Order Properties Concerning Tsallis Residual Entropy

Abstract

With the help of Tsallis residual entropy, we introduce Tsallis quantile entropy order between two random variables. We give necessary and sufficient conditions, study closure and reversed closure properties under parallel and series operations and show that this order is preserved in the proportional hazard rate model, proportional reversed hazard rate model, proportional odds model and record values model.

1. Introduction

The concept of entropy, defined mathematically by Shannon in [1], measures the uncertainty of a physical system and has applications in many scientific and technological areas such as physics, probability theory, statistics, communication theory and economics. This notion appeared from thermodynamics and statistical mechanics. In this theory, for a data communication system, we have three elements: a receiver, a communication channel and a source of data. Based on the signal that is received through the channel, Shannon tried to identify what sort of data were generated. Many methods on how to encode, compress and transmit the message were considered. In Shannon’s source coding theorem, also known as Shannon’s first theorem, an error-free encoding is established. This result is generalized especially for noisy channel in Shannon’s noisy coding theorem. In the last couple of years, Shannon entropy was intensively studied and many generalizations have appeared (Tsallis entropy, Rényi entropy, Varma entropy, Kaniadakis entropy, relative entropy, weighted entropy, cumulative entropy, etc.).

In [2], Tsallis used another formula instead of the classical algorithm which appears in Shannon entropy, defining, in this way, what we call today, Tsallis entropy. There are many applications of this new entropy, especially in physics, and, more precisely: superstatistics (see [3]), spectral statistics (see [4]), earthquakes (see [5,6,7]), stock exchanges (see [8,9]), plasma (see [10]), income distribution (see [11]), non-coding human DNA (see [12]), internet (see [13]), and statistical mechanics (see [2,14]). For more information about Tsallis entropy, we recommend reading [15].

Among the applications of other entropies (Rényi entropy, Varma entropy, Kaniadakis entropy, relative entropy, weighted entropy, etc.), we can list the following: Markov chains (see [16,17,18]), model selection (see [19,20]), combinatorics (see [21,22]), finance (see [23,24,25]), Lie symmetries (see [26,27]), and machine learning (see [28,29]).

There are several papers in which the authors compare random variables from the point of view of residual entropies: for Shannon residual entropy, see [30,31,32]; for Rényi residual entropy, see [33,34]; for Varma residual entropy, see [35]; and for Awad-Varma residual entropy, see [36]. Other orders between random variables can be found in [37,38,39,40,41,42,43,44].

Rao et al. [45] introduced an alternative measure to Shannon entropy, known as the cumulative residual entropy (CRE), by considering the survival function instead of the probability density function. Because the survival function is more regular than the probability density function, CRE is considered to be more stable and has more mathematical properties. Moreover, the distribution function exists even if the probability density function does not exist (see, e.g., generalized lambda, power-Pareto and Govindarajulu distributions). Sati and Gupta [46] introduced a cumulative residual Tsallis entropy and extended it to its dynamic form based on the residual lifetime. Rajesh and Sunoj [47] introduced an alternative form of the cumulative residual Tsallis entropy and proved some results with applications in reliability. Toomaj and Atabay [48] elaborated some further consequences of the alternative cumulative residual Tsallis entropy, introduced by Rajesh and Sunoj [47], including stochastic ordering, expressions and bounds and proposed a normalized version of the cumulative residual Tsallis entropy, which can be used as a dispersion measure in place of coefficient of variation. Kumar [49] obtained characterization results based on the dynamic cumulative residual Tsallis entropy. In many realistic situations, uncertainty is not necessarily related to the future and can refer to the past as well. For instance, if at time t, a system which is observed only at certain preassigned inspection times is found to be down, then the uncertainty of the system life relies on the past, i.e., on which instant in $(0,t)$ it has failed. A wide variety of research is available on entropy measures and its applications in past lifetime. For more detail, one can refer to Di Crescenzo and Longobardi [50], Di Crescenzo and Longobardi [51], Sachlas and Papaioannou [52] and Di Crescenzo and Toomaj [53]. Also, a study on the cumulative Tsallis entropy for past lifetime is available in Nair et al.’s work [54], Calì et al.’s work [55], Khammar and Jahanshahi’s work [56], Sunoj et al.’s study [57] and Alomani and Kayid’s study [58]. Baratpour and Khammar [59] studied Tsallis entropy of order statistics. The quantile-based approach has some advantages: it provides an alternative methodology in deriving the cumulative Tsallis entropy in past lifetime and facilitates the extension of domain of application of the cumulative Tsallis entropy in past lifetime to many flexible quantile functions which serve as useful lifetime models and which possess no probability distribution function.

The paper is organized as follows. After this Introduction section, in Section 2, Background and Notations, we present the main notions and notations used throughout the article. In Section 3, Fundamental Results, we present the main theorem (Theorem 1), which is used in all of our results. In this section, we also prove that the dispersive order and the convex transform order apply to the Tsallis quantile entropy order. In Section 4, Closure and Reversed Closure Properties, we show the closure and reversed closure properties of Tsallis quantile entropy order under parallel and series operations. In the last four sections, we show the preservation of Tsallis quantile entropy in some stochastic models: the proportional hazard rate model (Section 5—Preservation of Tsallis Quantile Entropy Order in the Proportional Hazard Rate Model), the proportional reversed hazard rate model (Section 6—Preservation of Tsallis Quantile Entropy Order in the Proportional Reversed Hazard Rate Model), the proportional odds model (Section 7—Preservation of Tsallis Quantile Entropy Order in the Proportional Odds Model) and the proportional record values model (Section 8—Preservation of Tsallis Quantile Entropy Order in the Record Values Model).

2. Background and Notations

Throughout this paper, we assume that all expectations are finite and all ratios and powers are well defined. For information on notions of probability theory, we recommend [60].

We consider X a non-negative random variable with an absolutely continuous cumulative distribution function $F_X$, a survival function ${\overline{F}}_X\stackrel{def}{=}1-F_X$ and a probability density function $f_X$ (X represents a living thing or the lifetime of a device).

Shannon entropy of X is given by

$$H_X=-{\mathbb{E}}_Z\left(log{f}_X\left(Z\right)\right),$$

where “log” is the natural logarithm function and Z is a non-negative random variable identically distributed like X.

Let $\alpha \in \mathbb{R}\setminus \left\{1\right\}$. Tsallis logarithm is given via

$${log}^T\left(x\right)=\left\{\begin{array}{cc}{\displaystyle \frac{x^{\alpha -1}-1}{\alpha -1}}\hfill & \mathrm{if}x>0\hfill \\ 0\hfill & \mathrm{if}x=0.\hfill \end{array}\right.$$

From this point onward, we assume that $\alpha >0$.

Tsallis entropy of X is defined by

$$H_X^T=-{\mathbb{E}}_Z\left({log}^T\left(f_X\left(Z\right)\right)\right).$$

In this paper, we work with Tsallis residual entropy, defined via

$$H_X^T\left(t\right)=-{\mathbb{E}}_Z\left({\displaystyle \frac{1}{{\overline{F}}_X\left(t\right)}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{{\overline{F}}_X\left(t\right)}}\right)|[Z>t]}\right) \mathrm{for} \mathrm{any} t\ge 0.$$

We recall that the quantile function of X is given by

$$Q_X\left(u\right)\stackrel{def}{=}{F}_X^{-1}\left(u\right)=inf\{x\in [0,\infty )|F_X\left(x\right)\ge u\} \mathrm{for} \mathrm{any} u\in [0,1].$$

We have $F_X\left(Q_X\left(u\right)\right)=u$ for any $u\in [0,1]$. Differentiating both sides of this equality with respect to u, we obtain $F_X^{\prime }\left(Q_X\left(u\right)\right)Q_X^{\prime }\left(u\right)=1$ for any $u\in [0,1]$. With the notation $q_X\left(u\right)=Q_X^{\prime }\left(u\right)$ for any $u\in [0,1]$, it follows that $q_X\left(u\right)f_X\left(Q_X\left(u\right)\right)=1$ for any $u\in [0,1]$.

Let ${\Psi }_X^T\left(u\right)=H_X^T\left(Q_X\left(u\right)\right)$ for any $u\in [0,1]$.

For any $u\in [0,1]$, we obtain

$$\begin{gathered} {\Psi }_X^T\left(u\right)=-{\mathbb{E}}_Z\left({\displaystyle \frac{1}{1-u}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{1-u}}\right)|[Z>Q_X\left(u\right)]}\right)= \\ -{\mathbb{E}}_U\left({\displaystyle \frac{1}{1-u}{log}^T\left({\displaystyle \frac{f_X\left(Q_X\left(U\right)\right)}{1-u}}\right)|[u<U<1]}\right), \end{gathered}$$

where U is a random variable uniformly distributed on $[0,1]$.

In this paper, we are concerned about comparing two absolutely continuous non-negative random variables from the point of view of Tsallis residual entropy. More precisely, if X and Y are absolutely continuous non-negative random variables, we compare ${\Psi }_X^T\left(u\right)$ and ${\Psi }_Y^T\left(u\right)$ for any $u\in [0,1]$.

In the proofs, we will make use of the lemma below.

Lemma 1 

(see [33]). Let $g:[0,\infty )\to [0,\infty )$ an increasing function and $h:[0,1]\times [0,\infty )\to \mathbb{R}$ such that

$${\mathbb{E}}_U\left(h(u,U)\left|\right[u<U<1]\right)\ge 0 \mathit{for} \mathit{any} u\in [0,1].$$

Then

$${\mathbb{E}}_U\left(h(u,U)g\left(U\right)\left|\right[u<U<1]\right)\ge 0 \mathit{for} \mathit{any} u\in [0,1].$$

3. Fundamental Results

Definition 1. 

We say that X is smaller than Y in Tsallis quantile entropy order (and denote by $X{\le }_{T}Y$) if ${\Psi }_X^T\left(u\right)\le {\Psi }_Y^T\left(u\right)$ for any $u\in [0,1]$.

In the last several years, stochastic orders and inequalities have been used intensively in many areas of probability and statistics, like reliability theory, queuing theory, survival analysis, biology, economics, insurance, actuarial science, operations research and management science. The simplest way of comparing two distribution functions is by a comparison of the associated means. Because this comparison is based on only two numbers (the means), it is sometimes not very informative. Moreover, the means to not exist is possible. In many applications, we have more detailed information concerning the comparison of two distribution functions than just the two means. If we compare two distribution functions with the same mean (or that are centered about the same value), we can compare the dispersion of these distributions. The simplest way of doing this is by the comparison of the associated standard deviations. But, again, the comparison depends on only two numbers (the standard deviations), which are at times not very informative. As mentioned above, it is possible for the standard deviations to not exist. The concept of stochastic orders plays a major role in the theory and practice of statistics. It generally refers to a set of relations that may hold between a pair of distributions of random variables. In reliability, the stochastic orders which compare life distributions based on different characteristics are used to study aging properties, to develop bounds on reliability functions, to compare the performance of policies and systems and to derive new inference procedures. Many of such orders are defined in terms of concepts based on distribution functions.

The theorem below is the main result of this paper.

Theorem 1. 

The following assertions are equivalent:

1. 

$X{\le }_{T}Y$.

2. 

${\mathbb{E}}_Z\left({\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0$ for any $t\ge 0$.

Proof. 

From Definition 1, $X{\le }_{T}Y$ if and only if

$$\begin{gathered} -{\mathbb{E}}_U\left({\displaystyle \frac{1}{1-u}{log}^T\left({\displaystyle \frac{f_X\left(Q_X\left(U\right)\right)}{1-u}}\right)|[u<U<1]}\right)\le \\ -{\mathbb{E}}_U\left({\displaystyle \frac{1}{1-u}{log}^T\left({\displaystyle \frac{f_Y\left(Q_Y\left(U\right)\right)}{1-u}}\right)|[u<U<1]}\right) \mathit{for} \mathit{any} u\in [0,1]. \end{gathered}$$

If we take $Q_X\left(U\right)=Z$ in the preceding inequality, the following equivalences are valid for any $u\in [0,1]$:

$$\begin{gathered} X{\le }_{T}Y⟺{\mathbb{E}}_Z\left({log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{{\overline{F}}_X\left(F_X^{-1}\left(u\right)\right)}}\right)|[Z>F_X^{-1}\left(u\right)]\right)\ge \\ {\mathbb{E}}_Z\left({log}^T\left({\displaystyle \frac{f_Y\left(Q_Y\left(F_X\left(Z\right)\right)\right)}{{\overline{F}}_X\left(F_X^{-1}\left(u\right)\right)}}\right)|[Z>F_X^{-1}\left(u\right)]\right)⟺ \\ {\mathbb{E}}_Z\left({log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{{\overline{F}}_X\left(F_X^{-1}\left(u\right)\right)}}\right)-{log}^T\left({\displaystyle \frac{f_Y\left(Q_Y\left(F_X\left(Z\right)\right)\right)}{{\overline{F}}_X\left(F_X^{-1}\left(u\right)\right)}}\right)|[Z>F_X^{-1}\left(u\right)]\right)\ge 0⟺ \\ {\mathbb{E}}_Z\left({\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>F_X^{-1}\left(u\right)]\right)\ge 0. \end{gathered}$$

In order to obtain the conclusion, it is sufficient to denote $F_X^{-1}\left(u\right)=t$. □

Definition 2 

(see [61]). We say that:

1. 

X is smaller than Y in the dispersive order (and write $X{\le }_{disp}Y$) if

$$f_X\left(x\right)\ge f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right) \mathit{for} \mathit{any} x\ge 0.$$

2. 

X is smaller than Y in the convex transform order (and write $X{\le }_{c}Y$) if the function

$${\displaystyle [0,\infty )\ni x⟼\frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)} \mathit{is} \mathit{non-negatively} \mathit{is} \mathit{increasing}.}$$

The dispersive order is a basic concept for comparing spread among probability distributions, with applications to order statistics, spacings and convolutions of independent random variables. The convex transform order is used to make precise comparisons between the skewness of probability distributions on the real line. From the point of view of the aging interpretation, this order can be seen as identifying aging rates in a way that also works when lifetimes do not start simultaneously (for more details concerning these two orders, the reader can consult [61]).

Theorem 2. 

If $X{\le }_{disp}Y$, then $X{\le }_{T}Y$.

Proof. 

Assume that $X{\le }_{disp}Y$. Then $f_X\left(x\right)\ge f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)$ for any $x\ge 0$, hence ${log}^T\left({\displaystyle \frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}}\right)\ge 0$ for any $x\ge 0$ and the conclusion follows Theorem 1. □

Theorem 3. 

If $X{\le }_{c}Y$ and $f_Y\left(0\right)\le f_X\left(0\right)$, then $X{\le }_{T}Y$.

Proof. 

Assume that $X{\le }_{c}Y$. Then the function ${\displaystyle [0,\infty )\ni x⟼\frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}}$ is non-negatively increasing; hence

$${\displaystyle \frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}\ge \frac{f_X\left(0\right)}{f_Y\left(0\right)}\ge 1 \mathrm{for} \mathrm{any} x\ge 0.}$$

With Theorem 1, we obtain the conclusion. □

4. Closure and Reversed Closure Properties

We consider $X_1,\dots ,X_n$ and $Y_1,\dots ,Y_n$ to be independent and identically distributed (i.i.d.) copies of X and Y, respectively, and

$$X_{1:n}=min\{X_1,\dots ,X_n\}, X_{n:n}=max\{X_1,\dots ,X_n\},$$

$$Y_{1:n}=min\{Y_1,\dots ,Y_n\}, Y_{n:n}=max\{Y_1,\dots ,Y_n\}.$$

Theorem 4. 

If $X{\le }_{T}Y$, then $X_{n:n}{\le }_T{Y}_{n:n}$.

Proof. 

Because $X{\le }_{T}Y$, we can determine with Theorem 1 that

$${\mathbb{E}}_Z\left({\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0\mathrm{for}\mathrm{any}t\ge 0.$$

(1)

It can be seen that, for any $x\ge 0,$

$$F_{X_{n:n}}\left(x\right)={\left(F_X\left(x\right)\right)}^n,$$

$$F_{Y_{n:n}}\left(x\right)={\left(F_Y\left(x\right)\right)}^n,$$

$$f_{X_{n:n}}\left(x\right)=n{\left(F_X\left(x\right)\right)}^{n-1}{f}_X\left(x\right),$$

$$f_{Y_{n:n}}\left(x\right)=n{\left(F_Y\left(x\right)\right)}^{n-1}{f}_Y\left(x\right),$$

$$F_{Y_{n:n}}^{-1}\left(F_{X_{n:n}}\left(x\right)\right)=F_Y^{-1}\left(F_X\left(x\right)\right),$$

$$f_{Y_{n:n}}\left(F_{Y_{n:n}}^{-1}\left(F_{X_{n:n}}\left(x\right)\right)\right)=n{\left(F_X\left(x\right)\right)}^{n-1}{f}_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)$$

and

$${\displaystyle \frac{f_{X_{n:n}}\left(x\right)}{f_{Y_{n:n}}\left(F_{Y_{n:n}}^{-1}\left(F_{X_{n:n}}\left(x\right)\right)\right)}=\frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}.}$$

Then

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{Y_{n:n}}\left(F_{Y_{n:n}}^{-1}\left(F_{X_{n:n}}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X_{n:n}}\left(Z\right)}{f_{Y_{n:n}}\left(F_{Y_{n:n}}^{-1}\left(F_{X_{n:n}}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)= \\ {\mathbb{E}}_Z\left({\left(n{\left(F_X\left(Z\right)\right)}^{n-1}\right)}^{\alpha }{\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right) \mathrm{for} \mathrm{any} \\ t\ge 0. \end{gathered}$$

Because the function

$$[0,\infty )\ni x⟼{\left(n{\left(F_X\left(x\right)\right)}^{n-1}\right)}^{\alpha } \mathrm{is} \mathrm{non-negatively} \mathrm{increasing},$$

it follows, via inequality (1) and Lemma 1, that

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{Y_{n:n}}\left(F_{Y_{n:n}}^{-1}\left(F_{X_{n:n}}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X_{n:n}}\left(Z\right)}{f_{Y_{n:n}}\left(F_{Y_{n:n}}^{-1}\left(F_{X_{n:n}}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \mathrm{any} \\ t\ge 0. \end{gathered}$$

The relationship $X_{n:n}{\le }_T{Y}_{n:n}$ follows Theorem 1. □

Theorem 5. 

If $X_{1:n}{\le }_T{Y}_{1:n}$, then $X{\le }_{T}Y$.

Proof. 

Because $X_{1:n}{\le }_T{Y}_{1:n}$, we have, by Theorem 1, that

$${\mathbb{E}}_Z\left({\left(f_{Y_{1:n}}\left(F_{Y_{1:n}}^{-1}\left(F_{X_{1:n}}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X_{1:n}}\left(Z\right)}{f_{Y_{1:n}}\left(F_{Y_{1:n}}^{-1}\left(F_{X_{1:n}}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0\mathrm{for}\mathrm{any}t\ge 0.$$

(2)

We can see that, for any $x\ge 0,$

$${\overline{F}}_{X_{1:n}}\left(x\right)={\left({\overline{F}}_X\left(x\right)\right)}^n,$$

$${\overline{F}}_{Y_{1:n}}\left(x\right)={\left({\overline{F}}_Y\left(x\right)\right)}^n,$$

$$f_{X_{1:n}}\left(x\right)=n{\left({\overline{F}}_X\left(x\right)\right)}^{n-1}{f}_X\left(x\right),$$

$$f_{Y_{1:n}}\left(x\right)=n{\left({\overline{F}}_Y\left(x\right)\right)}^{n-1}{f}_Y\left(x\right),$$

$$F_{Y_{1:n}}^{-1}\left(F_{X_{1:n}}\left(x\right)\right)=F_Y^{-1}\left(F_X\left(x\right)\right),$$

$$f_{Y_{1:n}}\left(F_{Y_{1:n}}^{-1}\left(F_{X_{1:n}}\left(x\right)\right)\right)=n{\left({\overline{F}}_X\left(x\right)\right)}^{n-1}{f}_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)$$

and

$${\displaystyle \frac{f_{X_{1:n}}\left(x\right)}{f_{Y_{1:n}}\left(F_{Y_{1:n}}^{-1}\left(F_{X_{1:n}}\left(x\right)\right)\right)}=\frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}.}$$

Then

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)= \\ {\mathbb{E}}_Z\left({\left(n{\overline{F}}_X{\left(Z\right))}^{n-1}\right)}^{-\alpha }{\left(f_{Y_{1:n}}\left(F_{Y_{1:n}}^{-1}\left(F_{X_{1:n}}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X_{1:n}}\left(Z\right)}{f_{Y_{1:n}}\left(F_{Y_{1:n}}^{-1}\left(F_{X_{1:n}}\left(Z\right)\right)\right)}}\right)|[Z>t]\right) \\ \mathrm{for} \mathrm{any} t\ge 0. \end{gathered}$$

Because the function

$$[0,\infty )\ni x⟼{\left(n{\overline{F}}_X{\left(x\right))}^{n-1}\right)}^{-\alpha } \mathrm{is} \mathrm{non-negatively} \mathrm{increasing},$$

it follows, via inequality (2) and Lemma 1, that

$${\mathbb{E}}_Z\left({\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \mathrm{any} t\ge 0.$$

By applying Theorem 1, we obtain that $X{\le }_{T}Y$. □

The natural step is to generalize the preceding two theorems from a finite number n to a random variable N.

We consider $X_1,X_2,\dots$ and $Y_1,Y_2,\dots$ as sequences of independent and identically distributed copies of X and Y, respectively. Let N be a positive integer random variable with the probability mass function $p_N\left(n\right)=P(N=n)$, $n=1,2,\dots$ and such that N is independent of $X_i^{\prime }s$ and $Y_i^{\prime }s$. Take

$$X_{1:N}=min\{X_1,\dots ,X_N\},X_{N:N}=max\{X_1,\dots ,X_N\}$$

and

$$Y_{1:N}=min\{Y_1,\dots ,Y_N\},Y_{N:N}=max\{Y_1,\dots ,Y_N\}.$$

Theorem 6. 

If $X{\le }_{T}Y$, then $X_{N:N}{\le }_T{Y}_{N:N}$.

Proof. 

Because $X{\le }_{T}Y$, we can determine by Theorem 1 that

$${\mathbb{E}}_Z\left({\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0\mathrm{for}\mathrm{any}t\ge 0.$$

(3)

One can see that, for any $x\ge 0,$

$${\displaystyle F_{X_{N:N}}\left(x\right)=\sum _{n=1}^{\infty }{\left(F_X\left(x\right)\right)}^n{p}_N\left(n\right),}$$

$${\displaystyle F_{Y_{N:N}}\left(x\right)=\sum _{n=1}^{\infty }{\left(F_Y\left(x\right)\right)}^n{p}_N\left(n\right),}$$

$$f_{X_{N:N}}\left(x\right)=\left[{\displaystyle \sum _{n=1}^{\infty }n{\left(F_X\left(x\right)\right)}^{n-1}{p}_N\left(n\right)}\right]·f_X\left(x\right)$$

and

$$f_{Y_{N:N}}\left(x\right)=\left[{\displaystyle \sum _{n=1}^{\infty }n{\left(F_Y\left(x\right)\right)}^{n-1}{p}_N\left(n\right)}\right]·f_Y\left(x\right).$$

It was proven in [61] that

$$F_{Y_{N:N}}^{-1}\left(F_{X_{N:N}}\left(x\right)\right)=F_Y^{-1}\left(F_X\left(x\right)\right) \mathrm{for} \mathrm{any} x\ge 0.$$

Hence, for any $x\ge 0,$

$$f_{Y_{N:N}}\left(F_{Y_{N:N}}^{-1}\left(F_{X_{N:N}}\left(x\right)\right)\right)=\left[{\displaystyle \sum _{n=1}^{\infty }n{\left(F_X\left(x\right)\right)}^{n-1}{p}_N\left(n\right)}\right]·f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)$$

and

$${\displaystyle \frac{f_{X_{N:N}}\left(x\right)}{f_{Y_{N:N}}\left(F_{Y_{N:N}}^{-1}\left(F_{X_{N:N}}\left(x\right)\right)\right)}=\frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}.}$$

Then

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{Y_{N:N}}\left(F_{Y_{N:N}}^{-1}\left(F_{X_{N:N}}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X_{N:N}}\left(Z\right)}{f_{Y_{N:N}}\left(F_{Y_{N:N}}^{-1}\left(F_{X_{N:N}}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)= \\ {\mathbb{E}}_Z\left({\left({\displaystyle \sum _{n=1}^{\infty }n{\left(F_X\left(Z\right)\right)}^{n-1}{p}_N\left(n\right)}\right)}^{\alpha }{\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right) \mathrm{for} \\ \mathrm{any} t\ge 0. \end{gathered}$$

Because the function

$$[0,\infty )\ni x⟼{\left({\displaystyle \sum _{n=1}^{\infty }n{\left(F_X\left(x\right)\right)}^{n-1}{p}_N\left(n\right)}\right)}^{\alpha } \mathrm{is} \mathrm{non-negatively} \mathrm{increasing},$$

it follows, via inequality (3) and Lemma 1, that

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{Y_{N:N}}\left(F_{Y_{N:N}}^{-1}\left(F_{X_{N:N}}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X_{N:N}}\left(Z\right)}{f_{Y_{N:N}}\left(F_{Y_{N:N}}^{-1}\left(F_{X_{N:N}}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \\ \mathrm{any} t\ge 0. \end{gathered}$$

The conclusion thus follows Theorem 1. □

Theorem 7. 

If $X_{1:N}{\le }_T{Y}_{1:N}$, then $X{\le }_{T}Y$.

Proof. 

Because $X_{1:N}{\le }_T{Y}_{1:N}$, we can determine by Theorem 1 that

$${\mathbb{E}}_Z\left({\left(f_{Y_{1:N}}\left(F_{Y_{1:N}}^{-1}\left(F_{X_{1:N}}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X_{1:N}}\left(Z\right)}{f_{Y_{1:N}}\left(F_{Y_{1:N}}^{-1}\left(F_{X_{1:N}}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0\mathrm{for}\mathrm{any}t\ge 0.$$

(4)

We can see that, for any $x\ge 0$,

$${\displaystyle {\overline{F}}_{X_{N:N}}\left(x\right)=\sum _{n=1}^{\infty }{\left({\overline{F}}_X\left(x\right)\right)}^n{p}_N\left(n\right),}$$

$${\displaystyle {\overline{F}}_{Y_{N:N}}\left(x\right)=\sum _{n=1}^{\infty }{\left({\overline{F}}_Y\left(x\right)\right)}^n{p}_N\left(n\right),}$$

$$f_{X_{1:N}}\left(x\right)=\left[{\displaystyle \sum _{n=1}^{\infty }n{\left({\overline{F}}_X\left(x\right)\right)}^{n-1}{p}_N\left(n\right)}\right]·f_X\left(x\right)$$

and

$$f_{Y_{1:N}}\left(x\right)=\left[{\displaystyle \sum _{n=1}^{\infty }n{\left({\overline{F}}_Y\left(x\right)\right)}^{n-1}{p}_N\left(n\right)}\right]·f_Y\left(x\right).$$

It was proven in [61] that

$$F_{Y_{1:N}}^{-1}\left(F_{X_{1:N}}\left(x\right)\right)=F_Y^{-1}\left(F_X\left(x\right)\right) \mathrm{for} \mathrm{any} x\ge 0.$$

Hence, for any $x\ge 0,$

$$f_{Y_{1:N}}\left(F_{Y_{1:N}}^{-1}\left(F_{X_{1:N}}\left(x\right)\right)\right)=\left[{\displaystyle \sum _{n=1}^{\infty }n{\left({\overline{F}}_X\left(x\right)\right)}^{n-1}{p}_N\left(n\right)}\right]·f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)$$

and

$${\displaystyle \frac{f_{X_{1:N}}\left(x\right)}{f_{Y_{1:N}}\left(F_{Y_{1:N}}^{-1}\left(F_{X_{1:N}}\left(x\right)\right)\right)}=\frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)},}$$

Then

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)= \\ {\mathbb{E}}_Z\left({\displaystyle \frac{{\left(f_{Y_{1:N}}\left(F_{Y_{1:N}}^{-1}\left(F_{X_{1:N}}\left(Z\right)\right)\right)\right)}^{\alpha -1}}{{\left({\displaystyle \sum _{n=1}^{\infty }n{\left({\overline{F}}_X\left(Z\right)\right)}^{n-1}{p}_N\left(n\right)}\right)}^{\alpha }}{log}^T\left({\displaystyle \frac{f_{X_{1:N}}\left(Z\right)}{f_{Y_{1:N}}\left(F_{Y_{1:N}}^{-1}\left(F_{X_{1:N}}\left(Z\right)\right)\right)}}\right)|[Z>t]}\right) \mathrm{for} \mathrm{any} \\ t\ge 0. \end{gathered}$$

Because the function

$$[0,\infty )\ni x⟼{\left({\displaystyle \sum _{n=1}^{\infty }n{\left({\overline{F}}_X\left(x\right)\right)}^{n-1}{p}_N\left(n\right)}\right)}^{-\alpha } \mathrm{is} \mathrm{non-negatively} \mathrm{increasing},$$

it follows, via inequality (4) and Lemma 1, that

$${\mathbb{E}}_Z\left({\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \mathrm{any} t\ge 0.$$

By Theorem 1, we conclude that $X{\le }_{T}Y$. □

5. Preservation of Tsallis Quantile Entropy Order in the Proportional Hazard Rate Model

We consider the following proportional hazard rate model (see [61]), namely for any $\theta >0$, for which we take $X\left(\theta \right)$ and $Y\left(\theta \right)$ as two absolutely continuous non-negative random variables with the survival functions ${\left({\overline{F}}_X\right)}^{\theta }$ and ${\left({\overline{F}}_Y\right)}^{\theta }$, respectively.

Theorem 8. 

1. 

If $\theta \ge 1$ and $X{\le }_{T}Y$, then $X\left(\theta \right){\le }_{T}Y\left(\theta \right)$.

2. 

If $0<\theta \le 1$ and $X\left(\theta \right){\le }_{T}Y\left(\theta \right)$, then $X{\le }_{T}Y$.

Proof. 

For any $x\ge 0$, we can obtain:

$${\overline{F}}_{X\left(\theta \right)}\left(x\right)={\left({\overline{F}}_X\left(x\right)\right)}^{\theta },$$

$${\overline{F}}_{Y\left(\theta \right)}\left(x\right)={\left({\overline{F}}_Y\left(x\right)\right)}^{\theta },$$

$$f_{X\left(\theta \right)}\left(x\right)=\theta {\left({\overline{F}}_X\left(x\right)\right)}^{\theta -1}{f}_X\left(x\right),$$

$$f_{Y\left(\theta \right)}\left(x\right)=\theta {\left({\overline{F}}_Y\left(x\right)\right)}^{\theta -1}{f}_Y\left(x\right),$$

$$F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(x\right)\right)=F_Y^{-1}\left(F_X\left(x\right)\right),$$

$$f_{Y\left(\theta \right)}\left(F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(x\right)\right)\right)=\theta {\left({\overline{F}}_X\left(x\right)\right)}^{\theta -1}{f}_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)$$

and

$${\displaystyle \frac{f_{X\left(\theta \right)}\left(x\right)}{f_{Y\left(\theta \right)}\left(F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(x\right)\right)\right)}=\frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}.}$$

Then:

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{Y\left(\theta \right)}\left(F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X\left(\theta \right)}\left(Z\right)}{f_{Y\left(\theta \right)}\left(F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)= \\ {\mathbb{E}}_Z\left({\left(\theta {\left({\overline{F}}_X\left(Z\right)\right)}^{\theta -1}\right)}^{\alpha }{\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \\ \mathrm{any} t\ge 0. \end{gathered}$$

1.

If $0<\theta \le 1$ and $X{\le }_{T}Y$, then the function

$$[0,\infty )\ni x⟼{\left(\theta {\left({\overline{F}}_X\left(x\right)\right)}^{\theta -1}\right)}^{\alpha } \mathrm{is} \mathrm{non-negatively} \mathrm{increasing}$$

and

$${\mathbb{E}}_Z\left({\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \mathrm{any} t\ge 0.$$

Using Lemma 1, we can determine that $X\left(\theta \right){\le }_{T}Y\left(\theta \right)$.

2.

If $\theta \ge 1$ and $X\left(\theta \right){\le }_{T}Y\left(\theta \right)$, then the function

$$[0,\infty )\ni x⟼{\left(\theta {\left({\overline{F}}_X\left(x\right)\right)}^{\theta -1}\right)}^{-\alpha } \mathrm{is} \mathrm{non-negatively} \mathrm{increasing}$$

and

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{Y\left(\theta \right)}\left(F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X\left(\theta \right)}\left(Z\right)}{f_{Y\left(\theta \right)}\left(F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \\ \mathrm{for} \mathrm{any} t\ge 0. \end{gathered}$$

Using Lemma 1, we can determine that $X{\le }_{T}Y$.

□

6. Preservation of Tsallis Quantile Entropy Order in the Proportional Reversed Hazard Rate Model

We consider the following proportional reversed hazard rate model (see [61]), namely for any $\theta >0$, for which we take $X\left(\theta \right)$ and $Y\left(\theta \right)$ as two absolutely continuous non-negative random variables with the distribution functions ${\left(F_X\right)}^{\theta }$ and ${\left(F_Y\right)}^{\theta }$, respectively.

Theorem 9. 

1. 

If $\theta \ge 1$ and $X{\le }_{T}Y$, then $X\left(\theta \right){\le }_{T}Y\left(\theta \right)$.

2. 

If $0<\theta \le 1$ and $X\left(\theta \right){\le }_{T}Y\left(\theta \right)$, then $X{\le }_{T}Y$.

Proof. 

We can determine for any $x\ge 0$:

$$F_{X\left(\theta \right)}\left(x\right)={\left(F_X\left(x\right)\right)}^{\theta },$$

$$F_{Y\left(\theta \right)}\left(x\right)={\left(F_Y\left(x\right)\right)}^{\theta },$$

$$f_{X\left(\theta \right)}\left(x\right)=\theta {\left(F_X\left(x\right)\right)}^{\theta -1}{f}_X\left(x\right),$$

$$f_{Y\left(\theta \right)}\left(x\right)=\theta {\left(F_Y\left(x\right)\right)}^{\theta -1}{f}_Y\left(x\right),$$

$$F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(x\right)\right)=F_Y^{-1}\left(F_X\left(x\right)\right),$$

$$f_{Y\left(\theta \right)}\left(F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(x\right)\right)\right)=\theta {\left(F_X\left(x\right)\right)}^{\theta -1}{f}_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)$$

and

$${\displaystyle \frac{f_{X\left(\theta \right)}\left(x\right)}{f_{Y\left(\theta \right)}\left(F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(x\right)\right)\right)}=\frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}.}$$

Then:

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{Y\left(\theta \right)}\left(F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X\left(\theta \right)}\left(Z\right)}{f_{Y\left(\theta \right)}\left(F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)= \\ {\mathbb{E}}_Z\left({\left(\theta {\left(F_X\left(Z\right)\right)}^{\theta -1}\right)}^{\alpha }{\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \\ \mathrm{any} t\ge 0. \end{gathered}$$

1.

If $\theta \ge 1$ and $X{\le }_{T}Y$, then the function

$$[0,\infty )\ni x⟼{\left(\theta {\left(F_X\left(x\right)\right)}^{\theta -1}\right)}^{\alpha } \mathrm{is} \mathrm{non-negatively} \mathrm{increasing}$$

and

$${\mathbb{E}}_Z\left({\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \mathrm{any} t\ge 0.$$

Using Lemma 1, we can determine that $X\left(\theta \right){\le }_{T}Y\left(\theta \right)$.

2.

If $0<\theta \le 1$ and $X\left(\theta \right){\le }_{T}Y\left(\theta \right)$, then the function

$$[0,\infty )\ni x⟼{\left(\theta {\left(F_X\left(x\right)\right)}^{\theta -1}\right)}^{-\alpha } \mathrm{is} \mathrm{non-negatively} \mathrm{increasing}$$

and

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{Y\left(\theta \right)}\left(F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X\left(\theta \right)}\left(Z\right)}{f_{Y\left(\theta \right)}\left(F_{Y\left(\theta \right)}^{-1}\left(F_{X\left(\theta \right)}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \\ \mathrm{for} \mathrm{any} t\ge 0. \end{gathered}$$

Using Lemma 1, we can determine that $X{\le }_{T}Y$.

□

7. Preservation of Tsallis Quantile Entropy Order in the Proportional Odds Model

We work with the following proportional odds model (see [62]), namely for any $\theta >0$, for which we take the proportional odds random variables $X_p$ and $Y_p$, defined by the survival functions ${\displaystyle {\overline{F}}_{X_p}\left(x\right)=\frac{\theta {\overline{F}}_X\left(x\right)}{1-(1-\theta ){\overline{F}}_X\left(x\right)}}$ and ${\displaystyle {\overline{F}}_{Y_p}\left(x\right)=\frac{\theta {\overline{F}}_Y\left(x\right)}{1-(1-\theta ){\overline{F}}_Y\left(x\right)}}$, respectively, for any $x\ge 0$.

Theorem 10. 

1. 

If $\theta \ge 1$ and $X{\le }_{T}Y$, then $X_p{\le }_T{Y}_p$.

2. 

If $0<\theta \le 1$ and $X_p{\le }_T{Y}_p$, then $X{\le }_{T}Y$.

Proof. 

For any $x\ge 0$ we have

$${\displaystyle {\overline{F}}_{X_p}\left(x\right)=\frac{\theta {\overline{F}}_X\left(x\right)}{1-(1-\theta ){\overline{F}}_X\left(x\right)},}$$

$${\displaystyle {\overline{F}}_{Y_p}\left(x\right)=\frac{\theta {\overline{F}}_Y\left(x\right)}{1-(1-\theta ){\overline{F}}_Y\left(x\right)},}$$

$${\displaystyle f_{X_p}\left(x\right)=\frac{\theta }{{\left(1-(1-\theta ){\overline{F}}_X\left(x\right)\right)}^2}·f_X\left(x\right),}$$

$${\displaystyle f_{Y_p}\left(x\right)=\frac{\theta }{{\left(1-(1-\theta ){\overline{F}}_Y\left(x\right)\right)}^2}·f_Y\left(x\right),}$$

$$F_{Y_p}^{-1}\left(F_{X_p}\left(x\right)\right)=F_Y^{-1}\left(F_X\left(x\right)\right),$$

$${\displaystyle f_{Y_p}\left(F_{Y_p}^{-1}\left(F_{X_p}\left(x\right)\right)\right)=\frac{\theta }{{\left(1-(1-\theta ){\overline{F}}_X\left(x\right)\right)}^2}·f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}$$

and

$${\displaystyle \frac{f_{X_p}\left(x\right)}{f_{Y_p}\left(F_{Y_p}^{-1}\left(F_{X_p}\left(x\right)\right)\right)}=\frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}.}$$

Then

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{Y_p}\left(F_{Y_p}^{-1}\left(F_{X_p}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X_p}\left(Z\right)}{f_{Y_p}\left(F_{Y_p}^{-1}\left(F_{X_p}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)= \\ {\mathbb{E}}_Z\left({\left({\displaystyle \frac{\theta }{{\left(1-(1-\theta ){\overline{F}}_X\left(Z\right)\right)}^2}}\right)}^{\alpha }{\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right) \\ \mathrm{for} \mathrm{any}t\ge 0. \end{gathered}$$

1.

Assume that $X{\le }_{T}Y$ and $\theta \ge 1$. Then

$${\mathbb{E}}_Z\left({\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \mathrm{any} t\ge 0$$

and the function

$$[0,\infty )\ni x⟼{\left({\displaystyle \frac{\theta }{{\left(1-(1-\theta ){\overline{F}}_X\left(x\right)\right)}^2}}\right)}^{\alpha } \mathrm{is} \mathrm{non-negatively} \mathrm{increasing}.$$

Hence, by Lemma 1, we obtain $X_p{\le }_T{Y}_p$.

2.

Assume that $X_p{\le }_T{Y}_p$ and $0<\theta \le 1$. Then

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{Y_p}\left(F_{Y_p}^{-1}\left(F_{X_p}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{X_p}\left(Z\right)}{f_{Y_p}\left(F_{Y_p}^{-1}\left(F_{X_p}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \mathrm{any} \\ t\ge 0 \end{gathered}$$

and the function

$$[0,\infty )\ni x⟼{\left({\displaystyle \frac{\theta }{{\left(1-(1-\theta ){\overline{F}}_X\left(x\right)\right)}^2}}\right)}^{-\alpha } \mathrm{is} \mathrm{non-negatively} \mathrm{increasing}.$$

Hence, by Lemma 1, we obtain $X{\le }_{T}Y$.

□

8. Preservation of Tsallis Quantile Entropy Order in the Record Values Model

Let $\{X_i$|$i\ge 1\}$ and $\{Y_i$|$i\ge 1\}$ be sequences of i.i.d. random variables from the random variables X and Y, respectively, with survival functions ${\overline{F}}_X$ and ${\overline{F}}_Y$, respectively, and density functions $f_X$ and $f_Y$, respectively. We consider the nth record times $T_n^X$ and $T_n^Y$, respectively, defined via $T_1^X=1$ and $T_{n+1}^X=min\{j>T_n^X$|$X_j>X_{T_n^X}\}$ for any $n\ge 1$ and $T_1^Y=1$ and $T_{n+1}^Y=min\{j>T_n^Y$|$Y_j>Y_{T_n^Y}\}$, respectively.

We denote $X_{T_n^X}\stackrel{def}{=}{R}_n^X$ and $Y_{T_n^Y}\stackrel{def}{=}{R}_n^Y$, respectively, and call them the nth record values (see [63]).

For any $x\ge 0$, we can obtain

$${\displaystyle {\overline{F}}_{R_n^X}\left(x\right)={\overline{F}}_X\left(x\right)\sum _{j=0}^{n-1}\frac{{\left({\Lambda }_X\left(x\right)\right)}^j}{j!}={\overline{\Gamma }}_n\left({\Lambda }_X\left(x\right)\right),}$$

$${\displaystyle {\overline{F}}_{R_n^Y}\left(x\right)={\overline{F}}_Y\left(x\right)\sum _{j=0}^{n-1}\frac{{\left({\Lambda }_Y\left(x\right)\right)}^j}{j!}={\overline{\Gamma }}_n\left({\Lambda }_Y\left(x\right)\right),}$$

$${\displaystyle f_{R_n^X}\left(x\right)=\frac{1}{\Gamma \left(n\right)}{\Lambda }_X^{n-1}\left(x\right)f_X\left(x\right)}$$

and

$${\displaystyle f_{R_n^Y}\left(x\right)=\frac{1}{\Gamma \left(n\right)}{\Lambda }_Y^{n-1}\left(x\right)f_Y\left(x\right),}$$

where ${\overline{\Gamma }}_n$ is the survival function of a Gamma random variable with the shape parameter n and the scale parameter 1, ${\Lambda }_X\left(x\right)=-log{\overline{F}}_X\left(x\right)$ is the cumulative failure rate function of X and ${\Lambda }_Y\left(x\right)=-log{\overline{F}}_Y\left(x\right)$ is the cumulative failure rate function of Y.

Theorem 11. 

Let $m,n\in \mathbb{N}\stackrel{def}{=}\{1,2,\dots \}$.

1. 

If $X{\le }_{T}Y$, then $R_n^X{\le }_T{R}_n^Y$.

2. 

If $n>m\ge 1$ and $R_m^X{\le }_T{R}_m^Y$, then $R_n^X{\le }_T{R}_n^Y$.

Proof. 

1.

If $X{\le }_{T}Y$, then

$${\mathbb{E}}_Z\left({\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \mathrm{any} t\ge 0.$$

We have, for any $x\ge 0$,

$$F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(x\right)\right)=F_Y^{-1}\left(F_X\left(x\right)\right),$$

$${\displaystyle f_{R_n^Y}\left(F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(x\right)\right)\right)=\frac{1}{\Gamma \left(n\right)}{\Lambda }_X^{n-1}\left(x\right)f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}$$

and

$${\displaystyle \frac{f_{R_n^X}\left(x\right)}{f_{R_n^Y}\left(F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(x\right)\right)\right)}=\frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}.}$$

Then

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{R_n^Y}\left(F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{R_n^X}\left(Z\right)}{f_{R_n^Y}\left(F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)= \\ {\mathbb{E}}_Z\left({\left({\displaystyle \frac{1}{\Gamma \left(n\right)}{\Lambda }_X^{n-1}\left(Z\right)}\right)}^{\alpha }{\left(f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_X\left(Z\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(Z\right)\right)\right)}}\right)|[Z>t]\right) \\ \mathrm{for} \mathrm{any} t\ge 0. \end{gathered}$$

Because the function

$$[0,\infty )\ni x⟼{\left({\displaystyle \frac{1}{\Gamma \left(n\right)}{\Lambda }_X^{n-1}\left(x\right)}\right)}^{\alpha } \mathrm{is} \mathrm{non-negatively} \mathrm{increasing},$$

we obtain via Lemma 1 that

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{R_n^Y}\left(F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{R_n^X}\left(Z\right)}{f_{R_n^Y}\left(F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \mathrm{any} \\ t\ge 0, \end{gathered}$$

i.e.,  $R_n^X{\le }_T{R}_n^Y.$

2.

If $R_m^X{\le }_T{R}_m^Y$, then

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{R_m^Y}\left(F_{R_m^Y}^{-1}\left(F_{R_m^X}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{R_m^X}\left(Z\right)}{f_{R_m^Y}\left(F_{R_m^Y}^{-1}\left(F_{R_m^X}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \mathrm{any} \\ t\ge 0. \end{gathered}$$

For any $x\ge 0$, we can determine

$$F_{R_m^Y}^{-1}\left(F_{R_m^X}\left(x\right)\right)=F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(x\right)\right)=F_Y^{-1}\left(F_X\left(x\right)\right),$$

$${\displaystyle \frac{f_{R_m^X}\left(x\right)}{f_{R_m^Y}\left(F_{R_m^Y}^{-1}\left(F_{R_m^X}\left(x\right)\right)\right)}=\frac{f_{R_n^X}\left(x\right)}{f_{R_n^Y}\left(F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(x\right)\right)\right)}=\frac{f_X\left(x\right)}{f_Y\left(F_Y^{-1}\left(F_X\left(x\right)\right)\right)}}$$

and

$${\displaystyle f_{R_n^Y}\left(F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(x\right)\right)\right)=\frac{\Gamma \left(m\right)}{\Gamma \left(n\right)}{\left({\Lambda }_X\left(x\right)\right)}^{n-m}{f}_{R_m^Y}\left(F_{R_m^Y}^{-1}\left(F_{R_m^X}\left(x\right)\right)\right).}$$

Then

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{R_n^Y}\left(F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{R_n^X}\left(Z\right)}{f_{R_n^Y}\left(F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)= \\ {\mathbb{E}}_Z\left({\left({\displaystyle \frac{\Gamma \left(m\right)}{\Gamma \left(n\right)}{\left({\Lambda }_X\left(Z\right)\right)}^{n-m}}\right)}^{\alpha }{\left(f_{R_m^Y}\left(F_{R_m^Y}^{-1}\left(F_{R_m^X}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{R_m^X}\left(Z\right)}{f_{R_m^Y}\left(F_{R_m^Y}^{-1}\left(F_{R_m^X}\left(Z\right)\right)\right)}}\right)|[Z>t]\right) \\ \mathrm{for} \mathrm{any} t\ge 0. \end{gathered}$$

Because the function

$$[0,\infty )\ni x⟼{\left({\displaystyle \frac{\Gamma \left(m\right)}{\Gamma \left(n\right)}{\left({\Lambda }_X\left(x\right)\right)}^{n-m}}\right)}^{\alpha } \mathrm{is} \mathrm{non-negatively} \mathrm{increasing}$$

and

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{R_m^Y}\left(F_{R_m^Y}^{-1}\left(F_{R_m^X}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{R_m^X}\left(Z\right)}{f_{R_m^Y}\left(F_{R_m^Y}^{-1}\left(F_{R_m^X}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \mathrm{any} \\ t\ge 0, \end{gathered}$$

using Lemma 1, we obtain that

$$\begin{gathered} {\mathbb{E}}_Z\left({\left(f_{R_n^Y}\left(F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(Z\right)\right)\right)\right)}^{\alpha -1}{log}^T\left({\displaystyle \frac{f_{R_n^X}\left(Z\right)}{f_{R_n^Y}\left(F_{R_n^Y}^{-1}\left(F_{R_n^X}\left(Z\right)\right)\right)}}\right)|[Z>t]\right)\ge 0 \mathrm{for} \mathrm{any} \\ t\ge 0, \end{gathered}$$

i.e., $R_n^X{\le }_T{R}_n^Y$.

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9. Conclusions

We introduced Tsallis quantile entropy order between two random variables, found necessary and sufficient conditions for it and proved closure and reversed closured properties of this order under parallel and series operations. We also showed that Tsallis quantile entropy order is preserved in some stochastic models, like proportional hazard rate model, proportional reversed hazard rate model, proportional odds model and record values model. In this way, there are generalized results from other papers working with Tsallis residual entropy instead of Shannon residual entropy (which is used in [30,31,32]), Rényi residual entropy (which is used in [33,34]), Varma residual entropy (which is used in [35]) and Awad-Varma residual entropy (which is used in [36]). The difference is that we work with Tsallis residual entropy instead of other residual entropies considered in the aforementioned papers.