Subhash Kochar

Portland State University, Portland, United States · Fariborz Maseeh Department of Mathematics and Statsitic

Let $(X_{1},\ldots,X_{n})$ be a random vector distributed according to a time-transformed exponential model . This is a special class of exchangeable models, which, in particular, includes multivariate distributions with Schur-constant survival functions. Let for $1\leq i\leq n$ , $X_{i:n}$ denote the corresponding i th-order statistic. We consider...

KeywordsExponential distributionKendall’s tauMore stochastically increasing (SI): TP 2 dependenceConcordance orderingCopulaPHR modelBivariate Pareto distributionBivariate Marshall-Olkin distribution

Let X = (X 1, …, X n) be a random vector of observations which may not be independent or identically distributed, and let $$\displaystyle T = \sum _{i=1}^n \theta _i X_i, $$ be a linear function of X 1, …, X n, where θ i, i = 1, …, n, are constants. In this chapter we stochastically compare statistics of the above type as the coefficients θ i’s var...

KeywordsSignaturesIFRDFRAStar orderLikelihood ratio orderHazard rate orderStochastic ordering

The differences between order statistics are called sample spacings, the sample range being a special case. They are of great interest in various areas of statistics, in particular, in characterizations of distributions, goodness-of-fit tests, life testing, reliability models, and auction theory. In the reliability context, they correspond to times...

A random variable X is said to have symmetric distribution (about a constant c) if the random variables X − c and c − X have the same distribution. The notion of skewness is intended to represent departure of a density from symmetry where one tail of the density is stretched out more than the other. The concept of skewness has been applied particul...

A stochastic order is a partial order that quantifies the concept of one random variable being bigger than (or more variable or more skewed) another in some probabilistic sense. In this chapter various types of stochastic orders which compare the magnitudes of two random variables and their properties are discussed. These include the (usual) stocha...

Most of the classical methods for comparing variability among distributions are based only upon summary statistics such as variance and standard deviation, which are very crude though convenient methods to use. It is possible that one variable is more spread out than the second initially, but later on it is the other way around. Variance and other...

Dependence among random variables is one of the most widely studied topics in the literature with lots of applications in diverse areas. Pearson’s coefficient of correlation, which is the traditional measure of dependence between random variables, has several limitations. It can detect only linear relationships between random variables. Several oth...

After giving an introduction to the area of stochastic orders, some basic definitions and notations are given in this chapter. Quantities like failure (hazard) rate, mean residual life function, and aging concepts like, IFR, IFRA, DMRL NBU, HNBUE, etc. are defined. A quick review of the distribution theory of order statistics and generalized order...

Order statistics and record values are compared according to various stochastic orders which are discussed in the previous chapters. It is shown that whereas order statistics are always increasing according to the stochastic and the hazard rate orders, that may not be the case with the likelihood ratio order when the observations are not identicall...

Let $X_1, \ldots, X_n$ be mutually independent exponential random variables with distinct hazard rates $\lambda _1, \ldots, \lambda _n$ and let $Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate $\bar \lambda = \sum _{i=1}^{n} \lambda _i/n$ . Also let $X_{1:n} and $Y_{1:n} be their associated order statistics....

Let $X_1, \ldots , X_n$ be mutually independent exponential random variables with distinct hazard rates $\lambda_1, \ldots , \lambda_n > 0$ and let $Y_1, \ldots, Y_n$ be a random sample from the exponential distribution with hazard rate $\bar \lmd = \sum_{i=1}^n \lmd_i/n$. Also let $X_{1:n} < \cdots < X_{n:n}$ and $Y_{1:n} < \cdots < Y_{n:n}$ be th...

We consider coherent systems with independent and identically distributed components. While it is clear that the system’s life will be stochastically larger when the components are replaced with stochastically better components, we show that, in general, similar results may not hold for hazard rate, reverse hazard rate, and likelihood ratio orderin...

Let X-lambda 1, X-lambda 2, ... ,X-lambda n be independent non negative random variables with X-lambda i similar to F(lambda(i)t), i = 1, ... , n, where lambda(i) > 0, i = 1, ... , n and F is an absolutely continuous distribution. It is shown that, under some conditions, one largest order statistic X-n:n(lambda) n is smaller than another one X-n:n(...

Assuming that the joint density of random variables $X_1,X_2,\ldots,X_n$ is arrangement increasing (AI), we obtain some stochastic comparison results on weighted sums of $X_i$'s under some further conditions. An application to optimal capital allocation is also given.

Let $X_{\lambda _{1}},X_{\lambda _{2}},\ldots ,X_{\lambda _{n}}$ be independent Weibull random variables with $X_{\lambda _{i}}\sim W(\alpha,\lambda _{i})$ where $\lambda _{i}>0$, for $i=1,\ldots ,n$. Let $X_{n:n}^{\lambda }$ denote the lifetime of the parallel system formed from $% X_{\lambda _{1}},X_{\lambda _{2}},\ldots ,X_{\lambda _{n}}$. We i...

Let $X_{\lambda _{1}},X_{\lambda _{2}},\ldots ,X_{\lambda _{n}}$ be independent nonnegative random variables with $X_{\lambda _{i}}\sim F(\lambda _{i}t)$, $i=1,\ldots ,n$, where $\lambda _{i}>0$, $i=1,\ldots ,n$ and $F$ is an absolutely continuous distribution. It is shown that, under some conditions, one largest order statistic $X_{n:n}^{\lambda }...

Assuming that the joint density of random variables is arrangement increasing (AI), we obtain some stochastic comparison results on weighted sums of ’s under some additional conditions. An application to optimal capital allocation is also given.

In this article, we focus on stochastic orders to compare the magnitudes of two parallel systems from Weibull distributions when one set of scale parameters majorizes the other. The new results obtained here extend some of those proved by Dykstra et al. (1997) and Joo and Mi (2010) from exponential to Weibull distributions. Also, we present some re...

Order statistics from heterogeneous samples have been extensively studied in the literature. However, most of the work focused on the effect of heterogeneity on the magnitude or dispersion of order statistics. In this paper, we study the skewness of order statistics from heterogeneous samples according to star ordering. The main results extend the...

The P–P plot is a powerful graphical tool to compare stochastically the magnitudes of two random variables. In this note, we introduce a new partial order, called P–P order based on P–P plots. For a pair of random variables (X 1, Y 1) and (X 2, Y 2) one can see the relative precedence of Y 2 over X 2 versus that of Y 1 over X 1 using P–P order. We...

Shaked and Shanthikumar [425] introduced the excess wealth transform and the related excess wealth order. A lot of research activities have taken place on this topic lately. In this paper, we discuss some recent developments of this transform and illustrate how to use this transform in extreme value analysis. We also summarize the applications of e...

Due to its wide range of applications, the distribution theory of convolutions of gamma random variables has attracted the attention from many researchers. In this paper, we review some of the latest developments on this problem.

We review some of the recent developments in the area of stochastic comparisons of order statistics and sample spacings. We consider the cases when the parent observations are identically as well as nonidentically distributed. But most of the time we will be assuming that the observations are independent. The case of independent exponentials with u...

In this paper, a new sufficient condition for comparing linear combinations of independent gamma random variables according to star ordering is given. This unifies some of the newly proved results on this problem. Equivalent characterizations between various stochastic orders are established by utilizing the new condition. The main results in this...

Kochar and Xu (2009) proved that a parallel system with heterogeneous exponential component lifetimes is more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. In this paper we extend this study to the general k-out-of-n systems for the case when there are only two...

Kochar and Xu (2009) proved that a parallel system with heterogeneous exponential component lifetimes is more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. In this paper we extend this study to the general k -out-of- n systems for the case when there are only tw...

Two sufficient conditions for comparing convolutions of heterogeneous gamma random variables in terms of star order are established. It is further shown that if the scale parameters of heterogeneous gamma random variables are more dispersed in the sense of majorization, then the convolutions are more dispersed according to the right spread order, w...

Independent random variables Xλ1,…,Xλn are said to belong to the scale family of distributions if Xλi∼F(λix), for i=1,…,n, where F is an absolutely continuous distribution function with hazard rate r and reverse hazard rate . We show that the hazard rate (reverse hazard rate) of a series (parallel) system consisting of components with lifetimes Xλ1...

In this paper we review some of the recently obtained results in the area of stochastic comparisons of sample spacings when the observations are not necessarily identically distributed. A few new results on necessary and sufficient conditions for various stochastic orderings among spacings are also given. The paper is concluded with some examples a...

A sufficient condition for comparing convolutions of heterogeneous exponential random variables in terms of right spread order is established. As a consequence, it is shown that a convolution of heterogeneous independent exponential random variables is more skewed than that of homogeneous exponential random variables in the sense of NBUE order. Thi...

In this article, mixture representations of survival functions of residual lifetimes of k-out-of-n systems are obtained when the components are independent but not necessarily identically distributed. Then we stochastically compare the residual lifetimes of k-out-of-n systems in one- and two-sample problems. In particular, the results extend some r...

Let Rn be the range of a random sample X1,...,Xn of exponential random variables with hazard rate [lambda]. Let Sn be the range of another collection Y1,...,Yn of mutually independent exponential random variables with hazard rates [lambda]1,...,[lambda]n whose average is [lambda]. Finally, let r and s denote the reversed hazard rates of Rn and Sn,...

A parallel system with heterogeneous exponential component lifetimes is shown to be more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. As a consequence, equivalent conditions for comparing the variabilities of the largest order statistics from heterogeneous and...

A parallel system with heterogeneous exponential component lifetimes is shown to be more skewed (according to the convex transform order) than the system with independent and identically distributed exponential components. As a consequence, equivalent conditions for comparing the variabilities of the largest order statistics from heterogeneous and...

In this paper, we introduce a new copula-based dependence order to compare the relative degree of dependence between two pairs of random variables. Relationship of the new order to the existing dependence orders is investigated. In particular, the new ordering is stronger than the partial ordering, more monotone regression dependence as developed b...

In our article [3] we have found a gap in the middle of the proof of Theorem 3.2. Therefore, we do not know whether Theorem 3.2 is true for the reverse hazard rate order. However, we could prove the following weaker result for the stochastic order.

There is a substantial literature on testing for the equality of the cumulative incidence functions associated with one specific cause in a competing risks setting across several populations against specific or all alternatives. In this paper we propose an asymptotically distribution-free test when the alternative is that the incidence functions ar...

Let X1,…,Xn be a random sample from an absolutely continuous distribution with non-negative support, and let Y1,…,Yn be mutually independent lifetimes with proportional hazard rates. Let also X(1)<⋯<X(n) and Y(1)<⋯<Y(n) be their associated order statistics. It is shown that the pair (X(1),X(n)) is then more dependent than the pair (Y(1),Y(n)), in t...

Unaware of the developments in each other’s areas, researchers in the disciplines of economics and reliability theory have been working independently. The objective of this paper is to point out some interesting relationships that exist between some of the notions in these two areas. In particular, we discuss notions of NBUE (New Better Than Used i...

In this note, we further study the properties of excess wealth (or right spread) order and the location independent riskier order. It is proved that if X is less variable than Y according to excess wealth order, then for k=0,1,…,n−1, where X0:n=Y0:n≡0. Similar results are obtained for location independent riskier order. An application in k-price bu...

Let X1, … , Xn be independent random variables with Xi having survival function λi, i = 1, … , n, and let Y1, … ,Yn be a random sample with common population survival distribution , where = ∑i=1nλi/n. Let Xn:n and Yn:n denote the lifetimes of the parallel systems consisting of these components, respectively. It is shown that Xn:n is greater than Yn...

This is a survey paper on recent results on stochastic comparisons of order statistics of n independent random variables differing in their scale parameters. Most of the results obtained so far are for the Weibull and the Gamma distributions.

This paper reviews some recent results on stochastic or- ders and dependence among order statistics when the observations are independent and follow the proportional hazard rates model.

Let Xi:n denote the ith order statistic of a random sample of size n from a continuous distribution with cdf F. Sufficient conditions are obtained on F so that Xj:m⩽⋆Xi:n (hence Xj:m⩽LorenzXi:n) for i⩽j and n-i⩾m-j.

A basic concept for comparing spread among probability distributions is that of dispersive ordering. Let X and Y be two random variables with distribution functions F and G, respectively. Let F −1 and G −1 be their right continuous inverses (quantile functions). We say that Y is less dispersed than X (Y≤ disp X) if G −1(β)−G −1(α)≤F −1(β)−F −1(α)...

In this paper we consider the problem of testing the equality of r(r⩾2) cumulative incidence functions against an ordered alternative, using the likelihood ratio approach. We assume a discrete time framework and obtain maximum likelihood estimators of the r cumulative incidence functions under the restriction that they are uniformly ordered. The as...

To compare two multivariate random vectors of the same dimension, we define a new stochastic order called upper orthant dispersive ordering and study its properties. We study its relationship with positive dependence and multivariate hazard rate ordering as defined by Hu, Khaledi, and Shaked (Journal of Mliltivariate Analysis, 2002). It is shown th...

Let S and T denote two survival functions, such that the func- tion µ(x) = S(x)=T(x) is non-increasing on the support of T. In this case S is called uniformly stochastically smaller than T: This con- cept is useful in reliability and life testing as it is equivalent to the hazard ordering. Rojo and Samaniego (1991, 1993) study consistent estimation...

To compare two multivariate random vectors of the same dimension, we define a new stochastic order called upper orthant dispersive ordering and study its properties. We study its relationship with positive dependence and multivariate hazard rate ordering as defined by Hu, Khaledi, and Shaked (Journal of Multivariate Analysis, 2002). It is show...

Generalized order statistics (gOSs) unify the study of order statistics, record values, k-records, Pfeifer's records and several other cases of ordered random variables. In this paper we consider the problem of comparing the degree of dependence between a pair of gOSs thus extending the recent work of Avérous et al. [2005. J. Multivariate Anal. 94,...

Given a random sample from a continuous variable, it is observed that the copula linking any pair of order statistics is independent of the parent distribution. To compare the degree of association between two such pairs of ordered random variables, a notion of relative monotone regression dependence (or stochastic increasingness) is considered. Us...

In the competing risks problem an important role is played by the cumulative incidence function (CIF), whose value at time t is the probability of failure by time t for a particular type of failure in the presence of other risks. Its estimation and asymptotic distribution theory have been studied by many. In some cases there are reasons to believe...

For nonnegative random variables X and Y we write X ≤ TTT Y if ∫ 0 F ⁻¹ ( p ) (1- F ( x ))d x ≤ ∫ 0 G ⁻¹ ( p ) (1- G ( x ))d x all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spr...

For nonnegative random variables X and Y we write X ≤_TTT Y if ∫₀^{F^-1(p)}(1-F(x))dx ≤ ∫₀^{G^-1(p)}(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic orde...

In this paper we propose two new classes of asymptotically distribution-free Renyi-type tests for testing the equality of two risks in a competing risk model with possible censoring. This work extends the work of Aly, Kochar and McKeague [1994, Journal of American Statistical Association, 89, 994-999] and many of the existing tests for this problem...

If X is a life distribution with finite mean then its mean residual life function (MRLF) is defined by M(x)=E[X−x|X>x]. It has been found to be a very intuitive way of describing the aging process. Suppose that M1 and M2 are two MRLFs, e.g., those corresponding to the control and the experimental groups in a clinical trial. It may be reasonable to...

Recently Bassan and Spizzichino (1999) have given some new concepts of multivariate ageing for exchangeable random variables, such as a special type of bivariate IFR, by comparing distributions of residual lifetimes of dependent components of different ages. In the same vein, we further study some properties of these concepts of IFR in the bivariat...

Recently Bassan and Spizzichino (1999) have given some new concepts of multivariate ageing for exchangeable random variables, such as a special type of bivariate IFR, by comparing distributions of residual lifetimes of dependent components of different ages. In the same vein, we further study some properties of these concepts of IFR in the bivariat...

We consider the competing risks problem with two risks and when the data are grouped or discrete. We firstly obtain nonparametric maximum likelihood estimates of the sub- survival functions corresponding to the two risks under the restriction that they are uni- formly ordered and then use them to derive the likelihood ratio statistic for testing th...

In this paper we review some of the results obtained recently in the area of stochastic comparisons of order statistics and sample spacings. We consider the cases when the parent observations are identically as well as non-identically distributed. But most of the time we shall be assuming that the observations are independent. The case of independe...

Lot a((i)) and b((i)) be the ith smallest components of a = (a(1), . . . , a(n)) and b = (b(1), . . . , b(n)) respectively, where a,b is an element of R+n. The vector a is said to be p-larger than b (denoted by a greater than or equal tob(p)) if Pi (k)(i-1) a((i)) less than or equal to Pi (k)(i=1) b((i)), for k = 1, . . . , n. Let U-1, . . . , U-n...

Let (Xi, Yi) i=1, 2, …, n be n independent and identically distributed random variables from some continuous bivariate distribution. If X(r) denotes the rth ordered X-variate then the Y-variate, Y[r], paired with X(r) is called the concomitant of the rth order statistic. In this paper we obtain new general results on stochastic comparisons and depe...

Some new results about the NBU(2) class of life distributions are obtained. Firstly, it is proved that the decreasing with time of the increasing concave ordering of the excess lifetime in a renewal process leads to the NBU(2) property of the interarrival times. Secondly, the NBU(2) class of life distributions is proved to be closed under the forma...

Some new results about the NBU(2) class of life distributions are obtained. Firstly, it is proved that the decreasing with time of the increasing concave ordering of the excess lifetime in a renewal process leads to the NBU(2) property of the interarrival times. Secondly, the NBU(2) class of life distributions is proved to be closed under the forma...

Consider a multivariate mixture model where the random variables X 1, ..., X n given (Θ1, ..., Θn ), are conditionally independent. Conditions are obtained under which different kinds of positive dependence hold among X i 's. The results obtained are applied to a variety of problems including the concomitants of order statistics and of record value...

We consider the three progressively more general sampling schemes without replacement from a finite population: simple random sampling without replacement, Midzuno sampling and successive sampling. We (i) obtain a lower bound on the expected sample coverage of a successive sample, (ii) show that the vector of first order inclusion probabilities div...

Let X 1 ,…, X n be independent exponential random variables with X i having hazard rate . Let Y 1 ,…, Y n be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏ i =1 n λ i ) 1/ n , the geometric mean of the λ i s. Let X n : n = max{ X 1 ,…, X n }. It is shown that X n : n is greater than Y n : n according to d...

Let X₁,...,X_n be independent exponential random variables with X_i having hazard rate λ_i, i = 1,...,n. Let Y₁,...,Y_n be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏_i=1ⁿλ_i)^1/n, the geometric mean...

In this paper, we study the dependence properties of spacings. It is proved that if X1,..., Xn are exchangeable random variables which are TP2 in pairs and their joint density is log-convex in each argument, then the spacings are MTP2 dependent. Next, we consider the case of independent but nonhomogeneous exponential random variables. It is s...

[Formula: see text] AMS (2000) Subject classification: 60E15, 62N05, 62D05.

In survival analysis and in the analysis of life tables an important biometric function of interest is the life expectancy at age $x, M(x)$, defined by $$M(x) = E[X - x|X > x],$$ ¶ where $X$ is a lifetime. $M$ is called the mean residual life function.In many applications it is reasonable to assume that $M$ is decreasing (DMRL) or increasing (IMRL)...

In the competing risks literature, one usually compares whether two risks are equal or whether one is "more serious." In this paper, we propose tests for the equality of two competing risks against an ordered alternative specified by their sub-survival functions. These tests are naturally developed as extensions of those based on hazard rates and c...

Let X-i : n denote the ith-order statistic of a random sample of size n from a continuous distribution with distribution function F. It is shown that if F is a decreasing failure rate (DFR) distribution, then X-i : n is less dispersed than X-j : m, for i less than or equal to j and n - i greater than or equal to m - j. Let Y-j : m denote the jth-or...

Various methods and criteria for comparing coherent systems are discussed. Theoretical results are derived for comparing systems of a given order when components are assumed to have independent and identically distributed lifetimes. All comparisons rely on the representation of a system's lifetime distribution as a function of the system's “signatu...

Let X1:n[less-than-or-equals, slant]X2:n[less-than-or-equals, slant]...[less-than-or-equals, slant]Xn:n denote the order statistics of a random sample of size n from a probability distribution with distribution function F. Similarly, let Y1:m[less-than-or-equals, slant]Y2:m[less-than-or-equals, slant]...[less-than-or-equals, slant]Ym:m denote the o...

Let Xλ1,…,Xλn be independent random variables such that Xλi has exponential distribution with hazard rate . It is shown that ∑i=1nXλi is more dispersed than if (λ1,…,λn) majorizes.

Let X1:n[less-than-or-equals, slant]X2:n[less-than-or-equals, slant]...[less-than-or-equals, slant]Xn:n denote the order statistics of a random sample X1,X2,...,Xn from a probability distribution with distribution function F. Similarly, let Y1:n[less-than-or-equals, slant]Y2:n[less-than-or-equals, slant]...[less-than-or-equals, slant]Yn:n denote th...

It is well known that the normalized spacings based on a random sample from a life distribution are independent and identically distributed if and only if the underlying life distribution is exponential. It is of interest to investigate the stochastic properties of the spacings when the parent observations do not necessarily constitute a random sam...

We consider the competing-risks problem without making any assumption concerning the independence of the risks. Maximum-likelihood estimates of the cause-specific hazard rates are obtained under the condition that their ratio is monotone. We also consider the likelihood-ratio test for testing the proportionality of two cause-specific hazard rates a...

In this paper we introduce a quantile dispersion measure. We use it to characterize different classes of ageing distributions. Based on the quantile dispersion measure, we propose a new partial ordering for comparing the spread or dispersion in two probability distributions. This new partial ordering is weaker than the well known dispersive orderin...

In this paper we introduce a quantile dispersion measure. We use it to characterize different classes of ageing distributions. Based on the quantile dispersion measure, we propose a new partial ordering for comparing the spread or dispersion in two probability distributions. This new partial ordering is weaker than the well known dispersive orderin...

Let X1, …, Xn be independent exponential random variables with Xi having hazard rate λi, i = 1, …, n. Let λ = (λ1, …, λn). Let Y1, …, Yn be a random sample of size n from an exponential distribution with common hazard rate . The purpose of this paper is to study stochastic comparisons between the largest order statistics Xn:n and Yn:n from these tw...

Recently, a new variability ordering, called right spread ordering or excess wealth ordering has been introduced. This new ordering is weaker than dispersive ordering. We show in this note that if X is less variable than Y in the sense of right spread ordering or convex ordering, then it implies that X1 - X2 is less variable than Y1 - Y2 according...

We consider the problem of testing the null hypothesis of no change against the alternative of exactly one change point. The proposed tests are based on generalized two-sample U- statistic processes. We drive the limiting null distributions of the proposed tests. Some applications in Statistical Reliability are given.

We study stochastic orders and dependence relations between order statistics from a linearly ordered finite population when using either simple random sampling without replacement (SRSWOR) or Midzuno sampling schemes. It is shown that when there are no multiplicities in the population, the density functions of order statistics, in the cases of SRSW...

A life distribution F is called NBUE-NWUE if for some t 0 ∈(0,∞), its mean residual life function e(t)=E F (X-t∣X≥t) satisfies e(t)<e(0) for 0<t<t 0 and e(t)>e(0) for t>t 0 . If the inequalities for e(t) are reversed on these time intervals, it is called NWUE-NBUE. Using a characterization of such distributions in terms of the scaled total-time-on-...