0.7,1)$. These cases are especially difficult to detect because, for
$a=1$, the OR is constant. As for (iii), we consider the W($a$) with shape
parameter $a\in[0.3,0.8]$, which exhibits a decreasing-increasing OR for
$a<1$; the B2 distribution, which has an increasing-decreasing OR for $a<1$
and $b>1$, (thus we consider $b=2$, $a\in[0.3,0.7]$); and the Birnbaum-
Saunders (BS) distribution with CDF
$\Phi\left(\frac{1}{a}\left(\sqrt{x}-\frac{1}{\sqrt{x}}\right)\right)$,
$x,a>0$ ($\Phi$ denotes the standard normal CDF), which has an increasing-
decreasing-increasing OR for $a\in[2,4]$ (this is especially critical to
detect for smaller values of $a$). The results are reported in Figure 4, which
show the rejection rates at level $\alpha=0.1$, corresponding to 500
simulation runs. The plots show that the performance of the tests is very
similar when $\mathcal{H}_{0}$ is true (LL($a$) with $a\geq 1$) and against
DOR alternatives (LL($a$) with $a\leq 1$). In particular, it can be seen that,
for both tests, in the IOR case the simulated type-I error probability is
always bounded by $\alpha$, whereas, in the DOR case, the simulated power is
always greater than $\alpha$. About ${\rm KT}$, this is formally established
by Theorem 3, which represents an advantage. However, it can be seen from
Figure 4 that ${\rm KS}$ remarkably outperforms ${\rm KT}$ when $F$ has a non-
monotone OR, which is typically the most critical case to detect. For both
tests, the simulated power increases with the sample size, confirming the
consistency properties established in Theorem 4 and Theorem 5.

Figure 4: Simulated power of ${\rm KS}$ (solid) and ${\rm KT}$ (dashed) under
LL (first row), W (second row), B2 (third row), and BS (fourth row)
alternatives.

To illustrate the behaviour of ${\rm KT}$ and ${\rm KS}$ under non-IOR
alternatives, we also consider the following CDF,

$F_{a,b}(x)=\begin{cases}\frac{x^{a}}{x^{a}+1}&0<x\leq 1,\\\
\frac{x^{b}}{x^{b}+1}&x>1,\end{cases}\qquad a,b>0.$

For $a,b\geq 1$, the OR of this distribution is increasing almost everywhere,
however, for $a>b$, it has a downward jump at $x=1$, so that $F_{a,b}$ cannot
be IOR. Figure 5 shows the remarkable distance between $\overline{T}_{n}$ and
$(\overline{T}_{n})$, and between $\mathbb{F}_{n}$ and
$\widetilde{\mathbb{F}}_{n}$, for a simulated sample of size 100 from
$F_{5,1}$. In both cases, these large distances lead to the rejection of
$\mathcal{H}_{0}$.

Figure 5: Simulated sample ($n=50$) from the non-IOR CDF $F_{5,1}$.
$\overline{T}_{n}$ and $(\overline{T}_{n})$ on the left side; $\mathbb{F}_{n}$
and $\widetilde{\mathbb{F}}_{n}$ on the right side. In both cases, the two
estimators (constrained and unconstrained) diverge for quantile values larger
than 0.2, approximately.

Funding. T.L. was supported by the Italian funds ex MURST 60% 2021, by the
Czech Science Foundation (GACR) under project 20-16764S and VŠB-TU Ostrava
under the SGS project SP2021/15. I.A. and P.E.O. were partially supported by
the Centre for Mathematics of the University of Coimbra UID/MAT/00324/2020,
funded by the Portuguese Government through FCT/MCTES and co-funded by the
European Regional Development Fund through the Partnership Agreement PT2020.

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