1. Introduction
Given a random sample from a probabilistic model, the largest and smallest order statistics play a fundamental role in statistical inference, especially when the domain of the probabilistic model that drives the data depends on an unknown parameter of interest. On the other hand, order statistics, in general, are the fundamental tools in reliability as they are usually related to the failure time of a tested item. Moreover, the largest order statistic plays a fundamental role in extreme value theory, and it is the basis for developing the extreme value distributions in cases where the size of the random sample approaches infinity. On the other hand, Shannon entropy of a continuous distribution is an omnipresent, global measure that quantifies and expresses the degree of uncertainty about the outcome of a random experiment. Large values of Shannon entropy are associated with high uncertainty or low information gain, which can be thought of as the purity of a system (see also [Reference Zaid, Attwa and Radwan33]). In this context, it would be of interest to study the behavior of the Shannon entropy and the extropy when they are applied to the distribution of the largest order statistic in the case of a finite or infinite number of independent and identically distributed (i.i.d.) observations.
To formulate the work, consider a random variable X with a continuous probability distribution function F and a respective probability density function f. The Shannon entropy associated with the distribution with density f is defined by
\begin{equation}\mathcal{H}(X)=-\int_{\mathbb{R}}f(x)\ln f(x)\,\mathrm{d}x. \end{equation}
It is a well-known and broadly applied quantity with applications in almost every field of science and engineering. On the other hand, the extropy of a random variable has received considerable attention in the last decade in the field of statistical information theory. In view of [Reference Zografos34, p. 296], and references therein, the extropy of the random variable X is defined by
\begin{equation}\mathcal{J}(X)=-\frac{1}{2}\int_{\mathbb{R}}f^{2}(x)\,\mathrm{d}x, \end{equation}
and it has been introduced in the statistical literature as the complementary dual of Shannon entropy, which is discussed and explained in detail in Section 4 in [Reference Lad, Sanfilippo and Agró19]. This measure is directly connected to other informational quantities, such as Onicescu information energy or the Golomb information function, discussed in [Reference Zografos34], and it is also related to the Rényi and Tsallis entropies for a specific value of the index appearing in their definitions. More information about extropy can be obtained from [Reference Lad, Sanfilippo and Agró19], [Reference Qiu25], [Reference Toomaj, Hashempour and Balakrishnan31], and the references therein.
In what follows, we consider a random sample
$X_{1},X_{2},\ldots,X_{n}$
from a continuous probability distribution function F with respective probability density function f, and let
$X_{(1)}\leq X_{(2)}\leq \cdots\leq X_{(n)}$
be the ordered sample, with
$X_{(n)}=\max \{X_{1},X_{2},\ldots,X_{n}\}$
the largest order statistics.
$X_{(n)}$
is a fundamental quantity in statistical inference and in reliability theory because it quantifies the greatest failure time of a set of items under test. It is well known that the density of
$X_{(n)}$
is given by
\begin{equation}f_{X_{(n)}}(x)=nF^{n-1}(x)f(x), \end{equation}
and it can be used to obtain the entropy and the extropy associated with the distribution of
$X_{(n)}$
, say,
$\mathcal{H}(X_{(n)})$
and
$\mathcal{J}(X_{(n)})$
. Because of the importance of
$X_{(n)}$
in statistical practice, several papers have been devoted to the study of the Shannon entropy and the extropy of order statistics. For example, Wong and Chen [Reference Wong and Chen32] studied the symmetry around the median of ordered statistics, while Baratpour et al. [Reference Baratpour, Ahmadi and Arghami3], Kayal [Reference Kayal17], and Shrahili and Kayid [Reference Shrahili and Kayid30] concentrated on several aspects of information for order statistics. A list of references for the existing literature on this subject is given in [Reference Qiu25], where some characterization results and bounds of the extropy of order statistics and record values are presented (see [Reference Qiu and Jia26]). Several properties of the information generating function of k-record values are studied in [Reference Chacko and Grace10], while an extropy-based inaccuracy measure of order statistics is presented and studied in [Reference Mohammadi and Hashempour21]. All the aforementioned papers and the references therein are gathered from the existing literature on some measures of information and order statistics and records.
Although a number of studies are available on analytic results and characterizations of several measures of information when they are applied in the distribution of order statistics, to the best of our knowledge no study has developed bounds of the Shannon entropy and the extropy of the largest order statistic and the respective normalized maxima. This is the gap that the present work is trying to fill.
In view of the above exposition, it is natural to be interested in the behavior of
$\mathcal{H}(X_{(n)})$
and
$\mathcal{J}(X_{(n)})$
, and their respective normalized versions, for finite sample size n and also when
$n\rightarrow \infty $
, just as it is customary in extreme value theory to seek the possible limit distributions of
$X_{(n)}$
(see [Reference Beirlant, Goegebeur, Teugels and Segers4, p. 46]). In this context, the interest is focused on the normalizing sequence
$\frac{X_{(n)}-b_{n}}{a_{n}}$
for some suitable sequences
$a_{n}$
and
$b_{n},n\geq 1$
, with
$a_{n}\gt 0$
, associated with
$X_{(n)}$
, when the i.i.d. random variables
$X_{1},X_{2},\ldots,X_{n}$
are governed by a log-concave distribution. In this setting, bounds for the Shannon entropy and the extropy of the normalizing sequence are derived in Section 2, where first the basic definitions and notation used in the subsequent sections are presented. A characterization of the exponential distribution is provided, and it is shown that it is the model that maximizes the Shannon entropy and the extropy of the sequence
$\frac{X_{(n)}-b_{n}}{a_{n}}$
when
$n\rightarrow \infty $
. Moreover, the convergence of
$\mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) $
and
$\mathcal{J}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) $
to the respective Shannon entropy and the extropy of the Gumbel distribution, discussed in [Reference Johnson16], [Reference Ravi and Saeb27], and [Reference Saeb29], is implicitly derived. Bounds for
$\mathcal{H}(X_{(n)})$
and
$\mathcal{J}(X_{(n)})$
and their limiting behavior are studied in Section 3, and a similar characterization of the exponential distribution is also provided. Conclusions are given in Section 4, and Appendix A provides the proofs of the results. In summary, this paper presents bounds for the Shannon entropy and the extropy of the largest order statistic and its associated normalized maxima when the sample, of finite or infinite size n, is from a log-concave density. The case of a large sample size (
$n\rightarrow \infty $
) is also examined in the context of extreme value theory, and a characterization of the exponential distribution is also provided in this setting.
2. Bounds for Shannon entropy and extropy in the extreme value regime
2.1. Shannon entropy and extropy of the largest order statistic
Before we proceed to the main results of this section, we introduce the key quantities that will be used in the sequel. In view of the above formulation, an application of (1) in the specific case (3) leads to the Shannon entropy of
$X_{(n)}$
, which can be obtained also by applying Corollary 1 in [Reference Zografos and Balakrishnan35], for
$\alpha =n$
and
$\beta=1$
. Then the Shannon entropy of
$X_{(n)}$
is given by
\begin{equation*}\mathcal{H}(X_{(n)})=\ln B(n,1)-(n-1)[\Psi (n)-\Psi (n+1)]-E_{Y}[\!\ln f(F^{-1}(Y))],\end{equation*}
where Y is a random variable that follows a beta distribution
$Y\sim\mathrm{Beta}(n,1)$
and
$B(\alpha,\beta )=\Gamma (\alpha )\Gamma (\beta )/\Gamma(\alpha +\beta ),$
with B and
$\Gamma $
being the beta and gamma functions, respectively. In this context, if we denote
\begin{equation}I(Y)=f(F^{-1}(Y)),\qquad Y\sim \mathrm{Beta}(n,1), \end{equation}
then
$\mathcal{H}(X_{(n)})$
is simplified as follows:
\begin{equation}\mathcal{H}(X_{(n)}) =1-\ln n-\frac{1}{n}-E_{Y}[\!\ln I(Y)],\end{equation}
or
\begin{equation}\mathcal{H}(X_{(n)}) =1-\ln n-\frac{1}{n}-\int _{0}^{1}ny^{n-1}\ln f(F^{-1}(y))\,\mathrm{d}y, \end{equation}
taking into account that for the digamma function
$\Psi $
it holds that
$\Psi (n+1)-\Psi (n)=\frac{1}{n}$
. Here
$\mathcal{H}(X_{(n)})$
in (5) or (6) represents the Shannon entropy of the extreme value distribution of the maximum
$X_{(n)}$
of a sequence of i.i.d. data
$X_{1},X_{2},\ldots,X_{n}$
.
The real function,
\begin{equation}I(t)=f(F^{-1}(t)),\qquad 0\lt t\lt 1, \end{equation}
is motivated by (4), and will play a fundamental role in the derivation of the main results in the following sections. It is the density quantile function, defined on p. 84 in [Reference David and Nagaraja12], and it has been investigated and exploited in a series of publications by Bobkov and his colleagues; see, for example, [Reference Bobkov5], [Reference Bobkov6], [Reference Bobkov and Ledoux7], and [Reference Bobkov and Madiman8]. If the density f, included in its definition, is log-concave then the function I, which is defined by (7), obeys nice properties, such as positiveness and concavity. In view of the said properties of I, the theoretical results that follow formulate bounds for the Shannon entropy
$\mathcal{H}(X_{(n)})$
, the extropy
$\mathcal{J}(X_{(n)})$
, and the respective normalized versions. In the statements of the said results, some well-known results from [Reference Abramowitz and Stegun1, pp. 255 and 258] and [Reference Mathai20, pp. 4 and 11], among other publications, will be used. In this frame,
\begin{equation}\gamma =\lim_{n\rightarrow \infty }(H_{n}-\ln n)=\lim_{n\rightarrow \infty}\left(\sum_{k=1}^{n}\frac{1}{k}-\ln n\right) \end{equation}
is the Euler constant, while
$H_{n}$
is the nth harmonic number, defined by
\begin{equation}H_{n}=\sum_{k=1}^{n}\frac{1}{k}. \end{equation}
In a similar manner, the extropy of a random variable is defined by (2), and it has been studied by Lad et al. [Reference Lad, Sanfilippo and Agró19]. In view of (2) and (3), the extropy of
$X_{(n)}$
is given by
\begin{equation}\mathcal{J}(X_{(n)})=-\frac{1}{2}\int_{\mathbb{R}}f_{X_{(n)}}^{2}(x)\,\mathrm{d}x=-\frac{n^{2}}{2}\int_{\mathbb{R}}F^{2n-2}(x)f^{2}(x)\,\mathrm{d}x. \end{equation}
Assuming that F is strictly increasing and using the transformation
$t=F(x) $
,
$0\lt t\lt 1,$
and
$x=F^{-1}(t)$
, where
$F^{-1}$
is the inverse of F and
$\frac{\mathrm{d}x}{\mathrm{d}t}=\frac{1}{f\left(F^{-1}(t)\right) }$
, we find that (10) leads to
\begin{equation*}\mathcal{J}(X_{(n)})=-\frac{n^{2}}{2}\int _{0}^{1}t^{2n-2}[f(F^{-1}(t)) ] ^{2}\frac{1}{f\left(F^{-1}(t)\right) }\,\mathrm{d}t,\end{equation*}
or
\begin{equation}\mathcal{J}(X_{(n)})=-\frac{n^{2}}{2}\int _{0}^{1}t^{2n-2}I(t)\,\mathrm{d}t.\end{equation}
2.2. Extreme value consideration
Although the distribution of the maximum of n,
$n\lt \infty,$
i.i.d. random variables is well defined in (3), the limit of
$X_{(n)}$
may degenerate when
$n\rightarrow \infty $
. To avoid this degeneracy, we consider the normalizing sequence
$({X_{(n)}-b_{n}})/{a_{n}}$
for some suitable sequences
$a_{n}$
and
$b_{n}$
,
$n\geq1$
, with
$a_{n}\gt 0$
. In this direction, it is well known from the area of the extreme value distributions that
$({X_{(n)}-b_{n}})/{a_{n}}$
converges in distribution to some non-degenerate limiting distribution G if and only if the distribution F is in the domain of attraction of the distribution G, which is denoted by
$F\in \mathcal{D}(G)$
(see [Reference Beirlant, Goegebeur, Teugels and Segers4, pp. 46–47] or [Reference Kotz and Nadarajah18, pp. 7–9]). The study of the behavior of Shannon entropy, extropy, and other information measures of the largest order statistic
$X_{(n)}$
and its normalized version has been considered in the recent literature, especially when
$n\rightarrow \infty $
. Recent advances on this topic include the work by Zaid et al. [Reference Zaid, Attwa and Radwan33], where the interest is focused on some measures of information when they are applied in the q-generalized extreme value distributions. Work by Saeb [Reference Saeb29] (see also [Reference Ravi and Saeb27]) concentrates on the convergence of the Shannon entropy of the normalized maxima of a random sample to the Shannon entropy of the limit max-stable law, and moreover shows that this convergence is equivalent to the convergence of the relative entropy to zero, under some conditions. Johnson [Reference Johnson16] recently introduced a type of score function—namely, the max-score function—that is used to prove convergence to the Gumbel distribution in the strong sense of relative entropy. As stated in the introduction in [Reference Johnson16], it provides, in contrast with the work by Ravi and Saeb [Reference Ravi and Saeb27] and Saeb [Reference Saeb29], ‘an elementary and direct proof under relatively simple conditions which hopefully gives some insight into why convergence to the Gumbel distribution occurs, rather than to necessarily provide the strongest possible result.’ In view of the above exposition, it is reasonable to study, in the sequel, the behavior of the Shannon entropy and the extropy of the normalizing sequence
$\frac{X_{(n)}-b_{n}}{a_{n}}$
, not only for
$n\rightarrow \infty $
but also in the case of finite sample size n. However, first, we present some prerequisites from the theory of extreme value distributions for the sake of completeness of the presentation and the results.
Given a random sample
$X_{1},X_{2},\ldots,X_{n}$
of size n from the continuous probability distribution function F, the normalizing sequence
$\frac{X_{(n)}-b_{n}}{a_{n}}$
converges in distribution to some non-degenerate limiting distribution G for some sequences
$a_{n}$
and
$b_{n}$
,
$n\geq 1$
, with
$a_{n}\gt 0$
, if and only if
$F\in \mathcal{D}(G)$
. Therefore, if
$F\in \mathcal{D}(G)$
then
\begin{equation}\lim_{n\rightarrow \infty }P\left(\frac{X_{(n)}-b_{n}}{a_{n}}\leq x\right)=G_{\xi }(x), \end{equation}
and in view of [Reference Beirlant, Goegebeur, Teugels and Segers4, p. 47] or [Reference de Haan and Ferreira13, p. 6], if
$F\in \mathcal{D}(G)$
then all extreme value distributions
\begin{equation}G_{\xi }(x)=\exp \left(\!-(1+\xi x\right) ^{-1/\xi }),\qquad 1+\xi x\gt 0,\, \xi \in\mathbb{R}, \end{equation}
can occur as limits of (12), where
$\xi $
is the extreme value index. For
$\xi =0$
, the right-hand side of (13) is interpreted as
\begin{equation}G_{0}(x)=\exp (\!-\mathrm{e}^{-x}),\qquad x\in\mathbb{R}. \end{equation}
The distribution function in (13) with respective density
\begin{equation}g_{\xi }(x)=(1+\xi x)^{-(\xi +1)/\xi }\exp (\!-(1+\xi x)^{-1/\xi }),\qquad 1+\xi x\gt 0,\, \xi \in\mathbb{R}, \end{equation}
is the well-known class of extreme value distributions, introduced, according to Kotz and Nadarajah [Reference Kotz and Nadarajah18, p. 61], by Jenkinson [Reference Jenkinson15], and it is also referred to as the von Mises type extreme value distributions or the von Mises–Jenkinson type distributions in [Reference Kotz and Nadarajah18]. The parameter
$\xi $
is the shape parameter of the model, and it may be used to model a wide range of tail behavior (see [Reference Kotz and Nadarajah18, p. 62] for more details). For
$\xi =0$
, the right-hand side of (15) is interpreted as
\begin{equation}g_{0}(x)=\exp (\!-(x+\mathrm{e}^{-x})),\qquad x\in\mathbb{R}. \end{equation}
In view of (12) and [Reference de Haan and Ferreira13, Corollary 1.2.4 and Remark 1.2.7, pp. 9–10 and 21–22], or [Reference Resnick28, Proposition 1.1, p. 40], or [Reference Kotz and Nadarajah18, pp. 8–9] and [Reference Beirlant, Goegebeur, Teugels and Segers4, p. 48, (2.2)], the extreme value distribution
$G_{\xi }$
of (13) and the norming constants
$a_{n}$
and
$b_{n}$
,
$n\geq 1$
, with
$a_{n}\gt 0$
, must be of one of the following three types:
-
(i)
$G_{1,\xi }(x)=\exp (\!-x^{-1/\xi })$
,
$x\gt 0,\, \xi \gt 0$
(Fréchet), with
$a_{n}=U(n)$
, with
$U(t)= F^{-1}\Big(1-\frac{1}{t}\Big)$
, and
$b_{n}=0.$
-
(ii)
$G_{2,\xi }(x)=\exp (\!-(\!-x)^{-1/\xi })$
,
$x\leq 0,\, \xi \lt 0$
(reversed Weibull), with
$a_{n}=x^{\ast }-U(n)=F^{-1}\left(1\right) -F^{-1}(1-\frac{1}{n})$
and
$b_{n}=x^{\ast }=F^{-1}(1)$
, with
$x^{\ast}=\sup \{x\,:\,F(x)\lt 1\}.$
-
(iii)
$G_{3,\xi }(x)=\exp (\!-\mathrm{e}^{-x})$
,
$x\in\mathbb{R},\, \xi =0$
(Gumbel distribution), with
$a_{n}=h(U(n))=h(F^{-1}(1-\frac{1}{n}) ) $
, with
$h(u)=\frac{1-F}{F^{\prime }}$
, and
$b_{n}=F^{-1}(1) -F^{-1}(1-\frac{1}{n}).$
Here
$F^{-1}\left(1\right) =x^{\ast }=\sup \{x\,:\,F(x)\lt 1\}$
, taking into account [Reference Bobkov and Ledoux7, p. 83]. The limiting behavior of the function
$U(n)=F^{-1}(1-\frac{1}{n}) $
is investigated in [Reference de Haan and Ferreira13, Lemma 1.2.9, p. 22], and
$\lim_{t\rightarrow \infty}U(t)\lt \infty $
if
$\xi \lt 0$
.
The following proposition investigates the log-concavity properties of
$g_{\xi }$
in (15), and it moreover provides some results related to the Shannon entropy and the extropy of the largest order statistic from the distribution in (13) or (15). The log-concavity of the extreme value distribution is studied in [Reference Müller and Rufibach22]. For the sake of completeness of the presentation, an elementary proof of the log-concavity and the whole proof of the following proposition are outlined in Appendix A.1.
Proposition 1. Let
$X_{1},X_{2},\ldots,X_{n}$
be a random sample of size n from the probability density function
$g_{\xi }$
in (15). Then we have the following.
-
(i) The generalized extreme value density function
$g_{\xi }$
, in (15), is log-concave for
$\xi =0$
or
$-1\lt \xi \lt 0$
. -
(ii) The function
$I(t)=g_{\xi }(G_{\xi }^{-1}(t)),$
$0\lt t\lt 1,$
in (7), for this distribution is given by and
\begin{equation*}I(t)=t(\!-\ln t)^{\xi +1},\qquad 0\lt t\lt 1,\, \xi \in\mathbb{R},\end{equation*}
\begin{equation*}I\left(\frac{1}{2}\right) =\frac{1}{2}(\!\ln 2)^{\xi +1},\qquad \xi \in\mathbb{R}.\end{equation*}
-
(iii) The Shannon entropy of
$X_{(n)}$
and the similar one of (15) are respectively
\begin{eqnarray*}\mathcal{H}\big(X_{(n)}\big) &\,=\,&1+\gamma +\xi \gamma +\xi \ln n,\qquad \xi \in\mathbb{R}, \\\mathcal{H}(G_{\xi }) &\,=\,&1+\gamma +\xi \gamma,\qquad \xi \in\mathbb{R}.\end{eqnarray*}
-
(iv) The extropy of
$X_{(n)}$
and the similar one of (15) are respectively
\begin{eqnarray*}\mathcal{J}\big(X_{(n)}\big) &\,=-\dfrac{\Gamma (\xi +2)}{2^{\xi +3}n^{\xi }},\qquad\xi \gt -2, \\[3pt]\mathcal{J}(G_{\xi }) &\,=-\dfrac{\Gamma (\xi +2)}{2^{\xi +3}},\qquad \xi \gt -2.\end{eqnarray*}
Remark 1. (a) Part (i) of the proposition investigates the log-concavity of the generalized extreme value distribution
$g_{\xi }$
in (15). The log-concavity for the case
$\xi =0$
is also included in Table 1 in [Reference Bagnoli and Bergstrom2]. The density
$g_{\xi }$
in (15) is log-concave for a specific range of values of the shape parameter
$\xi $
—namely, only in the case
$\xi =0$
or when
$-1\lt \xi \lt 0$
. This should not be surprising from the discussion about a number of non-regular situations associated with
$\xi $
provided on p. 62 in [Reference Kotz and Nadarajah18]. According to that discussion, ‘the experience with real-world data suggests that the condition
$-1/2\lt \xi \lt 1/2$
is almost always satisfied in practical applications – in particular in environmetrics’. This statement is also mentioned on p. 1442 in [Reference Müller and Rufibach22] to underline that the restriction of
$\xi =0$
or
$-1\lt \xi \lt 0$
is less restrictive than it seems at first sight. Hence, log-concavity of
$g_{\xi }$
is obeyed for values of
$\xi $
that are in harmony with applications of this model in practice.
(b) The explicit expression of
$\mathcal{H}(G_{\xi })$
is also provided by formula (19) in [Reference Zaid, Attwa and Radwan33] and in [Reference Saeb29]. The formula for
$\mathcal{J}(G_{\xi })$
does not appear in the literature to the best of the authors’ knowledge.
The following example is motivated by Example 1.2 in [Reference Johnson16], and it helps to illustrate the evaluation of the norming constant
$a_{n}$
,
$n\geq 1$
, with
$a_{n}\gt 0$
, in the case of an exponential distribution.
Example 1. Assume that a random sample
$X_{1},X_{2},\ldots,X_{n}$
of size n is obtained from an exponential distribution function
$F_{\theta }(x)=1-\mathrm{e}^{-\theta x}$
,
$x\gt 0$
. Then from [Reference Beirlant, Goegebeur, Teugels and Segers4, Table 2.3, p. 72], the exponential distribution is in the Gumbel domain with distribution function
$G_{0}(x)= \exp(\!-\mathrm{e}^{-x})$
,
$x\in\mathbb{R},\, \xi =0$
, according to (14). In this frame,
$F_{\theta }^{-1}(y)=-\frac{1}{\theta }\ln (1-y)$
and elementary algebraic manipulations lead to
\begin{equation*}a_{n}=h\left(F_{\theta }^{-1}\left(1-\frac{1}{n}\right) \right) =\frac{1-F_{\theta }\left(F_{\theta }^{-1}\left(1-\frac{1}{n}\right) \right) }{\theta \exp \left(\!-\theta F_{\theta }^{-1}\left(1-\frac{1}{n}\right)\right) }=\frac{1}{\theta }.\end{equation*}
Conversely, if
$a_{n}=\frac{1}{\theta }$
then
$\frac{1-F_{\theta }\left(F_{\theta }^{-1}\left(1-\frac{1}{n}\right) \right) }{\theta \exp \left(\!-\theta F_{\theta }^{-1}\left(1-\frac{1}{n}\right) \right) }=\frac{1}{\theta }$
or
$\exp \big(\!-\theta F_{\theta }^{-1}(1-y)\big) =y$
for
$y=\frac{1}{n}\in (0,1).$
The last equality entails that
$F_{\theta }^{-1}\left(z\right) =-\frac{1}{\theta }\ln (1-z)$
for
$z=1-y\gt 0$
, and then
$F_{\theta }(w)=1-\mathrm{e}^{-\theta w}$
,
$w\gt 0$
, which is an exponential distribution with parameter
$\theta \gt 0$
.
Having provided the above short exposition of extreme value distributions, for reasons given on pp. 46–47 in [Reference Beirlant, Goegebeur, Teugels and Segers4], we now concentrate on the behavior of the Shannon entropy
$\mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) $
and the extropy
$\mathcal{J}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) $
in two cases: when the sample size n is finite and when
$n\rightarrow \infty $
. In view of [Reference Beirlant, Goegebeur, Teugels and Segers4, p. 46], the standardization with
$b_{n}$
and
$a_{n}$
is natural in the case
$n\rightarrow \infty $
, since otherwise
$X_{(n)}\rightarrow x^{\ast }$
. On the other hand, it is well known from information-theoretic derivations that the Shannon entropy and the extropy are location-free but scale-dependent in the sense
\begin{eqnarray*}\mathcal{H}(cX+d) &\,=\,&\ln c+\mathcal{H}(X), \\\mathcal{J}(cX+d) &\,=\,&\frac{1}{c}\mathcal{J}(X),\end{eqnarray*}
with
$c,d\in\mathbb{R},\,c\gt 0,$
in view of Theorems 8.6.3 and 8.6.4 in [Reference Cover and Thomas11, pp. 253–254] and [Reference Toomaj, Hashempour and Balakrishnan31, p. 1337].
Hence, taking into account the above mentioned location-free but scale-dependent behavior of
$\mathcal{H}$
and
$\mathcal{J}$
, we immediately see that
\begin{equation}\mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) =-\ln a_{n}+\mathcal{H}\big(X_{(n)}\big), \end{equation}
and therefore
$\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})$
is directly connected with the limit, as
$n\rightarrow \infty $
, of the left-hand side of (17) and the similar limit of the sequence
$a_{n}$
as well. On the other hand, (5) and (17) lead to
\begin{equation*}\mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) =1-\ln a_{n}-\ln n-\frac{1}{n}-E_{Y}[\!\ln I(Y)], \end{equation*}
where I(Y),
$Y\sim \mathrm{Beta}(n,1)$
, is defined by (4). In a similar manner, the extropy
$\mathcal{J}$
satisfies
\begin{equation}\mathcal{J}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) =a_{n}\mathcal{J}(X_{(n)}) \end{equation}
for sequences
$\{a_{n}\,:\, n\geq 1\}$
and
$\{b_{n}\,:\, n\geq 1\}$
, with the first a sequence of positive numbers, and
$\mathcal{J}(X_{(n)})$
given by (11).
In the setting presented above, the following theorem states bounds for the Shannon entropy and the extropy of the normalizing sequence
$(X_{(n)}-b_{n})/ $
$a_{n}$
.
Theorem 1. Let
$X_{1},X_{2},\ldots,X_{n}$
be a random sample of size n from a continuous probability distribution function F with respective probability density function f, supported on some interval
$(\alpha,\beta )$
, finite or not, where f is positive and
$\ln f$
is concave. Then we have the following.
-
(i) The following inequalities are true:
and
\begin{align*}1&-\ln n-\frac{1}{n}-\ln a_{n}\leq \mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right)\\ &\quad \leq 1-\ln \left[ 2I\left(\frac{1}{2}\right) \right] -\ln n-\frac{1}{n}+\ln 2-\ln a_{n}+\sum_{k=1}^{n}\frac{1}{k}-\sum_{k=1}^{n-1}\frac{1}{k2^{k}}\end{align*}
\begin{equation*}1-\lim_{n\rightarrow \infty }(\!\ln (na_{n}))\leq \lim_{n\rightarrow \infty }\mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) \leq 1-\ln \left[2I\left(\frac{1}{2}\right) \right] +\gamma -\lim_{n\rightarrow \infty }(\!\ln a_{n}).\end{equation*}
-
(ii) It is valid that
and
\begin{align*}-\frac{na_{n}}{2}I\left(\frac{1}{2}\right) &\leq \mathcal{J}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) \\ &\leq -n^{2}a_{n}I\left(\frac{1}{2}\right)\left(\frac{1}{2n-1}-\frac{1}{2n}-\frac{1}{(2n-1)2^{2n-1}}+\frac{2}{2n2^{2n}}\right)\end{align*}
\begin{equation*}-\frac{1}{2}I\left(\frac{1}{2}\right) \lim_{n\rightarrow \infty}(na_{n})\leq \lim_{n\rightarrow \infty }\mathcal{J}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) \leq -\frac{1}{4}I\left(\frac{1}{2}\right)\lim_{n\rightarrow \infty }(a_{n}).\end{equation*}
The proof is presented in full detail in Appendix A.2.
Motivated by Theorem 1, we now investigate the attainment of the upper bound
$\lim_{n\rightarrow \infty }(\!-\ln a_{n}) + \mathcal{UB}^{\mathcal{H}}$
of
$\lim_{n\rightarrow \infty }\mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) $
and the upper bound
$\lim_{n\rightarrow\infty }(a_{n})\mathcal{UB}^{\mathcal{J}}$
of
$\lim_{n\rightarrow \infty }\mathcal{J}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) $
, where
\begin{equation}\mathcal{UB}^{\mathcal{H}}=1-\ln \left[ 2I\left(\frac{1}{2}\right) \right]+\gamma \end{equation}
and
\begin{equation}\mathcal{UB}^{\mathcal{J}}=-\frac{1}{4}I\left(\frac{1}{2}\right).\end{equation}
In view of the above formulation, we can now state the characterization of attainment of the upper bounds of Theorem 1. The detailed proof of the next theorem is provided in Appendix A.3.
Theorem 2. Let
$X_{1},X_{2},\ldots,X_{n}$
be a random sample of size n from a continuous probability distribution function
$F_{\theta }$
with respective probability density function
$f_{\theta }$
, supported on some interval
$(\alpha,\beta )$
, finite or not, where
$f_{\theta }$
is positive and
$\ln f_{\theta }$
is concave and they depend on a real parameter
$\theta $
. Then we have the following.
-
(i) The term
$\lim_{n\rightarrow \infty }\mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) $
attains its maximum value
$\lim_{n\rightarrow \infty }(\!-\ln a_{n})+\mathcal{UB}^{\mathcal{H}}$
, for
$\mathcal{UB}^{\mathcal{H}}$
given in (19), if and only if
$F_{\theta }$
is the distribution function of the exponential distribution. Moreover, if the above maximum value is attained then (21)
\begin{equation}\lim_{n\rightarrow \infty }\mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) =1+\gamma =\mathcal{H}(G_{0}). \end{equation}
-
(ii) The term
$\lim_{n\rightarrow \infty }\mathcal{J}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) $
attains its maximum value
$\lim_{n\rightarrow \infty }(a_{n})\mathcal{UB}^{\mathcal{J}}$
, for
$\mathcal{UB}^{\mathcal{J}}$
given in (20), if and only if
$F_{\theta }$
is the distribution function of the exponential distribution. Moreover, if the above maximum value is attained then (22)
\begin{equation}\lim_{n\rightarrow \infty }\mathcal{J}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) =-\frac{1}{8}=\mathcal{J}(G_{0}). \end{equation}
Remark 2. Theorem 2 is focused mainly on a characterization of the exponential distribution; however, it implicitly derives the convergence of
$\mathcal{H}\left((X_{(n)}-b_{n})/a_{n}\right) $
to the Shannon entropy of the Gumbel distribution
$G_{0}$
, presented in [Reference Johnson16], [Reference Ravi and Saeb27], and [Reference Saeb29]. Moreover, it implicitly extends this convergence for the extropy of the normalizing sequence
$(X_{(n)}-b_{n})/a_{n}$
, formulated in (22), which has not appeared in the respective literature to the best of our knowledge. The main assumption in Theorem 2 is that of the log-concavity of the underlined distribution that governs the data. The respective assumption in [Reference Ravi and Saeb27] and [Reference Saeb29] to prove the convergence in (21) is that of monotonicity (non-increasing) of the distribution that is considered to govern the data. Johnson [Reference Johnson16] proves the convergence in (21) subject to the assumption that the respective distribution has a von Mises representation, presented in (4.1) in [Reference Johnson16], a representation that is typically used in extreme value theory. Convergence of entropy in the Gumbel regime has been studied in the recent literature under different, relatively simple conditions for the density that drives the data. The said conditions are technical conditions necessary to derive the results in [Reference Ravi and Saeb27], [Reference Saeb29], and the present paper. The necessity of a von Mises representation of the density required in [Reference Johnson16] gives some insight into why convergence to the Gumbel distribution occurs, according to the author (see [Reference Johnson16, p. 245]). On the other hand, the assumption of log-concavity of a density function has been broadly discussed and used in several disciplines and contexts. It is related to several topics in reliability, and many univariate distributions used in practice obey log-concavity (see [Reference Bagnoli and Bergstrom2]). Moreover, log-concavity of the underlined density is an essential prerequisite in several topics, such as isoperimetric problems [Reference Bobkov6] and concentration inequalities [Reference Boucheron and Thomas9].
3. Bounds for Shannon entropy and extropy
This section presents bounds for the Shannon entropy and the extropy of the largest order statistic. The said bounds are stated in the following proposition, and they are derived in the finite and infinite (
$n\rightarrow \infty $
) sample size cases.
Proposition 2. Let
$X_{1},X_{2},\ldots,X_{n}$
be a random sample of size n from a continuous probability distribution function F with respective probability density function f, supported on some interval
$(\alpha,\beta )$
, finite or not, where f is positive and
$\ln f$
is concave. Then, for I defined by (7):
-
(i) it holds that
and
\begin{equation*}1-\ln n-\frac{1}{n}\leq \mathcal{H}\big(X_{(n)}\big)\leq 1-\ln \left[ 2I\left(\frac{1}{2}\right) \right] -\ln n-\frac{1}{n}+\ln 2+\sum_{k=1}^{n}\frac{1}{k}-\sum_{k=1}^{n-1}\frac{1}{k2^{k}}\end{equation*}
where
\begin{equation*}-\infty \leq \lim_{n\rightarrow \infty }\mathcal{H}\big(X_{(n)}\big)\leq 1-\ln \left[2I\left(\frac{1}{2}\right) \right] +\gamma,\end{equation*}
$\gamma $
is the Euler constant, defined in (8) and (9);
-
(ii) it is valid that
and
\begin{equation*}-\frac{n}{2}I\left(\frac{1}{2}\right) \leq \mathcal{J}\big(X_{(n)}\big)\leq-n^{2}I\left(\frac{1}{2}\right) \left(\frac{1}{2n-1}-\frac{1}{2n}-\frac{1}{(2n-1)2^{2n-1}}+\frac{2}{2n2^{2n}}\right)\end{equation*}
\begin{equation*}-\infty \leq \lim_{n\rightarrow \infty }\mathcal{J}\big(X_{(n)}\big)\leq -\frac{1}{4}I\left(\frac{1}{2}\right)\!.\end{equation*}
The proof of Proposition 2 is essentially derived in Appendix A.2 in the proofs of Theorem1 by (A21) and (A29). Lower bounds for the extropy of order statistics and record values are obtained in [Reference Qiu25] in a different setting.
Table 1 provides examples related to the behavior of the entropy and extropy of the largest order statistic of Proposition 2, and it can accompany Theorem 1 of the respective normalized versions, in view of (17) and (18). The exact form of
$\mathcal{H}(X_{(n)})$
and its limiting value
$\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})$
for specific distributions are given in Table 1. The explicit form of the function I(t), defined by (7), is also given for each of these distributions. The second column includes, in addition, the upper bound
$\mathcal{UB}^{\mathcal{H}}=1-\ln \left[ 2I\left(\frac{1}{2}\right) \right]+\gamma $
of
$\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})$
, defined by (19). A subscript is used in the notation
$\mathcal{UB}^{\mathcal{H}}$
, in (19), with the first letter of the distribution considered each time. The last column of Table 1 provides the exact form of
$\mathcal{J}(X_{(n)}),$
its limiting value
$\lim_{n\rightarrow \infty }\mathcal{J}(X_{(n)}),$
and the upper bound
$\mathcal{UB}^{\mathcal{J}}$
in (20) for the same specific distributions. The derivations of the formulas are the result of standard algebraic manipulations that take into account also results from [Reference Zografos and Balakrishnan35, pp. 348–350] and standard properties of special functions such as
\begin{equation*}\Psi (n+1)-\Psi (n)=\frac{1}{n}=[H_{n}-\gamma ]-[H_{n-1}-\gamma ],\end{equation*}
where
$\Psi $
is the digamma function,
$H_{n}$
is the nth harmonic number, defined by (9), and
$\gamma $
is the Euler constant, related to
$H_{n}$
by
$\gamma =\lim_{n\rightarrow \infty }(H_{n}-\ln n)=0.57721\ldots$
, given in (8) (see, e.g., [Reference Mathai20, p. 11, (1.4.6)]).
Analytic expressions for
$\mathcal{H}(X_{(n)})$
,
$\mathcal{J}(X_{(n)})$
,
$\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)}$
,
$\mathop{\lim}\limits_{n\rightarrow \infty }\mathcal{J}(X_{(n)})$
,
$\mathcal{UB}^{\mathcal{H}}$
, and
$\mathcal{UB}^{\mathcal{J}}$
for specific distributions.
Table 1 Long description
The table has six columns and five rows, each row representing a different distribution: Uniform, Exponential, Logistic, Pareto, and Power function. The columns are labeled as follows: Distribution and I(t), H(X(n)), lim n→∞ H(X(n)), and UBH in (19), J(X(n)), lim n→∞ J(X(n)), and UBJ in (20). Each row provides specific analytic expressions for the entropy and extropy of the largest order statistic for each distribution. For example, the Uniform distribution has f(x) = 1/θ, F(x) = x/θ, and I(t) = 1/θ. The table also includes limiting values and upper bounds for both entropy and extropy. The values are calculated using standard algebraic manipulations and properties of special functions.
It should be mentioned at this point that the expression for
$\mathcal{H}(X_{(n)})$
of the exponential distribution can be obtained also by using a similar expression, derived on p. 693 in [Reference Nadarajah and Kotz23], for the beta exponential distribution.
Remark 3. We observe that the inequality
$\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})\leq \mathcal{UB}^{\mathcal{H}}$
, which is stated in part (i) Proposition 2, is valid in all cases the except in the case of the Pareto distribution. This is expected as the Pareto distribution is not log-concave, and hence the basic assumption of log-concavity of the theoretical results is violated for this distribution. Moreover, we observe that
$\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})$
and
$\lim_{n\rightarrow \infty }\mathcal{J}(X_{(n)})$
attain their maximum values
$\mathcal{UB}_{\exp }^{\mathcal{H}}$
and
$\mathcal{UB}_{\exp }^{\mathcal{J}}$
, respectively, only in the case of the exponential distribution. This point motivates the investigation of a characterization of the exponential distribution in terms of the attainment of the upper bounds (19) and (20) by the limit, as
$n\rightarrow\infty,$
of the Shannon entropy and the extropy of the largest order statistic. A similar characterization of an exponential-type distribution is derived in [Reference Chacko and Grace10] by means of the information generating function of records. The characterization of the exponential distribution, mentioned in this remark, is formulated in the following theorem.
Theorem 3. Let
$X_{1},X_{2},\ldots,X_{n}$
be a random sample of size n from a continuous probability distribution function
$F_{\theta }$
with respective probability density function
$f_{\theta }$
, supported on some interval
$(\alpha,\beta )$
, finite or not, where
$f_{\theta }$
is positive and
$\ln f_{\theta }$
is concave. Let, moreover,
$f_{\theta }$
depend on a real parameter
$\theta $
. Then:
-
(i)
$\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})$
attains its maximum value
$\mathcal{UB}^{\mathcal{H}}$
, given in (19), if and only if
$F_{\theta }$
is the distribution function of the exponential distribution; -
(ii)
$\lim_{n\rightarrow \infty }\mathcal{J}(X_{(n)})$
attains its maximum value
$\mathcal{UB}^{\mathcal{J}}$
, given in (20), if and only if
$F_{\theta }$
is the distribution function of the exponential distribution.
The proof of Theorem 3 is essentially derived in Appendix A.3. Intuitively speaking, the above characterization concludes that the exponential distribution maximizes the uncertainty, as it is quantified by the Shannon entropy and the extropy, which is associated with the maximum value in a sample of size n, when
$n\rightarrow \infty $
.
4. Conclusions
Information-theoretic inequalities occupy a precious role in information theory as they provide a tool to approximate a quantity if it is impossible to evaluate and, on the other hand, they usually lead to important characterizations in achieving the bounds. The Cramér–Rao inequality provides a lower bound for the variance of an unbiased estimator in terms of Fisher’s information measure, while the maximum entropy principle is a fundamental principle in information theory in terms of Shannon entropy. Both provide precious tools in the theory of estimation.
In this paper, two information-theoretic measures are considered, the Shannon entropy and the extropy, and they are applied to the distribution of the largest order statistic and its normalized version in the extreme value theory setup. Bounds of these measures are derived, and a characterization of the exponential distribution is provided as the model that maximizes the uncertainty that is associated with the maximum value in a large-size random sample from a log-concave distribution. It is moreover proved that the Shannon entropy and the extropy of the normalized maxima of a large-size sample converge to the respective measures for the Gumbel distribution, something considered in recent literature (see [Reference Johnson16] and [Reference Saeb29]). The results derived in this paper provide more insight into the behavior of two omnipresent information-theoretic quantities in the extreme value theory regime. The characterization of the exponential distribution could potentially be used to advance statistical analysis, for instance, in developing goodness-of-fit tests.
Appendix A. Proofs of the theoretical results
A.1. Proof of Proposition 1
-
(i) We can immediately see that the second derivative
is negative for
\begin{equation*}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\ln g_{\xi }(x)=(\xi +1)(1+\xi x)^{-2}\left[ \xi -(1+\xi x)^{-1/\xi }\right]\end{equation*}
$-1\lt \xi \lt 0$
, and hence
$g_{\xi }$
in (15) is log-concave for
$-1\lt \xi \lt 0$
. The case
$\xi =0$
is known from Table 1 in [Reference Bagnoli and Bergstrom2].
-
(ii) It is simple to obtain the inverse of the distribution function
$G_{\xi }$
in (13), and it is given by Straightforward manipulations then lead to
\begin{equation*}G_{\xi }^{-1}(t)=-\frac{1}{\xi }+\frac{1}{\xi }\left(\!-\frac{1}{\ln t}\right)^{\xi }\!,\qquad 0\lt t\lt 1\quad\text{and}\quad \xi \neq 0.\end{equation*}
$I(t)=g_{\xi }(G_{\xi }^{-1}(t))$
,
$0\lt t\lt 1$
, by our taking into account (15) and
$G_{\xi }^{-1}.$
In a similar manner, if
$\xi =0$
then in view of (14),
\begin{equation*}G_{0}^{-1}(t)=-\ln (\!-\ln t),\qquad 0\lt t\lt 1,\end{equation*}
and taking into account (16), we obtain
\begin{eqnarray*}I(t) &\,=\,&g_{0}\big(G_{0}^{-1}(t)\big)=\exp \big(\!-\big(\!-\ln (\!-\ln t)+\mathrm{e}^{\ln (\!-\ln t)}\big) \big), \\&\,=\,&-t\ln t,\qquad 0\lt t\lt 1.\end{eqnarray*}
\begin{equation}\mathcal{H}(X_{(n)})=1-\ln n-\frac{1}{n}-\int _{0}^{1}ny^{n-1}\ln (y(\!-\ln y)^{\xi +1}) \,\mathrm{d}y. \end{equation}
However,
\begin{equation}\int _{0}^{1}ny^{n-1}\ln (y(\!-\ln y)^{\xi +1})\,\mathrm{d}y=n\int _{0}^{1}y^{n-1}\ln y\,\mathrm{d}y+n(\xi +1)\int _{0}^{1}y^{n-1}\ln (\!-\ln y)\,\mathrm{d}y. \end{equation}
In view of 4.253(1) on p. 540 of [Reference Gradshteyn and Ryzhik14],
\begin{equation*}\int _{0}^{1}x^{\mu -1}\ln x\,\mathrm{d}x=B(\mu,1)[\Psi (\mu )-\Psi (\mu +1)],\qquad \mu \gt 0,\end{equation*}
and hence
\begin{equation}\int _{0}^{1}y^{n-1}\ln y\,\mathrm{d}y=B(n,1)[\Psi (n)-\Psi (n+1)]=\frac{\Gamma(n)\Gamma (1)}{\Gamma (n+1)}\left(\!-\frac{1}{n}\right) =-\frac{1}{n^{2}}.\end{equation}
It remains to obtain the last integral on the right-hand side of (A2). In view of 4.325(8) on p. 570 of [Reference Gradshteyn and Ryzhik14],
\begin{equation*}\int _{0}^{1}x^{\mu -1}\ln \ln \frac{1}{x}\,\mathrm{d}x=-\frac{1}{\mu }(\gamma+\ln \mu ),\qquad \mu \gt 0,\end{equation*}
and then
\begin{equation}\int _{0}^{1}y^{n-1}\ln (\!-\ln y)\,\mathrm{d}y=-\frac{1}{n}(\gamma +\ln n).\end{equation}
\begin{equation}\int _{0}^{1}ny^{n-1}\ln (y(\!-\ln y)^{\xi +1}) \,\mathrm{d}y=-\frac{1}{n}-(\xi +1)(\gamma +\ln n). \end{equation}
\begin{equation*}\mathcal{H}(X_{(n)})=1-\ln n-\frac{1}{n}+\frac{1}{n}+(\xi +1)(\gamma +\ln n),\end{equation*}
which leads to the desired result.
$\mathcal{H}(G_{\xi })$
is immediately obtained from
$\mathcal{H}(X_{(n)})$
for
$n=1$
because the density
$g_{\xi }$
of
$G_{\xi }$
is obtained from the density of
$X_{(n)}$
for
$n=1$
.
-
(iv) As in (11),
with
\begin{equation*}\mathcal{J}\big(X_{(n)}\big)=-\frac{n^{2}}{2}\int _{0}^{1}t^{2n-2}I(t)\,\mathrm{d}t,\end{equation*}
$I(t)=t(\!-\ln t)^{\xi +1},$
$0\lt t\lt 1,$
in view of part (ii). Hence, (A6)From 4.272(6), on p. 551 of [Reference Gradshteyn and Ryzhik14],
\begin{equation}\mathcal{J}(X_{(n)})=-\frac{n^{2}}{2}\int _{0}^{1}t^{2n-1}\left(\!\ln \frac{1}{t}\right) ^{\xi +2-1}\,\mathrm{d}t. \end{equation}
(A7)Then, from (A6) and (A7),
\begin{equation}\int _{0}^{1}x^{\nu -1}\left(\!\ln \frac{1}{x}\right) ^{\mu -1}\,\mathrm{d}x=\frac{1}{\nu ^{\mu }}\Gamma (\mu ),\qquad\nu,\mu \gt 0. \end{equation}
which leads to the desired result.
\begin{equation*}\mathcal{J}(X_{(n)})=-\frac{n^{2}}{2}\frac{1}{(2n)^{\xi +2}}\Gamma (\xi +2),\qquad \xi \gt -2,\end{equation*}
$\mathcal{J}(G_{\xi })$
is obtained from
$\mathcal{J}(X_{(n)})$
for
$n=1$
.
A.2. Proof of Theorem 1
A.2.1. Proof of part (i): the case of entropy
$\mathcal{H}$
As in (17),
\begin{equation}\mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) =-\ln a_{n}+\mathcal{H}(X_{(n)}), \end{equation}
and it is therefore enough to obtain, at first, bounds for
$\mathcal{H}(X_{(n)}).$
Because f is log-concave,
$\ln f$
is concave, and then, in view of [Reference Bobkov and Madiman8, p. 1532], the function
$I(t)=f(F^{-1}(t))$
,
$0\lt t\lt 1$
, defined by (7), is positive and concave on (0,1) (see also [Reference Bobkov5], Proposition A.1). Given that
$I(t)$
is positive and concave,
$\ln I(t)$
is also concave, and then Jensen’s inequality leads to
\begin{equation*}E_{Y}[\!\ln I(Y)]\leq \ln I\left(E_{Y}(Y)\right),\qquad Y\sim \mathrm{Beta}(n,1),\end{equation*}
and therefore
\begin{equation}E_{Y}[\!\ln I(Y)]\leq \ln I\left(\frac{n}{n+1}\right)\!, \end{equation}
because
$E_{Y}(Y)=n/(n+1)$
, for
$Y\sim \mathrm{Beta}(n,1)$
.
On the other hand, taking into account [Reference Bobkov and Madiman8, p. 1532], we have
\begin{equation*}\min \{t,1-t\}\leq I(t)\leq 1,\qquad0\lt t\lt 1,\end{equation*}
which leads to
\begin{equation}-\ln (n+1)\leq \ln I\left(\frac{n}{n+1}\right) \leq 0. \end{equation}
The inequalities (A9) and (A10) entail that
\begin{equation*}-E_{Y}[\!\ln I(Y))]\geq -\ln I\left(\frac{n}{n+1}\right) \geq 0,\end{equation*}
which in association with (5) gives
\begin{equation}\mathcal{H}\big(X_{(n)}\big)\geq 1-\ln n-\frac{1}{n}. \end{equation}
In view of (4.7) on p. 1916 of [Reference Bobkov6],
\begin{equation*}2I\left(\frac{1}{2}\right) \min \{t,1-t\}\leq I(t),\qquad 0\lt t\lt 1,\end{equation*}
and then
\begin{equation*}2I\left(\frac{1}{2}\right) \min \{Y,1-Y\}\leq I(Y),\qquad Y\sim \mathrm{Beta}(n,1),\end{equation*}
or
\begin{equation}\ln \left[ 2I\left(\frac{1}{2}\right) \right] +E_{Y}[\!\ln (\!\min\{Y,1-Y\})]\leq E_{Y}[\!\ln I(Y)],\qquad Y\sim \mathrm{Beta}(n,1). \end{equation}
However,
\begin{eqnarray}E_{Y}[\!\ln (\!\min \{Y,1-Y\})] &\,=\,&\int _{0}^{1}\ln (\!\min\{y,1\!-y\})f_{\mathrm{Beta}(n,1)}(y)\,\mathrm{d}y=\int _{0}^{1}\ln (\!\min\{y,1\!-y\})ny^{n-1}\,\mathrm{d}y \notag \\&\,=\,&\int _{0}^{1/2}\ln (\!\min\{y,1\!-y\})ny^{n-1}\,\mathrm{d}y+\int _{1/2}^{1}\ln (\!\min \{y,1\!-y\})ny^{n-1}\,\mathrm{d}y\notag \\&\,=\,&\int _{0}^{1/2}\ln (y)ny^{n-1}\,\mathrm{d}y+\int _{1/2}^{1}\ln (1\!-y)ny^{n-1}\,\mathrm{d}y. \end{eqnarray}
In view of 2.723(1) on p. 238 of [Reference Gradshteyn and Ryzhik14],
\begin{equation}\int _{0}^{1/2}\ln (y)ny^{n-1}\,\mathrm{d}y=-\frac{1}{2^{n}}\left(\!\ln 2+\frac{1}{n}\right). \end{equation}
Let us now work on the second integral. In view of 2.728(1) on p. 239 of [Reference Gradshteyn and Ryzhik14],
\begin{equation}\int _{1/2}^{1}\ln (1-y)ny^{n-1}\,\mathrm{d}y=\left[ y^{n}\ln (1-y)+\int \frac{y^{n}}{1-y}\,\mathrm{d}y\right] _{1/2}^{1}\!. \end{equation}
Using 2.111, 3
$^{8}$
on p. 69 of [Reference Gradshteyn and Ryzhik14], we obtain
\begin{eqnarray}\int \frac{y^{n}}{1-y}\,\mathrm{d}y &\,=\,&-\frac{y^{n}}{n}-\frac{y^{n-1}}{n-1}-\frac{y^{n-2}}{n-2}-\cdots+(\!-1)^{n-1}\frac{y}{1\cdot (\!-1)^{n}}+\frac{(\!-1)^{n}}{(\!-1)^{n+1}}\ln (1-y) \notag \\&\,=\,&-\left(\frac{y^{n}}{n}+\frac{y^{n-1}}{n-1}+\frac{y^{n-2}}{n-2}+\cdots+\frac{y}{1}+\ln (1-y)\right)\!. \end{eqnarray}
In this frame, (A15) and (A16) entail that
\begin{align}&\int _{1/2}^{1}\ln (1-y)ny^{n-1}\,\mathrm{d}y \notag\\&\quad =\left[ y^{n}\ln (1-y)-\frac{y^{n}}{n}-\frac{y^{n-1}}{n-1}-\frac{y^{n-2}}{n-2}-\cdots-\frac{y}{1}-\ln (1-y)\right]_{1/2}^{1} \notag \\&\quad =\left[ [y^{n}\ln (1-y)]|_{y=1}-[\!\ln (1-y)]|_{y=1}-\left(\frac{1}{n}+\frac{1}{n-1}+\frac{1}{n-2}+\cdots+\frac{1}{1}\right) \right] \notag \\&\qquad -\left[ -\left(\frac{1}{2^{n}}-1\right) \ln 2-\left(\frac{1}{2^{n}n}+\frac{1}{2^{n-1}(n-1)}+\frac{1}{2^{n-2}(n-2)}+\cdots+\frac{1}{2\cdot 1}\right)\right] \notag \\&\quad = -\sum_{k=1}^{n}\frac{1}{k}+\sum_{k=1}^{n}\frac{1}{k2^{k}}+\left(\frac{1}{2^{n}}-1\right) \ln 2. \end{align}
In view of (A13), (A14), and (A17),
\begin{equation*}E_{Y}[\!\ln (\!\min \{Y,1-Y\})]=-\frac{1}{2^{n}}\left(\!\ln 2+\frac{1}{n}\right)-\sum_{k=1}^{n}\frac{1}{k}+\sum_{k=1}^{n}\frac{1}{k2^{k}}+\left(\frac{1}{2^{n}}-1\right) \ln 2,\end{equation*}
or
\begin{eqnarray}E_{Y}[\!\ln (\!\min \{Y,1-Y\})] &\,=\,&-\frac{1}{2^{n}}\ln 2-\frac{1}{n2^{n}}-\sum_{k=1}^{n}\frac{1}{k}+\sum_{k=1}^{n}\frac{1}{k2^{k}}+\frac{1}{2^{n}}\ln 2-\ln 2 \notag \\&\,=\,&-\ln 2-\sum_{k=1}^{n}\frac{1}{k}+\sum_{k=1}^{n-1}\frac{1}{k2^{k}}. \end{eqnarray}
Then, in view of (A12) and (A18),
\begin{equation*}\ln \left[ 2I\left(\frac{1}{2}\right) \right] +E_{Y}[\!\ln (\!\min\{Y,1-Y\})]=\ln \left[ 2I\left(\frac{1}{2}\right) \right] -\ln 2-\sum_{k=1}^{n}\frac{1}{k}+\sum_{k=1}^{n-1}\frac{1}{k2^{k}}\leq E_{Y}[\!\ln I(Y)],\end{equation*}
with
$Y\sim \mathrm{Beta}(n,1)$
, and
\begin{equation}-E_{Y}[\!\ln I(Y)]\leq -\ln \left[ 2I\left(\frac{1}{2}\right) \right] +\ln 2+\sum_{k=1}^{n}\frac{1}{k}-\sum_{k=1}^{n-1}\frac{1}{k2^{k}},\qquad Y\sim \mathrm{Beta}(n,1). \end{equation}
\begin{eqnarray}\mathcal{H}(X_{(n)}) &\,=\,&1-\ln n-\frac{1}{n}-E_{Y}[\!\ln I(Y)) \notag \\&\leq &1-\ln \left[ 2I\left(\frac{1}{2}\right) \right] -\ln n-\frac{1}{n}+\ln 2+\sum_{k=1}^{n}\frac{1}{k}-\sum_{k=1}^{n-1}\frac{1}{k2^{k}}. \end{eqnarray}
In summary, if f is log-concave then from (A11) and (A20) we conclude that
\begin{equation}1-\ln n-\frac{1}{n}\leq \mathcal{H}(X_{(n)})\leq 1-\ln \left[ 2I\left(\frac{1}{2}\right) \right] -\ln n-\frac{1}{n}+\ln 2+\sum_{k=1}^{n}\frac{1}{k}-\sum_{k=1}^{n-1}\frac{1}{k2^{k}}. \end{equation}
In view of (A21) and (A8), the bounds for
$\mathcal{H}((X_{(n)}-b_{n})/a_{n})$
have been shown.
Let us now consider the case
$n\rightarrow \infty $
. It is well known (see, e.g., [Reference Abramowitz and Stegun1, pp. 255 and 258] and [Reference Mathai20, pp. 4 and 11]) that
\begin{equation}\gamma =\lim_{n\rightarrow \infty }(H_{n}-\ln n)=\lim_{n\rightarrow \infty}\left(\sum_{k=1}^{n}\frac{1}{k}-\ln n\right)\!, \end{equation}
where
$\gamma $
is the Euler constant, while
$H_{n}$
is the nth harmonic number, defined by
$H_{n}=\sum_{k=1}^{n}\frac{1}{k}.$
Then it remains to obtain
$\lim_{n\rightarrow \infty }\sum_{k=1}^{n-1}\frac{1}{k2^{k}}$
. Working in this direction and on the basis of the sum of an infinite geometric series, it is true that
\begin{equation*}\sum_{k=1}^{\infty }\left(\frac{z}{2}\right) ^{k-1}=\frac{1}{1-(z/2)}=\frac{2}{2-z},\qquad |z|\lt 2.\end{equation*}
Integrating both sides, we obtain
\begin{equation*}\sum_{k=1}^{\infty }\frac{1}{2^{k-1}}\int_{0}^{1}\left(\frac{z^{k}}{k}\right) ^{\prime }\,\mathrm{d}z=2\int_{0}^{1}\frac{1}{2-z}\,\mathrm{d}z,\end{equation*}
or
\begin{equation*}\sum_{k=1}^{\infty }\frac{1}{2^{k-1}}\frac{1}{k}=-2\int_{0}^{1}[\!\ln (2-z)]^{\prime }\,\mathrm{d}z,\end{equation*}
which gives
\begin{equation*}\sum_{k=1}^{\infty }\frac{1}{2^{k-1}}\frac{1}{k}=2\ln 2,\end{equation*}
and therefore
\begin{equation}\sum_{k=1}^{\infty }\frac{1}{k2^{k}}=\ln 2. \end{equation}
A.2.2. Proof of part (ii): the case of extropy
$\mathcal{J}$
As in (18),
\begin{equation}\mathcal{J}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) =a_{n}\mathcal{J}(X_{(n)}), \end{equation}
and it is therefore enough to obtain, at first, bounds for
$\mathcal{J}(X_{(n)}).$
In view of [Reference Bobkov6, p. 1916],
\begin{equation*}2I\left(\frac{1}{2}\right) \min \{t,1-t\}\leq I(t)\leq I\left(\frac{1}{2}\right) +u\left(t-\frac{1}{2}\right),\qquad 0\lt t\lt 1,\, \text{}|u|\leq2I\left(\frac{1}{2}\right)\!.\end{equation*}
Hence, taking into account (11), we have
\begin{equation*}\mathcal{J}(X_{(n)})=-\frac{n^{2}}{2}\int _{0}^{1}t^{2n-2}I(t)\,\mathrm{d}t,\end{equation*}
and the previous inequality becomes
\begin{align}&-\frac{n^{2}}{2}I\left(\frac{1}{2}\right) \int _{0}^{1}t^{2n-2}\,\mathrm{d}t-\frac{n^{2}}{2}u\int _{0}^{1}t^{2n-2}\left(t-\frac{1}{2}\right)\,\mathrm{d}t \nonumber\\&\quad \leq \mathcal{J}\big(X_{(n)}\big)\leq -n^{2}I\left(\frac{1}{2}\right)\int _{0}^{1}t^{2n-2}\min \{t,1-t\}\,\mathrm{d}t. \end{align}
Calculation of the integrals gives
\begin{equation}\int _{0}^{1}t^{2n-2}\,\mathrm{d}t=\left. \frac{t^{2n-1}}{2n-1}\right\vert_{0}^{1}=\frac{1}{2n-1}, \end{equation}
\begin{equation}\int _{0}^{1}t^{2n-2}\left(t-\frac{1}{2}\right) \,\mathrm{d}t=\left. \frac{t^{2n}}{2n}\right\vert _{0}^{1}-\frac{1}{2}\left. \frac{t^{2n-1}}{2n-1}\right\vert_{0}^{1}=\frac{1}{2n}-\frac{1}{2}\frac{1}{2n-1}=\frac{1}{2}\frac{n-1}{n(2n-1)}. \end{equation}
On the other hand,
\begin{eqnarray}\int _{0}^{1}t^{2n-2}\min \{t,1-t\}\,\mathrm{d}t&\,=\,&\int _{0}^{1/2}t^{2n-2}t\,\mathrm{d}t+\int _{1/2}^{1}t^{2n-2}(1-t)\,\mathrm{d}t\notag \\&\,=\,&\left. \frac{t^{2n}}{2n}\right\vert _{0}^{1/2}+\left. \frac{t^{2n-1}}{2n-1}\right\vert _{1/2}^{1}-\left. \frac{t^{2n}}{2n}\right\vert _{1/2}^{1}\notag \\&\,=\,&\frac{1}{2n2^{2n}}+\frac{1}{2n-1}-\frac{1}{(2n-1)2^{2n-1}}-\frac{1}{2n}+\frac{1}{2n2^{2n}} \notag \\&\,=\,&\frac{1}{2n-1}-\frac{1}{(2n-1)2^{2n-1}}-\frac{1}{2n}+\frac{2}{2n2^{2n}}.\end{eqnarray}
Hence, in view of (A25)–(A28), we obtain
\begin{align*}&-\frac{n^{2}}{2(2n-1)}I\left(\frac{1}{2}\right) -\frac{n(n-1)}{4(2n-1)}u \\ &\quad\leq \mathcal{J}\big(X_{(n)}\big)\leq -n^{2}I\left(\frac{1}{2}\right) \left(\frac{1}{2n-1}-\frac{1}{2n}-\frac{1}{(2n-1)2^{2n-1}}+\frac{2}{2n2^{2n}}\right)\!,\end{align*}
and taking into account that
$|u|\leq 2I\left(\frac{1}{2}\right) $
(see [Reference Bobkov6, p. 1916]), we finally get
\begin{align*}&-\frac{n^{2}}{2(2n-1)}I\left(\frac{1}{2}\right) -\frac{n(n-1)}{4(2n-1)}2I\left(\frac{1}{2}\right) \\ &\quad\leq \mathcal{J}\big(X_{(n)}\big)\leq -n^{2}I\left(\frac{1}{2}\right) \left(\frac{1}{2n-1}-\frac{1}{2n}-\frac{1}{(2n-1)2^{2n-1}}+\frac{2}{2n2^{2n}}\right)\!,\end{align*}
or
\begin{equation*}-\frac{2n^{2}-n}{2(2n-1)}I\left(\frac{1}{2}\right) \leq \mathcal{J}\big(X_{(n)}\big)\leq -n^{2}I\left(\frac{1}{2}\right) \left(\frac{1}{2n-1}-\frac{1}{2n}-\frac{1}{(2n-1)2^{2n-1}}+\frac{2}{2n2^{2n}}\right)\!,\end{equation*}
or
\begin{equation}-\frac{n}{2}I\left(\frac{1}{2}\right) \leq \mathcal{J}\big(X_{(n)}\big)\leq-n^{2}I\left(\frac{1}{2}\right) \left(\frac{1}{2n-1}-\frac{1}{2n}-\frac{1}{(2n-1)2^{2n-1}}+\frac{2}{2n2^{2n}}\right)\!. \end{equation}
By this last inequality and (A24), the proof has been completed.
Let us now show the case
$n\rightarrow \infty $
. The proof follows immediately in view of the fact that
$\lim_{n\rightarrow \infty}\left(\frac{1}{(2n-1)2^{2n-1}}\right) =0$
,
$\lim_{n\rightarrow \infty}\left(\frac{1}{2n2^{2n}}\right) =0$
, and
$\lim_{n\rightarrow \infty}\left\{-n^{2}\left(\frac{1}{2n-1}-\frac{1}{2n}\right) \right\} =-\frac{1}{4}$
.
A.3. Proof of Theorem 2
A.3.1. Proof of part (i): the case of entropy
$\mathcal{H}$
As in (17),
\begin{equation*}\mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) =-\ln a_{n}+\mathcal{H}\big(X_{(n)}\big),\end{equation*}
and it is therefore enough to prove the characterization, at first, for
$\mathcal{H}(X_{(n)}).$
(
$\Longrightarrow $
) Let the random sample come from an exponential distribution with density
$f_{\theta }(x)=\theta \mathrm{e}^{-\theta x}$
,
$x\gt 0$
, and parameter
$\theta \gt 0$
. Then standard algebraic manipulations lead to the desired result (see also Table 1 in Section 3).
(
$\Longleftarrow $
) Let
$\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})=\mathcal{UB}^{\mathcal{H}}$
. Then, from (6) and (19),
\begin{equation*}\lim_{n\rightarrow \infty }\mathcal{H}\big(X_{(n)}\big)=\lim_{n\rightarrow \infty}\bigg(1-\ln n-\frac{1}{n}-\int _{0}^{1}ny^{n-1}\ln f_{\theta}\big(F_{\theta }^{-1}(y) \big)\,\mathrm{d}y\bigg) =1-\ln \left[ 2I_{\theta }\left(\frac{1}{2}\right) \right] +\gamma.\end{equation*}
Taking into account (8),
$\lim_{n\rightarrow \infty }\ln n=\lim_{n\rightarrow \infty }H_{n}-\gamma $
, and hence
$\lim_{n\rightarrow\infty }\mathcal{H}(X_{(n)})=\mathcal{UB}^{\mathcal{H}}$
entails that
\begin{equation*}1-\lim_{n\rightarrow \infty }(\!\ln n)-\lim_{n\rightarrow \infty }\frac{1}{n}-\lim_{n\rightarrow \infty }\int _{0}^{1}ny^{n-1}\ln I_{\theta}(y))\,\mathrm{d}y=1-\ln \left[ 2I_{\theta }\left(\frac{1}{2}\right) \right] +\gamma,\end{equation*}
or
\begin{equation*}\lim_{n\rightarrow \infty }H_{n}+\lim_{n\rightarrow \infty}\int _{0}^{1}ny^{n-1}\ln I_{\theta }(y)\,\mathrm{d}y=\ln \left[ 2I_{\theta}\left(\frac{1}{2}\right) \right]\!,\end{equation*}
or, taking into account that
$ny^{n-1},0\lt y\lt 1,$
is the density of
$\mathrm{Beta}(n,1)$
, it is valid that
\begin{equation*}\int _{0}^{1}ny^{n-1}\ln \left[ 2I_{\theta }\left(\frac{1}{2}\right)\right] \,\mathrm{d}y=\ln \left[ 2I_{\theta }\left(\frac{1}{2}\right) \right]\!,\end{equation*}
and therefore the equation
$\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})=\mathcal{UB}^{\mathcal{H}}$
entails that
\begin{equation}\lim_{n\rightarrow \infty }\bigg\{H_{n}+\int _{0}^{1}ny^{n-1}\ln \frac{I_{\theta }(y)}{2I_{\theta }\left(1/2\right) }\,\mathrm{d}y\bigg\} =0,\end{equation}
where
\begin{equation*}I_{\theta }(y)=f_{\theta }\big(F_{\theta }^{-1}(y)\big),\qquad 0\lt y\lt 1,\end{equation*}
in view of (7). Moreover, in view of well-known properties of the beta distribution (see [Reference Zografos and Balakrishnan35, Lemma 1(c)] and [Reference Nawa and Nadarajah24, p. 2, (3)]), for
$Y\sim \mathrm{Beta}(n,1),$
\begin{equation*}E[\!\ln (1-Y)]=\Psi (1)-\Psi (n+1)=-H_{n},\end{equation*}
and therefore
\begin{equation}H_{n}=-E[\!\ln (1-Y)]=-\int _{0}^{1}ny^{n-1}\ln (1-y)\,\mathrm{d}y. \end{equation}
In view of (A30) and (A31),
$\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})=\mathcal{UB}^{\mathcal{H}}$
entails that
\begin{equation}\lim_{n\rightarrow \infty }\int _{0}^{1}ny^{n-1}\ln \frac{I_{\theta}(y)}{2I_{\theta }\left(1/2\right) (1-y)}\,\mathrm{d}y=0. \end{equation}
Equation (A32) can be written as follows:
\begin{equation}\lim_{n\rightarrow \infty }\int _{0}^{1}ny^{n-1}\ln \frac{I_{\theta}(y)}{2I_{\theta }\left(1/2\right) \min \{y,1-y\}}\,\mathrm{d}y+\lim_{n\rightarrow\infty }\int _{0}^{1}ny^{n-1}\ln \frac{\min \{y,1-y\}}{1-y}\,\mathrm{d}y=0.\end{equation}
Consider now the integral on the right-hand side of (A33), which is simplified as follows:
\begin{eqnarray*}\int _{0}^{1}ny^{n-1}\ln \frac{\min \{y,1-y\}}{1-y}\,\mathrm{d}y&\,=\,&\int _{0}^{1/2}ny^{n-1}\ln \frac{y}{1-y}\,\mathrm{d}y+\int_{1/2}^{1}ny^{n-1}\ln \frac{1-y}{1-y}\,\mathrm{d}y \notag \\&\,=\,&\int _{0}^{1/2}ny^{n-1}\ln y\,\mathrm{d}y-\int _{0}^{1/2}ny^{n-1}\ln (1-y)\,\mathrm{d}y. \end{eqnarray*}
In view of 2.723(1) on p. 238 of [Reference Gradshteyn and Ryzhik14] or the proof of Theorem 1 (see (A14)),
\begin{equation}\int _{0}^{1/2}ny^{n-1}\ln y\,\mathrm{d}y=-\frac{1}{2^{n}}\ln 2-\frac{1}{n2^{n}}.\end{equation}
Moreover, in view of 2.728(1) on p. 239 of [Reference Gradshteyn and Ryzhik14]
\begin{eqnarray}\int _{0}^{1/2}ny^{n-1}\ln (1-y)\,\mathrm{d}y &\,=\,&\left[ y^{n}\ln (1-y)+\int \frac{y^{n}}{1-y}\,\mathrm{d}y\right] _{0}^{1/2} \notag \\&\,=\,&\frac{1}{2^{n}}\ln \frac{1}{2}+\left[ \int \frac{y^{n}}{1-y}\,\mathrm{d}y\right]_{0}^{1/2}\!. \end{eqnarray}
However, from 2.111, 3
$^{8}$
on p. 69 of [Reference Gradshteyn and Ryzhik14] or (A16)
\begin{equation*}\int \frac{y^{n}}{1-y}\,\mathrm{d}y=-\left(\frac{y^{n}}{n}+\frac{y^{n-1}}{n-1}+\frac{y^{n-2}}{n-2}+\cdots+\frac{y}{1}+\ln (1-y)\right)\!,\end{equation*}
and therefore
\begin{equation}\left[ \int \frac{y^{n}}{1-y}\,\mathrm{d}y\right] _{0}^{1/2}=-\left(\frac{1}{n2^{n}}+\frac{1}{(n-1)2^{n-1}}+\cdots+\frac{1}{2}+\ln \frac{1}{2}\right)\!.\end{equation}
\begin{equation*}\int _{0}^{1}ny^{n-1}\ln \frac{\min \{y,1-y\}}{1-y}\,\mathrm{d}y=\frac{1}{(n-1)2^{n-1}}+\frac{1}{(n-2)2^{n-2}}+\cdots+\frac{1}{2}+\ln \frac{1}{2},\end{equation*}
or
\begin{equation}\int _{0}^{1}ny^{n-1}\ln \frac{\min \{y,1-y\}}{1-y}\,\mathrm{d}y=\sum_{k=1}^{n-1}\frac{1}{k2^{k}}-\ln 2. \end{equation}
However, in the proof of Theorem 1 it was shown (see (A23)) that
\begin{equation}\ln 2=\sum_{k=1}^{\infty }\frac{1}{k2^{k}}. \end{equation}
\begin{equation*}\int _{0}^{1}ny^{n-1}\ln \frac{\min \{y,1-y\}}{1-y}\,\mathrm{d}y=\sum_{k=1}^{n-1}\frac{1}{k2^{k}}-\sum_{k=1}^{\infty }\frac{1}{k2^{k}},\end{equation*}
and therefore
\begin{equation}\lim_{n\rightarrow \infty }\int _{0}^{1}ny^{n-1}\ln \frac{\min\{y,1-y\}}{1-y}\,\mathrm{d}y=\lim_{n\rightarrow \infty }\sum_{k=1}^{n-1}\frac{1}{k2^{k}}-\sum_{k=1}^{\infty }\frac{1}{k2^{k}}=0. \end{equation}
Summarizing all the above, from (A32), (A33), and (A39),
\begin{equation}\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})=\mathcal{UB}^{\mathcal{H}}\text{entails that }\lim_{n\rightarrow \infty}\int _{0}^{1}ny^{n-1}\ln \frac{I_{\theta }(y)}{2I_{\theta }\left(1/2\right) \min \{y,1-y\}}\,\mathrm{d}y=0. \end{equation}
Equation (A40) means that
$\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})=\mathcal{UB}^{\mathcal{H}}$
entails that
\begin{equation*}\int _{0}^{1}ny^{n-1}\ln \frac{I_{\theta }(y)}{2I_{\theta }\left(1/2\right) \min \{y,1-y\}}\,\mathrm{d}y=0\end{equation*}
for large n
$(n\rightarrow \infty )$
. Given that
$ny^{n-1}$
,
$0\lt y\lt 1,$
is the density of the
$\mathrm{Beta}(n,1)$
distribution, the function inside the last integral is non-negative because
$I_{\theta }(y)\geq $
$2I_{\theta }\left(1/2\right)\min \{y,1-y\}$
, in view of (4.7) on p. 1916 of [Reference Bobkov6]. Therefore,
\begin{equation}\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})=\mathcal{UB}^{\mathcal{H}}\text{entails that }I_{\theta }(y)=2I_{\theta }\left(1/2\right) \min\{y,1-y\}\text{ for large }n\, (n\rightarrow \infty ). \end{equation}
On the other hand, application of (7) for
$t=F(y)$
leads to
\begin{equation}I(F(y))=f(F^{-1}(F(y)))=f(y). \end{equation}
Without any loss of generality, suppose that
$\min\{y,1-y\}=1-y,0\lt y\lt 1.$
Then (A41) and (A42) give
\begin{equation*}\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})=\mathcal{UB}^{\mathcal{H}}\text{entails that }I_{\theta }(F_{\theta }(y))=2I_{\theta }\left(1/2\right) \min \{F_{\theta }(y),1-F_{\theta }(y)\}\text{ for large }n\, (n\rightarrow \infty ),\end{equation*}
or
\begin{equation}\lim_{n\rightarrow \infty }\mathcal{H}(X_{(n)})=\mathcal{UB}^{\mathcal{H}}\text{entails that }f_{\theta }(y)=2I_{\theta }\left(1/2\right)(1-F_{\theta }(y)), \quad 0\lt y\lt 1,\text{ for large }n\, (n\rightarrow\infty ). \end{equation}
But
$f_{\theta }(y)=2I_{\theta }\left(1/2\right) (1-F_{\theta }(y))$
in (A43) means that
$F_{\theta }(y)=1-[ 2I_{\theta }(1/2) ] ^{-1}f_{\theta }(y)$
or
$(\mathrm{d}/\mathrm{d}y)F_{\theta }(y)=-[2I_{\theta }(1/2) ] ^{-1}f_{\theta }^{^{\prime }}(y)$
or
$f_{\theta }^{^{\prime }}(y)/f_{\theta }(y)=-2I_{\theta }\left(1/2\right) $
. Hence,
\begin{equation*}\frac{\mathrm{d}}{\mathrm{d}y}\ln f_{\theta }(y)=-2I_{\theta }\left(1/2\right) \text{ or }\ln f_{\theta }(y)=-2I_{\theta }\left(1/2\right) y+c\text{ or }f_{\theta}(y)=\mathrm{e}^{c}\exp \{-2I_{\theta }\left(1/2\right) y\},\end{equation*}
and given that
$\int f_{\theta }(y)\,\mathrm{d}y=1,$
we obtain
$c=2I_{\theta}\left(1/2\right) $
, and then
\begin{equation*}f_{\theta }(y)=2I_{\theta }\left(1/2\right) \exp \{-2I_{\theta }\left(1/2\right) y\}.\end{equation*}
However, if
$g(y)=\lambda \mathrm{e}^{-\lambda y}$
then
$I_{\lambda}(t)=g(G^{-1}(t))=\lambda (1-t)$
,
$0\lt t\lt 1$
, and therefore
$2I_{\lambda}(1/2)=\lambda.$
Therefore, taking into account that the exponential distribution is identifiable, we finally obtain that
$\lim_{n\rightarrow\infty }\mathcal{H}(X_{(n)})=\mathcal{UB}^{\mathcal{H}}$
entails that
$F_{\theta }$
is the distribution function of the exponential distribution for large n
$(n\rightarrow \infty )$
, and the theorem has been proved, in view of (17).
Moreover, if
$F_{\theta }$
is the distribution function of the exponential distribution with parameter
$\theta $
, then
$a_{n}=\frac{1}{\theta },$
in view of Example 1, and it can be easily shown (see also Table 1 in Section3) that
$\mathcal{UB}_{\exp }^{\mathcal{H}}=1-\ln \theta +\gamma,$
where
$\mathcal{UB}^{\mathcal{H}}$
was defined in (19). Then
$\mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) $
attains its upper bound, and
\begin{eqnarray*}\lim_{n\rightarrow \infty }\mathcal{H}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) &\,=\,&\ln \theta +1-\ln \theta +\gamma \\&\,=\,&1+\gamma =\mathcal{H}\left(G_{0}\right)\!,\end{eqnarray*}
in view of Proposition 1(iii).
A.3.2. Proof of part (ii): the case of extropy
$\mathcal{J}$
As in (18),
\begin{equation*}\mathcal{J}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) =a_{n}\mathcal{J}(X_{(n)}),\end{equation*}
and it is therefore enough to prove the characterization, at first, for
$\mathcal{J}(X_{(n)}).$
(
$\Longrightarrow $
) Let the random sample come from the exponential distribution with parameter
$\theta \gt 0$
. Then standard algebraic manipulations lead to the desired result (see also Table 1 in Section 3).
(
$\Longleftarrow $
) Let
$\lim_{n\rightarrow \infty }\mathcal{J}(X_{(n)})=\mathcal{UB}^{\mathcal{J}}$
. Then, as in (11),
\begin{equation*}\mathcal{J}(X_{(n)})=-\frac{n^{2}}{2}\int _{0}^{1}t^{2n-2}I_{\theta}(t)\,\mathrm{d}t,\end{equation*}
and in view of (20),
\begin{equation}\lim_{n\rightarrow \infty }\mathcal{J}\big(X_{(n)}\big)=\mathcal{UB}^{\mathcal{J}}\text{leads to }\lim_{n\rightarrow \infty }\bigg\{n^{2}\int _{0}^{1}t^{2n-2}I_{\theta }(t)\,\mathrm{d}t\bigg\} =\frac{1}{2}I_{\theta }\left(\frac{1}{2}\right). \end{equation}
If we take into account that the density of
$Y\sim \mathrm{Beta}(2n-1,1)$
is given by
$(2n-1)y^{2n-2}, 0\lt y\lt 1,$
(A44) entails that
\begin{equation*}\lim_{n\rightarrow \infty }\bigg\{\frac{n^{2}}{2n-1}\int _{0}^{1}I_{\theta }(t)(2n-1)t^{2n-2}\,\mathrm{d}t\bigg\} =\int _{0}^{1}\frac{1}{2}I_{\theta}\left(\frac{1}{2}\right) (2n-1)t^{2n-2}\,\mathrm{d}t.\end{equation*}
Hence,
$\lim_{n\rightarrow \infty }\mathcal{J}(X_{(n)})=\mathcal{UB}^{\mathcal{J}}$
leads to
\begin{equation}\lim_{n\rightarrow \infty }\bigg\{\int _{0}^{1}\left[ I_{\theta }(t)-\frac{1}{2}\frac{2n-1}{n^{2}}I_{\theta }\left(\frac{1}{2}\right) \right]\frac{n^{2}}{2n-1}(2n-1)t^{2n-2}\,\mathrm{d}t\bigg\} =0. \end{equation}
In view of (4.7) on p. 1916 of [Reference Bobkov6],
$I_{\theta }(t)\geq $
$2I_{\theta }\left(1/2\right) \min \{t,1-t\}$
,
$0\lt t\lt 1$
. On the other hand, for large n,
$n\rightarrow \infty $
,
\begin{equation*}2\min \{t,1-t\}\geq \frac{1}{2}\frac{2n-1}{n^{2}},\qquad 0\lt t\lt 1,\end{equation*}
because the right-hand side converges to zero. Hence, for large n,
$n\rightarrow \infty,$
\begin{equation*}2\min \{t,1-t\}I_{\theta }\left(\frac{1}{2}\right) \geq \frac{1}{2}\frac{2n-1}{n^{2}}I_{\theta }\left(\frac{1}{2}\right),\qquad 0\lt t\lt 1.\end{equation*}
Therefore, for large n,
$n\rightarrow \infty,$
and
$0\lt t\lt 1,$
\begin{equation}I_{\theta }(t)-\frac{1}{2}\frac{2n-1}{n^{2}}I_{\theta }\left(\frac{1}{2}\right) \geq I_{\theta }(t)-2\min \{t,1-t\}I_{\theta }\left(\frac{1}{2}\right) \geq 0. \end{equation}
If we take into account, in view of (A46), that
\begin{equation*}I_{\theta }(t)-\frac{1}{2}\frac{2n-1}{n^{2}}I_{\theta }\left(\frac{1}{2}\right) \geq 0,\end{equation*}
(A45) means that
\begin{equation}I_{\theta }(t)-\frac{1}{2}\frac{2n-1}{n^{2}}I_{\theta }\left(\frac{1}{2}\right) =0\quad \text{for large }n,\, n\rightarrow \infty. \end{equation}
Equations (A46) and (A47) are summarized to
\begin{equation}I_{\theta }(t)-2\min \{t,1-t\}I_{\theta }\left(\frac{1}{2}\right) =0,\quad \text{for large }n,\, n\rightarrow \infty. \end{equation}
Then, in view of (A44) and (A48),
\begin{equation*}\lim_{n\rightarrow \infty }\mathcal{J}(X_{(n)})=\mathcal{UB}^{\mathcal{J}}\text{entails that }I_{\theta }(y)=2I_{\theta }\left(1/2\right) \min\{t,1-t\}\text{ for large }n, \, n\rightarrow \infty,\end{equation*}
and the proof is completed by following the steps in the proof of Theorem 2(i) just after (A41) and (18).
Moreover, to show the convergence of the extropy of the normalizing sequence to the respective measure of the Gumbel distribution, it is clear that if
$\lim_{n\rightarrow \infty }\mathcal{J}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) $
attains its maximum
$\mathcal{UB}_{\exp }^{\mathcal{J}}\lim_{n\rightarrow \infty }(a_{n})$
, then
\begin{eqnarray*}\lim_{n\rightarrow \infty }\mathcal{J}\left(\frac{X_{(n)}-b_{n}}{a_{n}}\right) &\,=\,&-\frac{\theta }{8}\lim_{n\rightarrow \infty }\frac{1}{\theta } \\&\,=\,&-\frac{1}{8}=\mathcal{J}\left(G_{0}\right)\!,\end{eqnarray*}
in view of Example 1, where
$a_{n}=\frac{1}{\theta }$
for the exponential distribution (see Table 1 in Section 3, where
$\mathcal{UB}_{\exp }^{\mathcal{J}}=-\frac{\theta }{8}$
), and Proposition 1(iv).
Acknowledgements
The author thanks the editors and two anonymous reviewers for careful reading of a previous version of the paper and for valuable comments and suggestions that have considerably improved this work.
Funding information
There are no funding bodies to thank in relation to the creation of this article.
Competing interests
There were no competing interests to declare which arose during the preparation or publication process for this article.
H(X(n))
J(X(n))
limn→∞H(X(n)
limn→∞J(X(n))
UBH
UBJ
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