Cumulative Residual Extropy of Minimum Ranked Set Sampling with Unequal Samples
Mohammad Reza Kazemia, Saeid Tahmasebib, Camilla Calì, Maria Longobardi
aa r X i v : . [ m a t h . S T ] A ug Cumulative Residual Extropy of Minimum Ranked Set Sampling withUnequal Samples
Mohammad Reza Kazemi a , Saeid Tahmasebi b , Camilla Cal`ı c , Maria Longobardi c, ∗ a Department of Statistics, Faculty of Science, Fasa University, Fasa, Iran b Department of Statistics, Persian Gulf University , Bushehr, Iran c Department of Biology, University of Napoli Federico II, Napoli, Italy
Abstract
Recently, an alternative measure of uncertainty called cumulative residual extropy (CREX) was proposedby Jahanshahi et al. (2019). In this paper, we consider uncertainty measures of minimum ranked setsampling procedure with unequal samples (MinRSSU) in terms of CREX and its dynamic version andwe compare the uncertainty and information content of CREX based on MinRSSU and simple randomsampling (SRS) designs. Also, using simulation, we study on new estimators of CREX for MinRSSU andSRS designs in terms of bias and mean square error. Finally, we provide a new discrimination measureof disparity between the distribution of MinRSSU and parental data SRS.
Keywords:
Cumulative residual extropy, Discrimination measure, Minimum ranked setsampling, Stochastic ordering.
1. Introduction
Ranked set sampling (RSS) design is a cost-effective sampling for situations where taking actualmeasurements on units is expensive but ranking units is easy. For the first time, based on the RSSsampling design, McIntyre (1952) provided a more efficient estimator of the population mean comparingto the simple random sampling (SRS) counterpart. To learn more about this concept, the readers can referto Patil et al. (1999). There are many available studies that have developed and generalized the methodof sampling used in RSS scheme and they efficiently estimate the population parameter comparing to theSRS scheme. Recently, Qiu & Eftekharian (2020) studied information content of minimum ranked setsampling procedure with unequal samples (MinRSSU) as useful modification of RSS procedure in termsof extropy. In the MinRSSU, we draw m simple random samples, where the size of the i th samples is i , i = 1 , ..., m . The one-cycle MinRSSU involves an initial ranking of m samples of size m as follows:1 : X (1:1)1 → ˜ X = X (1:1)1 X (1:2)2 X (2:2)2 → ˜ X = X (1:2)2 ... ... ... . . . ... ... ... m : X (1: m ) m X (2: m ) m · · · X ( m : m ) m → ˜ X m = X (1: m ) m ∗ Corresponding author
Email address: [email protected] (Maria Longobardi)
Preprint submitted to Journal of L A TEX Templates August 21, 2020 here X ( i : i ) j denotes the i th order statistic from the j th SRS of size i . The resulting sample is calledone-cycle MinRSSU of size m and denoted by X ( m ) MinRSSU = { ˜ X i , i = 1 , . . . , m } . The parameter m shouldbe kept small because the ranking should not be difficult in this sense and the ranking may be done forexample by using an easily measurable covariate, then it is not difficult to identify the minimum of rankedindividuals in each subset. Note that ˜ X i has the same distribution as X (1) i which is the smallest orderstatistic in a set of size i with probability density function (pdf) f (1) i ( x ) = if ( x )[1 − F ( x )] i − and survivalfunction ¯ F (1) i ( x ) = [1 − F ( x )] i = ¯ F i ( x ), where f ( . ), F ( . ) and ¯ F ( . ) are the underlying pdf, cumulativedistribution function (cdf) and survival function. In reliability theory, ˜ X i measures the lifetime of a seriessystem.Several authors have worked on measures of information for RSS and its variants. Jozani & Ahmadi(2014) explored the notions of information content of RSS data and compared them with their counter-parts in SRS data. Tahmasebi et al. (2016) obtained some results of residual (past) entropy for rankedset samples. Eskandarzadeh et al. (2016) studied information measures for record ranked set sampling.Eskandarzadeh et al. (2018) considered information measures of maximum ranked set sampling procedurewith unequal samples in terms of Shannon entropy, R´enyi entropy and Kullback-Leibler information,instead Tahmasebi et al. (2020) in terms of Tsallis entropy. More recently, Qiu & Eftekharian (2020)studied information content of MinRSSU in terms of extropy and Raqab & Qiu (2019) considered theproblems of uncertainty and information content of RSS data based on extropy measure and the relatedmonotonic properties and stochastic comparisons.Let X denotes a continuous random variable with pdf f . Frank et al. (2015) introduced a new measuretermed by extropy associated with X as J ( X ) = − Z + ∞−∞ [ f ( x )] dx = − Z f ( F − ( u )) du, (1.1)where F − ( . ) is the quantile function of X . Qiu (2017) explored some characterization results, monotoneproperties, and lower bounds of extropy of order statistics and record values. Also, Qiu & Eftekharian(2020) and Raqab & Qiu (2019) considered the information measure of extropy J ( X ) based on MinRSSUand RSS schemes, respectively and compared the results with their counterpart under SRS design. This paper is organized as follows: Section 2 deals with the results of cumulative residual extropy (CREX)for MinRSSU data by comparing to its counterpart under SRS data. In Section 3, new estimators areproposed for CREX in SRS and MinRSSU designs using empirical approach. Also, by using simulationstudy, the behavior of estimators of CREX in MinRSSU and SRS are compared in terms of bias and meansquare error. Furthermore, we show that how MinRSSU scheme can efficiently reduce the uncertainty measure comparing to SRS design. Section 4 provides a new discrimination measure of disparity betweenthe distribution of MinRSSU and parental data SRS. Section 5 concludes the paper.
2. Cumulative residual extropy of MinRSSU
Let X denotes the lifetime of a system with survival function ¯ F . Recently, a new measure of informa-tion is proposed by Jahanshahi et al. (2019) with substituting the function ¯ F in extropy formula (1.1). ξ J ( X ) = − Z + ∞ ¯ F ( x ) dx. (2.1)Note that −∞ < ξ J ( X ) ≤
0. If the CREX of X is less than that of another random variable, say Y , i.e. ξ J ( X ) ≤ ξ J ( Y ), then X has less uncertainty than Y . Now let ξ J ( X ) < + ∞ . Then, for the MinRSSUand SRS designs, we have ξ J ( X ( m ) MinRSSU ) = − m Y i =1 [ − ξ J ( X (1: i ) )] = − m Y i =1 Z + ∞ ¯ F i ( x ) dx (2.2)= − m Y i =1 Z (1 − u ) i f ( F − ( u )) du = − m Y i =1 E (cid:20) (1 − U ) i f ( F − ( U )) (cid:21) , (2.3)and ξ J ( X ( m ) SRS ) = − (cid:20)Z + ∞ ¯ F ( x ) dx (cid:21) m = −
12 [ − ξ J ( X )] m . (2.4)To compare the above measures, let us consider the following examples. Example 2.1. If U ∼ U nif orm (0 , , then ξ J ( U ( m ) MRSSU ) = − m Y i =1 i + 1 = − (cid:18) √ π m Γ( m + ) (cid:19) < ξ J ( U ( m ) SRS ) = − . (cid:18) (cid:19) m . (2.5) Example 2.2. If Z is exponentially distributed with mean λ . Then, we have ξ J ( Z ( m ) MRSSU ) = − m Y i =1 iλ < ξ J ( Z ( m ) SRS ) = − (cid:18) λ (cid:19) m . (2.6) Example 2.3.
Let X is finite range distribution with ¯ F ( x ) = (1 − ax ) b , < x < a , a > , b > . Then,we have ξ J ( X ( m ) MRSSU ) = − m Y i =1 a (1 + 2 ib ) < ξ J ( X ( m ) SRS ) = − (cid:18) a (1 + 2 b ) (cid:19) m . (2.7) Theorem 2.1.
Let X ( m ) MinRSSU be the MinRSSU from population X with pdf f and cdf F . Then, ξ J ( X ( m ) MinRSSU ) ≤ ξ J ( X ( m ) SRS ) for m > .Proof. Since ¯ F ( x ) ≥ ¯ F i ( x ) for i ≥
1, we have (cid:18)Z + ∞ ¯ F ( x ) dx (cid:19) m ≤ m Y i =1 Z + ∞ ¯ F i ( x ) dx. The proof follows by recalling (2.2) and (2.4).
Remark 2.1. If f ( F − ( u )) ≥ , < u < , then ξ J ( X ( m ) MinRSSU ) is increasing in m ≥ .Proof. From (2.2), we get ξ J ( X ( m +1) MinRSSU ) ξ J ( X ( m ) MinRSSU ) = Z (1 − u ) m +2 f ( F − ( u )) du ≤ m + 3 ≤ . The result follows readily, since the extropy is negative .3n the following, we provide some results on the cumulative residual extropy of X ( m ) MinRSSU in termsof stochastic ordering properties. Now, we state important properties of ξ J ( X ( m ) MinRSSU ) using thestochastic ordering. For that we present the following definitions:
Definition 2.2. (Shaked and Shanthikumar, 2007) Let X and Y be two non-negative random variables with pdfs f and g , cdfs F and G , and hazard functions λ X ( x ) = f ( x )¯ F ( x ) and λ Y ( y ) = g ( y )¯ G ( y ) , respectively.Then1. X is said to be smaller than Y in the usual stochastic order (denoted by X ≤ st Y ) if P ( X ≥ x ) ≤ P ( Y ≥ x ) for all x ∈ R .2. X is smaller than Y in the hazard rate order (denoted by X ≤ hr Y ) if λ X ( x ) ≥ λ Y ( x ) for all x . X is smaller than Y in the dispersive order (denoted by X ≤ disp Y ) if f ( F − ( u )) ≥ g ( G − ( u )) for all u ∈ (0 , , where F − and G − are right continuous inverses of F and G , respectively.4. X is said to have decreasing failure rate (DFR) if λ X ( x ) is decreasing in x .5. X is smaller than Y in the convex transform order (denoted by X ≤ c Y ) if G − F ( x ) is a convexfunction on the support of X . X is smaller than Y in the star order (denoted by X ≤ ∗ Y ) if G − F ( x ) x is increasing in x ≥ .7. X is smaller than Y in the superadditive order (denoted by X ≤ su Y ) if G − ( F ( t + u )) ≥ G − ( F ( t )) + G − ( F ( u )) for t ≥ , u ≥ . 8. X is said to have an increasing reversed hazard rate (IRHR) if ˜ λ X ( x ) = f ( x ) F ( x ) is increasing in x . Theorem 2.3. If X ≤ st Y , then ξ J ( X ( m ) MinRSSU ) ≥ ξ J ( Y ( m ) MinRSSU ) , m > . Proof.
By the assumption of the stochastic order, ¯ F i ( x ) ≤ ¯ G i ( x ) for all x ≥
0. Now using (2.2), for m >
1, we get the desired result.
Theorem 2.4.
Let X and Y be two non-negative random variable. If X ≤ disp Y , then ξ J ( X ( m ) MinRSSU ) ≥ ξ J ( Y ( m ) MinRSSU ) for m > .Proof. By the assumption of the dispersive order , f ( F − ( u )) ≥ g ( G − ( u )) for all u ∈ (0 , for m > Theorem 2.5. If X ≤ hr Y , and X or Y is DFR, then ξ J ( X ( m ) MinRSSU ) ≥ ξ J ( Y ( m ) MinRSSU ) for m > .Proof. If X ≤ hr Y , and X or Y is DFR, then X ≤ disp Y , due to Bagai & Kochar (1986). Thus, fromTheorem (2.4) the desired result follows. Theorem 2.6.
Let X and Y be two non-negative random variable with pdf ’s f and g , respectively, such that f (0) ≥ g (0) > . If X ≤ su Y ( X ≤ ∗ Y or X ≤ c Y ) , then ξ J ( X ( m ) MinRSSU ) ≥ ξ J ( Y ( m ) MinRSSU ) for m > .Proof. If X ≤ su Y ( X ≤ ∗ Y or X ≤ c Y ), then X ≤ disp Y , due to Ahmed et al. (1986). So, from Theorem(2.4) the desired result follows. 4 roposition 2.7. Let X ( m ) MinRSSU and X ( m ) RSS be MinRSSU and RSS data from distribution X with DFRageing property, respectively. Then for m > we have ξ J ( X ( m ) MinRSSU ) ≥ ξ J ( X ( m ) SRS ) . Remark 2.2.
Let ϕ be a non-negative function such that the derivative ϕ ′ ( x ) ≥ for all x . Then X ≤ disp ϕ ( X ) . Thus, ξ J ( X ( m ) MinRSSU ) ≥ ξ J ( ϕ ( X ) ( m ) MinRSSU ) . If X ≤ disp Y , Theorem 3.B.26 in Shaked & Shanthikumar (2007) claims that X i : m ≤ disp Y i : m , i = , , ..., m . Thus, according to Theorems 4.5 and 4.6 in Qiu (2017), we obtain the following Proposition . Proposition 2.8.
Let X ( m ) MinRSSU be a sample from MinRSSU design.(i) If X is DFR ageing property, then ξ J ( X (1) m ) is increasing in m ≥ .(ii) If X is IRHR property, then ξ J ( X (1) m ) is decreasing in m ≥ . Proposition 2.9.
Let Y ( m ) MinRSSU = a X ( m ) MinRSSU + b with a > and b ≥ . Then, ξ J ( Y ( m ) MinRSSU ) = aξ J ( X ( m ) MinRSSU ) . Proposition 2.10.
Let X be a symmetric random variable with respect to the finite mean µ = E ( X ) .Then ξ J ( X ( m ) MinRSSU ) = CJ ( X ( m ) MinRSSU ) , where CJ ( X ) = − R + ∞ [ F X ( x )] dx is the cumulative extropy (see Jahanshahi et al. (2019)). Let X be the random lifetime of a system, recall that X [ t ] = [ X − t | X ≤ t ] describes the residual lifetime of a system. For all t ≥ µ ( t ) = E [ X − t | X ≥ t ] = 1¯ F ( t ) Z + ∞ t ¯ F ( x ) dx. Now, we can define a generalized measure of cumulative residual extropy as ξ J ( X ; t ) = − Z + ∞ t (cid:20) ¯ F ( x )¯ F ( t ) (cid:21) dx. (2.8)Note that ξ J ( X ; t ) ≥ − µ ( t ) / (2 ¯ F ( t )). Moreover , we have ξ J ( X ( m ) SRS ; t ) = −
12 [ − ξ J ( X, t )] m . (2.9)Under the MinRSSU design, it is clear that ξ J ( X ( m ) MinRSSU ; t ) = − m Y i =1 [ − ξ J ( X ( i : i ) ; t )] = − m Y i =1 Z + ∞ t (cid:20) ¯ F ( x )¯ F ( t ) (cid:21) i dx = − m Y i =1 E (cid:20) U i ¯ F ( t ) f ( F − (1 − U ¯ F ( t ))) (cid:21) . (2.10) Theorem 2.11.
Let X be a random lifetime variable with cdf F ( · ) , Then, for m > ξ J ( X ( m ) MinRSSU ; t ) ≤ ξ J ( X ( m ) SRS ; t ) . (2.11) Proof.
The proof is similar to Theorem 2.1. Remark 2.3. If f ( F − (1 − u ¯ F ( t ))) ≥ , < u < , then ξ J ( X ( m ) MinRSSU ; t ) is increasing in m ≥ . . Results on empirical measure of CREX This section focuses on the estimation of the ξ J ( X ) based on the SRS and MinRSSU schemes. Let X (1) ≤ X (2) ≤ ... ≤ X ( n ) be the order statistics of the random sample X , X , ..., X n from cdf F. Thenthe empirical measure of F is defined asˆ F n ( x ) = , x < X (1) , kn , X ( k ) ≤ x ≤ X ( k +1) , k = 1 , , ..., n − , x > X ( n ) . Thus, the empirical measure of ξ J ( X ) is obtained by replacing the distribution function F by theempirical distribution function ˆ F n as V n = − Z ˆ¯ F n ( x ) dx = − n − X k =1 Z X ( k +1) X ( k ) (cid:18) − kn (cid:19) dx = − n − X k =1 U k +1 (cid:18) − kn (cid:19) , (3.1)where U k +1 = X ( k +1) − X ( k ) , k = 1 , ..., n −
1. Jahanshahi et al. (2019) showed that V n almost surelyconverges to the CREX of X , i.e. V n a.s. → ξ J ( X ) , as n → + ∞ . The problem of estimation of ξ J ( X ) based on MinSSU scheme can be deduced in the same line of the es-timation based on SRS design V n . In this part, we assume that instead of one-cycle MinRSSU, the processis repeated l cycles to have a sample of size n = ml . In this case, the resulting MinRSSU is denoted by (cid:8) X (1: i ) j , i = 1 , ..., m ; j = 1 , ..., l (cid:9) , where X (1: i ) j is the lowest order statistic from the i th sample in the j thcycle. Let Y (1) , ..., Y ( n ) be the ordered values of the MinRSSU design (cid:8) X (1: i ) j , i = 1 , ..., m ; j = 1 , ..., l (cid:9) . Then, the natural estimation of ξ J ( X ) based on the MinRSSU, can be obtained as R n = − n − X k =1 Z k +1 (cid:18) − kn (cid:19) , where Z k +1 = Y ( k +1) − Y ( k ) , k = 1 , , ..., n −
1. Our preliminary computations and simulations showedthat R n has some deficiencies to be unbiased and have low mean square error (MSE) to estimate the ξ J ( X ) . We observed that the term (cid:0) − kn (cid:1) should be slightly modified so that the estimator has optimalproperties to estimate ξ J ( X ) . We propose to modify this term with (cid:16) − kn + m + w (cid:17) , where w is a numberthat resulting estimator has optimally low bias and MSE. The resulting estimator of the ξ J ( X ) has thefollowing form R m,n = − n − X k =1 Z k +1 (cid:18) − kn + m + w (cid:19) . (3.2) In the previous section, the estimator V n of ξ J ( X ) is a linear function of sample spacing U k +1 = X ( k +1) − X ( k ) , k = 1 , ..., n −
1. The asymptotic distribution of this linear function of sample spacing can6e found, for example in Di Crescenzo & Longobardi (2009) and Tahmasebi (2019) only for exponentialand standard uniform distributions. So we provide another estimator for ξ J ( X ) which is a linear functionof order statistics. Proposition 3.1.
Let X be an absolutely continuous non-negative random variable with survival function ¯ F , then ξ J ( X ) = − Z + ∞ x ¯ F ( x ) dF ( x ) . (3.3) Proof.
By (2.1) and Fubini’s theorem, we obtain − Z + ∞ x ¯ F ( x ) dF ( x ) = − Z + ∞ (cid:18)Z x dt (cid:19) ¯ F ( x ) f ( x ) dx = − Z + ∞ (cid:18)Z + ∞ t ¯ F ( x ) f ( x ) dx (cid:19) dt = − Z + ∞ ¯ F ( t ) dt. Hence, the proof is completedThe new estimator can be obtained replacing ¯ F ( x ) with ˆ¯ F n ( x ) in (3.3), so c ξ J ( X ) has the followingform c ξ J ( X ) = − Z + ∞ x ˆ¯ F n ( x ) d ˆ F n ( x ) = − n n X i =1 (cid:18) − in (cid:19) X ( i ) . (3.4)Let J ( x ) = (1 − x ) . Then (3.4) has the form c ξ J ( X ) = − n n X i =1 J (cid:18) in (cid:19) X ( i ) , which is a linear function of order statistics. The natural estimation of ξ J ( X ) based on the MinRSSU,can be obtained as c ξ J ( Y ) := c ξ J ( X MinRSSU ) = − n n X i =1 J (cid:18) in (cid:19) Y ( i ) . Stigler (1974) showed that asymptotic distribution of such a linear combination is normal distribution. The results of Stigler (1974) also hold if the independent observations are not identically distributed.These properties help us to obtain the asymptotic distribution of c ξ J ( X ) for both SRS (observations areindependent and identical) and MinRSSU (observations are only independent) designs. Theorem 3.2.
Assume that E (cid:0) X (cid:1) < + ∞ . Then √ n (cid:16) c ξ J ( X ) − ξ J ( X ) (cid:17) d → N (cid:0) , σ ( J, F ) (cid:1) , √ n (cid:16) c ξ J ( Y ) − ξ J ( X ) (cid:17) d → N (cid:16) , σ MinRSSU ( J, ˜ F , K ) (cid:17) , where σ ( J, F ) = Z + ∞ Z + ∞ J ( F ( x )) J ( F ( y ))[ F ( min ( x, y )) − F ( x ) F ( y )] dxdy,σ MinRSSU ( J, ˜ F , K ) = Z + ∞ Z + ∞ J ( ˜ F ( x )) J ( ˜ F ( y )) K ( x, y ) dxdy, nd ˜ F ( . ) and K ( x, y ) are given as ˜ F ( x ) = 1 m m X i =1 F (1) i ( x ) ,K ( x, y ) = 1 m m X i =1 [ F (1) i ( min ( x, y )) − F (1) i ( x ) F (1) i ( y )] . Proof.
For ˜ F ( x ) and K ( x, y ), we have˜ F ( x ) = lim n → + ∞ ln m X i =1 F (1) i ( x ) = 1 m m X i =1 F (1) i ( x ) ,K ( x, y ) = lim n → + ∞ ln m X i =1 [ F (1) i ( min ( x, y )) − F (1) i ( x ) F (1) i ( y )]= 1 m m X i =1 [ F (1) i ( min ( x, y )) − F (1) i ( x ) F (1) i ( y )] , where as before l is the size of the cycle of MinRSSU design with n = ml . The rest of the proof is doneby using the results of Stigler (1974) for both SRS and MinRSSU designs.As we formerly stated for providing the estimator R m,n , some adjusted forms of estimator c ξ J ( Y ) canbe used since the estimator c ξ J ( Y ) has some deficiencies to estimate CREX which need to be fixed. Weagain observed that the the term (cid:0) − in (cid:1) needs to be slightly adjusted so that the resulting estimatorhas optimal properties to estimate CREX. This new term has the following form1 − in + ψ ( m, w ) , (3.5)where the function ψ ( ., . ) is a challenging factor that can be specifically determined for each given distri-bution and consequently the resulting estimator has optimal low bias and MSE. In this case, the resultingestimator has the following form c ξ J m,n ( Y ) = − n n X i =1 J (cid:18) in + ψ ( m, w ) (cid:19) Y ( i ) . (3.6)In the next section, we determine the form of the function ψ ( ., . ) for exponential, uniform and betadistributions and we show that the optimal choice of this function can reduce the bias and MSE in the estimate of CREX. In advance, we explain the role of the parameter w and function ψ ( m, w ) for which R m,n and c ξ J m,n ( X MinRSSU ) have optimally low bias and MSE. For this purpose, we examine some distributionsto obtain the function ψ ( m, w ) and optimal value of w in R m,n and c ξ J m,n ( Y ). In Tables 1-3, for four estimators R n , R m,n , c ξ J ( Y ) and c ξ J m,n ( Y ), we compute the bias and MSE to estimate the parameter ξ J ( X ) for some different values of w . Here, the exponential ( Exp ( λ )) , uniform ( U nif (0 , b )) and beta( Beta ( α, , α >
1) distributions are considered. For each configuration, the simulation study was car-ried out with 5000 repetitions. The number of cycle and size of the sample in each cycle are taken as l = 2 , m = 2 , ...,
5, respectively. We compute the bias and root of MSE (RMSE) of each estimator
8f parameter ξ J ( X ). In Tables 1-3, it can be seen that results of biases and RMSEs of estimator c ξ J ( Y )are not comparable to those of c ξ J m,n ( Y ) for different values of w . We intuitively obtain the form of thefunction ψ ( m, w ) for different distributions. We found that the function ψ has the form 5 m − k m + w with k = 3,..., k = 0, 3 m − (2 k m + 1) + w with k = − k = 2 and m − w for exponential, uniformand beta distributions, respectively. It is observed that changing the value of l has no effect on the whole results verified from biases and RMSEs. In all tables, we see that choosing the proper function ψ ( m, w )and parameter w as the challenging factors can improve the efficiency of estimators R m,n and c ξ J m,n ( Y )against R n and c ξ J ( Y ), respectively, in estimating the parameter ξ J . Table 1:
The biases and MSEs of the different estimators: Exponential distribution
Results based on R m,n Results based on c ξ J m,n ( Y ) l = 2 l = 3 l = 2 l = 3 m w Bias RMSE Bias RMSE m w
Bias RMSE Bias RMSE2 -2 0.321 0.407 0.370 0.477 2 -11 0.632 0.650 0.536 0.562-1 0.131 0.354 0.133 0.460 -10 0.402 0.483 0.354 0.4270 -0.051 0.423 -0.069 0.580 -9 0.263 0.435 0.223 0.3701 -0.216 0.556 -0.234 0.733 -8 0.171 0.434 0.125 0.361 R n c ξ J ( Y ) 0.402 0.483 0.354 0.4273 -1 0.124 0.367 0.164 0.304 3 -7 0.033 0.374 0.006 0.3070 -0.017 0.429 0.040 0.315 -6 0.012 0.382 -0.020 0.3171 0.124 0.367 0.164 0.304 -5 -0.006 0.391 -0.044 0.3282 -0.253 0.632 -0.191 0.472 -4 -0.022 0.399 -0.065 0.340 R n c ξ J ( Y ) 0.452 0.489 0.430 0.4584 0 0.024 0.383 0.118 0.291 4 -3 0.029 0.329 -0.007 0.2691 -0.077 0.452 0.029 0.313 -2 0.020 0.332 -0.019 0.2742 -0.170 0.532 -0.057 0.363 -1 0.012 0.334 -0.030 0.2783 -0.256 0.616 -0.139 0.427 0 0.004 0.337 -0.040 0.283 R n c ξ J ( Y ) 0.509 0.527 0.489 0.5035 1 -0.004 0.390 0.114 0.281 5 1 0.072 0.277 0.050 0.2322 -0.082 0.448 0.046 0.296 2 0.067 0.277 0.043 0.2333 -0.156 0.512 -0.021 0.329 3 0.063 0.278 0.036 0.2334 -0.224 0.578 -0.085 0.373 4 0.058 0.278 0.030 0.234 R n c ξ J ( Y ) 0.554 0.563 0.540 0.547
4. Discrimination information
This section considers a new discrimination measure of disparity between the distribution of MinRSSUand parental data SRS. Raqab & Qiu (2019) defined the discrimination information between the densityfunction of the i th order statistic f ( i ) m and the underlying density function f as D m (cid:0) f ( i ) m : f (cid:1) = 12 Z + ∞−∞ f ( i ) m ( x ) (cid:0) f ( i ) m ( x ) − f ( x ) (cid:1) dx. (4.1)Analogously to (4.1), we define the discrimination information between the survival function of thesmallest ordered statistic ¯ F (1) i and the underlying survival function ¯ F as D (cid:0) ¯ F (1) i : ¯ F (cid:1) = − Z + ∞−∞ ¯ F (1) i ( x ) (cid:0) ¯ F (1) i ( x ) − ¯ F ( x ) (cid:1) dx. (4.2)9 able 2: The biases and MSEs of the different estimators: Uniform distribution
Results based on R m,n Results based on c ξ J m,n ( Y ) l = 2 l = 3 l = 2 l = 3 m w Bias RMSE Bias RMSE m w
Bias RMSE Bias RMSE2 -2 0.360 0.392 0.298 0.333 2 -4 0.098 0.275 0.058 0.227-1 0.249 0.305 0.212 0.264 -3 0.045 0.281 -0.001 0.2360 0.158 0.253 0.134 0.214 -2 0.005 0.293 -0.049 0.2541 0.085 0.233 0.066 0.186 -1 -0.026 0.306 -0.088 0.275 R n c ξ J ( Y ) 0.284 0.340 0.238 0.2933 -1 0.191 0.243 0.201 0.237 3 -2 -0.006 0.247 -0.023 0.2000 0.125 0.204 0.147 0.196 -1 -0.026 0.255 -0.047 0.2091 0.067 0.183 0.096 0.165 0 -0.043 0.263 -0.068 0.2202 0.016 0.181 0.055 0.148 1 -0.058 0.270 -0.087 0.231 R n c ξ J ( Y ) 0.318 0.350 0.290 0.3174 0 0.14 0.195 0.175 0.204 4 0 -0.012 0.205 -0.008 0.1641 0.092 0.171 0.135 0.173 1 -0.023 0.209 -0.021 0.1682 0.049 0.159 0.098 0.148 2 -0.034 0.213 -0.034 0.1723 0.018 0.159 0.062 0.131 3 -0.043 0.217 -0.046 0.177 R n c ξ J ( Y ) 0.352 0.369 0.337 0.3505 1 0.116 0.17 0.175 0.198 5 2 0.020 0.181 0.028 0.1462 0.079 0.153 0.144 0.172 3 0.013 0.183 0.019 0.1463 0.044 0.144 0.114 0.150 4 0.006 0.184 0.010 0.1474 0.012 0.144 0.085 0.132 5 -0.001 0.186 0.002 0.148 R n c ξ J ( Y ) 0.387 0.398 0.375 0.383 The discrimination information in (4.2) may be rewritten in a simpler way. It can be shown that D (cid:0) ¯ F (1) i : ¯ F (cid:1) = − (cid:2) E ( X (1)2 i ) − E ( X (1) i +1 ) (cid:3) , (4.3)where X (1) j is the smallest order statistic in a random sample of size j . Example 4.1.
Let U ∼ U nif orm (0 , . We know that the order statistics from the standard uniformdistribution follow the beta distribution. Then discrimination information based on (4.2) is D (cid:0) ¯ F (1) i : ¯ F (cid:1) = − (cid:20) i + 1 − i + 2 (cid:21) = i −
12 (2 i + 1) ( i + 2) . In the following theorem, we obtain the discrimination information D between MinRSSU and SRSdesigns. Theorem 4.1.
For X ( m ) MinRSSU and X ( m ) SRS , we have D (cid:16) X ( m ) MinRSSU : X ( m ) SRS (cid:17) = − m Y i =1 E (cid:0) X (1)2 i (cid:1) − m Y i =1 E (cid:0) X (1) i +1 (cid:1)! . Proof.
From (2.1) and (4.2) we have D (cid:16) X ( m ) MinRSSU : X ( m ) SRS (cid:17) = − m Y i =1 Z + ∞−∞ ¯ F i ( x ) − m Y i =1 Z + ∞−∞ ¯ F i +1 ( x ) ! = − m Y i =1 E (cid:0) X (1)2 i (cid:1) − m Y i =1 E (cid:0) X (1) i +1 (cid:1)! . The proof is completed. 10 able 3:
The biases and MSEs of the different estimators: Beta distribution
Results based on R m,n Results based on c ξ J m,n ( Y ) l = 2 l = 3 l = 2 l = 3 m w Bias RMSE Bias RMSE m w
Bias RMSE Bias RMSE2 -2 0.296 0.298 0.276 0.279 2 -3 0.202 0.204 0.146 0.149-1 0.280 0.283 0.264 0.267 -2 0.108 0.116 0.081 0.0890 0.267 0.271 0.253 0.257 -1 0.053 0.070 0.035 0.0551 0.257 0.261 0.244 0.248 0 0.015 0.053 0.000 0.045 R n c ξ J ( Y ) 0.108 0.116 0.081 0.0893 -1 0.251 0.255 0.238 0.242 3 -3 0.096 0.102 0.079 0.0850 0.241 0.245 0.230 0.234 -2 0.052 0.064 0.045 0.0561 0.233 0.238 0.222 0.227 -1 0.019 0.045 0.017 0.0392 0.225 0.231 0.216 0.221 0 -0.007 0.043 -0.006 0.037 R n c ξ J ( Y ) 0.096 0.102 0.079 0.0854 0 0.227 0.231 0.218 0.222 4 -3 0.059 0.068 0.056 0.0631 0.220 0.225 0.212 0.216 -2 0.031 0.047 0.033 0.0442 0.214 0.219 0.206 0.211 -1 0.007 0.038 0.013 0.0333 0.208 0.213 0.201 0.206 0 -0.012 0.041 -0.004 0.031 R n c ξ J ( Y ) 0.095 0.100 0.082 0.0875 1 0.217 0.221 0.210 0.214 5 -3 0.043 0.053 0.047 0.0542 0.211 0.216 0.205 0.209 -2 0.022 0.039 0.030 0.0403 0.206 0.210 0.200 0.204 -1 0.004 0.034 0.015 0.0314 0.201 0.205 0.196 0.201 0 -0.012 0.036 0.002 0.028 R n c ξ J ( Y ) 0.097 0.101 0.088 0.091 Example 4.2.
Let U ∼ U nif orm (0 , . Using the results of Example (4.1) , the discrimination informa-tion D for the MinRSSU and SRS designs of the same size m is D (cid:16) U ( m ) MinRSSU : U ( m ) SRS (cid:17) = − m Y i =1 i + 1 − m Y i =1 i + 2 ! .
5. Conclusion
This paper has introduced the uncertainty measure of the cumulative residual extropy based on theMinRSSU and SRS data. Several results of the CREX measure including stochastic orders were obtainedfor MinRSSU and SRS data. Also, we provided two estimators of CREX measure for both SRS andMinRSSU data. Furthermore, it was shown that MinRSSU scheme can efficiently reduce the uncertaintymeasure of CREX. Also, by providing a discrimination measure, we derived the distance size between
MinRSSU and SRS data.
Acknowledgements
C. Cal`ı and M. Longobardi are partially supported by the GNAMPA research group of INDAM(Istituto Nazionale di Alta Matematica) and MIUR-PRIN 2017, Project ”Stochastic Models for ComplexSystems” (No. 2017JFFHSH). eferencesReferences Ahmed, A. N., Alzaid, A., Bartoszewicz, J., & Kochar, S. C. (1986). Dispersive and superadditiveordering.
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