Ordered Variables and Their Concomitants under Extropy via COVID-19 Data Application

Mohamed, Mohamed S.; Abdulrahman, Alanazi Talal; Almaspoor, Zahra; Yusuf, M.

doi:https://doi.org/10.1155/2021/6491817

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Dynamic Analysis, Learning, and Robust Control of Complex Systems

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Volume 2021 | Article ID 6491817 | https://doi.org/10.1155/2021/6491817

Ordered Variables and Their Concomitants under Extropy via COVID-19 Data Application

Mohamed S. Mohamed,¹Alanazi Talal Abdulrahman,²Zahra Almaspoor,³and M. Yusuf⁴

Academic Editor: Ning Cai

Received19 Apr 2021

Accepted21 Jun 2021

Published06 Jul 2021

Abstract

Extropy, as a complementary dual of entropy, has been discussed in many works of literature, where it is declared for other measures as an extension of extropy. In this article, we obtain the extropy of generalized order statistics via its dual and give some examples from well-known distributions. Furthermore, we study the residual and past extropy for such models. On the other hand, based on Farlie–Gumbel–Morgenstern distribution, we consider the residual extropy of concomitants of m-generalized order statistics and present this measure with some additional features. In addition, we provide the upper bound and stochastic orders of it. Finally, nonparametric estimation of the residual extropy of concomitants of m-generalized order statistics is included using simulated and real data connected with COVID-19 virus.

1. Introduction

Shannon [1] introduced a well-known vintage measure of uncertainty called Shannon entropy. This information theoretic entropy manipulates in diverse fields such as financial analysis, computer science, and medical research. The extropy proposed by Lad et al. [2] is an accomplishment to notions of information based on entropy. They exhibited that entropy has a complementary dual function known as “extropy.” In the view of extropy in discrete density, the extropy measure is neatly closer to when the range of possibilities increases (as a result of larger N). Therefore, to realize extropy for a continuous density, the extropy of a nonnegative continuous random variable (r.v.) X, with probability density function (PDF) is defined as

The extropy measure has been developed for ordered variables. Qiu [3] was the first to apply extropy for order statistics and record values and present several of their properties. After that, the researchers manifested to present extension measures of extropy. Qiu and Jia [4] investigated the connotation of residual extropy of a nonnegative r.v. as

Qiu et al. [5] presented a mixed systems lifetime via extropy and obtained some features and bounds of it. Recently, Jose and Sathar [6, 7] exploited the residual and past extropy of k-records, respectively, emerging from any continuous distribution. For extra studies on extropy, see Qiu and Jia [8], Yang et al. [9], Noughabi and Jarrahiferiz [10], Raqab and Qiu [11], and Lad et al. [12].

Krishnan et al. [13] presented the past extropy, for a fixed , for past lifetime of r.v. as follows:

Jahanshahi et al. [14] proposed cumulative residual extropy (CREX). For a nonnegative r.v. X with an absolutely continuous survival function , the CREX is given by

In analogy with Jahanshahi et al. [14], Abdul Sathar and Dhanya [15] introduced the CREX and refer to it as survival extropy. Moreover, they conduct the dynamic survival extropy as

It is easy to observe that extropy and its related measures are constantly negative.

The connotation of generalized order statistics () that contains all forms of ordered random observations was first proposed by Kamps [16]. Let , , be parameters such that , for all . For a subclass of (called ), where , the PDF of the , , can be written aswhere , , for ,

We will write , and for all , since .

Based on descending ordered r.v.’s, Pawlas and Szynal [17] and Burkschat et al. [18] presented the dual generalized order statistics (). By the same manner and parameters in , when , the PDF of is defined by

The concept of concomitants of ordinary order statistic was derived by David et al. [19]. Let , , be pairs of independent r.v.’s drawn from some bivariate distributions with cumulative distribution function (CDF) . Let be the order statistic, then the r.v. concerned with is called the concomitant of order statistics and is specified by .

The Farlie–Gumbel–Morgenstern (FGM) family is an extremely supple class of bivariate family; it was primarily derived by Morgenstern [20], which is set by CDF and PDF, respectively, as follows:where , and , are the marginal CDF’s and PDF’s of and , respectively, . If the dependent parameter , then and are not dependent. Beg and Ahsanullah [21] introduced the PDF of the concomitant of , , under the FGM family as follows:wherewith parameters , , , such that , for all .

Throughout this paper, we propose the extropy of and and study those models for the related measures of extropy. In the second part of the paper, we deal with the concomitants of of FGM family to extract the residual extropy and give some of its properties. The paper is organized as follows: Section 2 contains extropy of and obtained from uniform distribution. Moreover, we obtain them for any distribution in terms of the obtained extropy from uniform distribution. Meanwhile, we produce some examples of some well-known distributions. Furthermore, we obtain the lower bound of the extropy of in terms of the mode. In addition, the residual and past extropy of and is considered in Section 3. In Section 4, we derive the residual extropy of concomitants of of FGM family and discuss its relation with the stop-loss transform and Gini index. Besides, we consider this model in terms of its upper bound and produce some examples on it. Finally, in Section 5, real-life data connected with the COVID-19 virus is applied for the nonparametric estimation of residual extropy of concomitants of order statistics under the FGM family.

2. Extropy of m-Generalized Order Statistics and Its Dual

In this section, we discuss the extropy of and for uniform distribution and for any distribution, which depends on beta function and its generalized first kind.

Based on uniform distribution , the extropy of the , , , can be obtained by the following theorem.

Theorem 1. Suppose the nonnegative continuous r.v. X be emerging from distribution. Let be its ; then, from (1) and (6), the extropy of is given by

Proof. From (1) and (6), we havewhere . Using the transformation , thuswhere is the beta function. Therefore, we can reduce the beta function in (15) as follows:Furthermore,and the result is taken after.
In the following theorem, we derive the extropy of , , from any continuous distribution based on extropy of emerging from .

Theorem 2. Let X be a continuous r.v. that is nonnegative with CDF . Then, from (1), (6), and (13), the extropy of the , , , is given bywhere has generalized beta of first kind distribution (i.e., ) and has PDF:

Proof. From (1) and (6), we havePutting , thuswhich proves the theorem.
From the previous theorem, we show that the extropy of is the product of extropy of emerging from distribution and expectation of the first kind generalized beta distributed r.v.
Now, we will give some special cases on Theorem 2 by the following examples.

Example 1. Suppose the nonnegative continuous r.v. X arising from exponential distribution, denoted by , with CDFTherefore,Thus,Thence,By the same manner in (17), we can reduce (25) as

Example 2. Suppose the nonnegative continuous r.v. X arising from Pareto distribution with CDFTherefore,Thus,Thence,

In the next corollary, the lower bound for the extropy of will be obtained in terms of the extropy of emerging from and the mode of the distribution.

Corollary 1. Suppose that M is the mode of the r.v. X such that which exists; therefore,

2.1. Extropy of Dual m-Generalized Order Statistics

Based on distribution, from (1) and (8) and Theorem 1, we can obviously see that the extropy of , , is the same as the extropy of the . On the other hand, we can obtain the extropy of , , , for any distribution from the following theorem.

Theorem 3. Let X be a continuous r.v. that is nonnegative with CDF distribution. Then, from (1), (8), and (13), the extropy of the , , , is given bywhere with PDF defined in (19).

Proof. From (1) and (8), we havePutting , thuswhere the extropy of obtained in (13), which proves the theorem.

3. Residual and Past Extropy of m-Generalized Order Statistics and Its Dual

Under the condition of parameters in (6), Kamps [16] obtained the CDF of , , , as follows:where . Furthermore, Shahbaz et al. [22] derived the CDF of both and , , , , respectively, in terms of incomplete beta function ratio as follows:where is incomplete beta function ratio, , .

In the next theorems, we obtain the residual and past extropy of and emerging from any distribution.

Theorem 4. Let X be a continuous r.v. that is nonnegative with CDF distribution. Then, from (2), (6), and (36), the residual extropy of , , , is given bywhere with PDF defined in (19) and defined in (36).

Theorem 5. Let X be a continuous r.v. that is nonnegative with CDF distribution. Then, from (3), (6), and (36), the past extropy of the , , , is given bywhere with PDF defined in (19) and defined in (36).

Theorem 6. Let X be a continuous r.v. that is nonnegative with CDF distribution. Then, from (2), (8), and (37), the residual extropy of the , , , is given bywhere with PDF defined in (19) and defined in (37).

Theorem 7. Let X be a continuous r.v. that is nonnegative with CDF distribution. Then, from (3), (8), and (37), the past extropy of the , , , is given bywhere with PDF defined in (19) and defined in (37).

4. Residual Extropy of Concomitants of m-Generalized Order Statistics

In this section, we will discuss the residual extropy of concomitants of under the FGM family. From equation (9), the conditional CDF of Y given is given by

Under the FGM family with conditional CDF given by equation (42), Mohie El-Din et al. [23] presented the CDF of the concomitant of , , as follows:where is the PDF of . Therefore,

From (4) and (44), the residual extropy of concomitants of is given by

Furthermore, we can write (45) in terms of the moments as follows:where and is the order statistic of a random sample of size of the variate, .

4.1. Stop-Loss Transform and Gini Coefficient

In this section, we will present related to stop-loss transform and Gini index.

Definition 1. The stop-loss transform of the nonnegative r.v. Y is defined as

Definition 2. Suppose X and Y are independent r.v.’s and have the same distribution as X. Then, the Gini index or Gini coefficient is given by(see Wang [24] for more details).

Remark 1. From Jahanshahi et al. [14], based on (4) and Definitions 1 and 2, we havewhere is the mean residual life function.

Theorem 8. Let Y be a continuous r.v. that is nonnegative with CREX defined in (4). Then, the cumulative residual extropy of concomitants of , , can be expressed aswhere

Proof. The proof directly follows from (45) and Remark 1.

4.2. Upper Bound of

In another view of the cumulative residual extropy of concomitants of , , we present it depending on its upper bound.

Theorem 9. Let Y be a continuous r.v. that is nonnegative with survival function . Based on FGM family with CDF given by (9), utilizing (4), the residual extropy of the concomitants , , , can be expressed by

Proof. Using Bernoulli’s inequality, from (44), we havewhere is defined in (12).
From the previous theorem, we can conclude the following remark.

Remark 2. Since , then the residual extropy of the concomitant of given in (53) can be considered in following cases:(1)If , then .(2)If and or and , then(3)If and or and , then

Theorem 10. Let Y be a continuous r.v. that is nonnegative with survival function . Then, we have the following:(1)From Remark 2, under the conditions on the parameters in (54), we get(2)From Remark 2, under the conditions on the parameters in (55), we getwhere is the cumulative residual entropy presented by Rao et al. [25]:

Proof. The proof directly follows from inequality and ; see Rao et al. [25].
In the next remark, we will give an application of the previous results, taking the order statistics (with and ) as a special case under the FGM family.

Remark 3. For order statistic ( and ), we have . Moreover, by controlling the values of n and r, we can observe the following:(1), if is odd and (2), if is even, and is replaced with (3), if is even and (4), if is odd and

Example 3. Let Y be a continuous r.v. that is nonnegative arising from with CDF defined in (22); then, utilizing (58) and (59), we haveFor order statistics (with and ), we can apply some numerical values as follows:(1)For , , , and , we have , , and (2)For , , , and , we have , , andwhich assure Theorem 10.

Example 4. Let Y be a continuous r.v. that is nonnegative arising from uniform distribution , ; then, utilizing (58) and (59), we haveFor order statistics (with and ), we can apply some numerical values as follows:(1)For , , , and , we have , , , and (2)For , , , and , we have , , andTherefore, (1.) and (2.) assure Theorem 10.

Theorem 11. Let Y be a continuous r.v. that is nonnegative with survival function . From Remark 2, under the conditions on the parameters in (54), for order statistics (with and ) as a special case of , we denotewhere , , .

Proof. A method of obtaining an upper bound for the mean of the maximum of n identically distributed nonnegative r.v.’s is given by Lai and Robbins [26] and Gravey [27]. If , , thenThus, from (54),

Example 5. Suppose follows with survival function . From (63), we have . Therefore, using (62) and from Remark 2, under the conditions on the parameters in (54), with , and , we have and , which assure the previous results.

4.3. Stochastic Orders

Definition 3. is known to be smaller than in the usual stochastic order, denoted by if and only if , for all . For more details, see Shaked and Shanthikumar [28].

Theorem 12. Let and be two nonnegative continuous r.v.’s with ’s and and finite mean and , respectively. If , then we have the following:(1)From Remark 2, under the conditions on the parameters in (54), we get(2)From Remark 2, under the conditions on the parameters in (55), we get

Proof. First, from Theorem 7 of Jahanshahi et al. [14], if , then . Therefore, the proof of (1) isThe proof of (2) isNow, we will give an application of the last theorem as follows.

Example 6. Let and be two r.v.’s of power function distribution with ’s and , respectively. For order statistics (with and ). From Theorem 12, we have the following:(1)For , , and , we have , , and (2)For , , and , we have , , and which assure the previous results.

5. Nonparametric Estimation

In this section, we obtain a nonparametric estimation of the residual extropy of concomitants of under the FGM family by the empirical data. Let be a random sample from a population with CDF F and its empirical estimator . From (45), the empirical residual extropy of concomitants of is given bywhere , is the empirical CDF, and are the associated order statistics of the random sample.

To estimate , we consider the first empirical estimator as follows:where , , , and . Moreover, the second empirical estimator (kernel-smoothed estimator) is given bywhere and is a bandwidth parameter; see Nadaraya [29].

In the following examples, we apply the proposed methods to explain the performance of the empirical and kernel estimators.

Example 7. Let be a random sample of uniform distribution . According to Pyke [30], the sample spacing follows the beta distribution . Hence, from (70) and (71), we have

Example 8. Let be a random sample of . According to Pyke [30], the sample spacing follows . Hence, from (70) and (71), we have

Based on order statistics , Table 1 presents the mean and variance of and from and , respectively, by using different values of sample size . In Table 1, for fixed n and r increases, we conclude that the mean decreases and the variance increases.

5.1. Data Application

In the following, we illustrate our empirical estimators in real and simulated data for .

Example 9. We refer to Kasilingam et al.’s [31] research to understand the spreading patterns of the COVID-19 virus; they used exponential growth modelling and identifies countries that have shown early signs of containment until 26th of March 2020. The data represent the percentage of serious cases of infections in 42 countries listed as follows: 1.56, 8.51, 2.17, 0.37, 1.09, 9.84, 4.95, 3.18, 11.37, 2.81, 6.22, 1.87, 0.00, 0.00, 9.05, 2.44, 1.38, 4.17, 3.74, 1.37, 2.33, 7.80, 2.10, 0.47, 2.54, 0.92, 0.09, 0.18, 1.72, 1.02, 0.62, 2.34, 0.50, 2.37, 3.65, 0.59, 5.76, 2.14, 0.88, 0.95, 4.17, and 2.25.

We use Kolmogorov–Smirnov (K–S) test to check the fitting of the data for , which implies that the K-S statistic is 0.076282 with value 0.9674. Thus, it is admitted to fit the data by ; furthermore, see Figure 1. Based on , Figures 2 and 3 present the real-life and simulated data, respectively. Therefore, we can conclude that by decreasing and increasing , the empirical estimators approach the theoretical value and vice versa.

(a)

(b)

(c)

(d)

(a)

(b)

(c)

(d)

6. Conclusion

In this communication, we introduced the extropy of and arising from any distribution and wrote them in terms of the extropy of and of distribution, respectively. Furthermore, examples of the obtained models for exponential and Pareto distributions are provided. Also, we produced the lower bound of the extropy of emerging from any continuous distribution in terms of the extropy of of distribution and the mode of the parent PDF. Moreover, residual and past extropy of and are derived. Meanwhile, the residual extropy of concomitants of , , of FGM distribution is presented. The measure is discussed in terms of stop-loss transform and Gini coefficient. An alternative view of depending on its upper bound is considered. Besides, some examples and numerical results and stochastic orders of are obtained. Finally, we considered the problem of estimating by proposing two different empirical estimators of CDF. We concluded that the proposed estimators are affected by sample size n, r, and and generally the first empirical estimator is more accurate than the second estimator.

Data Availability

All the data sets are provided within the main body of the paper.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright

Copyright © 2021 Mohamed S. Mohamed et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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