Stochastic Comparisons of Order Statistics and Spacings: A Review

Abstract

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 unequal scale parameters as well as the proportional hazard rate model is discussed in detail.

1. Introduction

The simplest and the most popular method of comparing the magnitudes of two random variables is through their means and medians. It may happen that in some cases the median of 𝑋 is larger than that of 𝑌, while the mean of 𝑋 is smaller than the mean of 𝑌. However, this confusion will not arise if the random variables are stochastically ordered. Similarly, the same may happen if one would like to compare the variability of 𝑋 with that of 𝑌 based only on numerical measures like standard deviation, and so forth. Besides, these characteristics of distributions might not exist in some cases. In most cases one can express various forms of knowledge about the underlying distributions in terms of their survival functions, quantile functions, hazard rate functions, mean residual functions, and other suitable functions of probability distributions. These methods are much more informative than those based only on few numerical characteristics of distributions. Comparisons of random variables based on such functions usually establish partial orders among them. We call them as stochastic orders.

Stochastic models are usually sufficiently complex in various fields of statistics. Obtaining bounds and approximations for their characteristics is of practical importance. That is, the approximation of a stochastic model either by a simpler model or by a model with simple constituent components might lead to convenient bounds and approximations for some particular and desired characteristics of the model. The study of changes in the properties of a model, as the constituent components vary, is also of great interest. Accordingly, since the stochastic components of models involve random variables, the topic of stochastic orders among random variables plays an important role in these areas. Books by Muller and Stoyan [1] and Shaked and Shanthikumar [2] give excellent treatment of this topic.

Order statistics and spacings are of great interest in many areas of statistics and they have received a lot of attention from many researchers. Let 𝑋₁,…,𝑋_𝑛 be 𝑛 random variables. The 𝑖th order statistic, the 𝑖th smallest of X_𝑖’s, is denoted by 𝑋_𝑖∶𝑛. In reliability engineering, an 𝑛 component system that works if and only if at least 𝑘 of the 𝑛 components work is called a 𝑘-out-of-𝑛 system. The lifetime of a 𝑘-out-of-𝑛 system can be represented as 𝑋_{𝑛−𝑘+1∶𝑛}. A parallel system is a 1-out-of-𝑛 system while a series system is an 𝑛-out-of-𝑛 system. Thus, the study of lifetimes of 𝑘-out-of-𝑛 systems is equivalent to the study of the stochastic properties of order statistics. Spacings, the differences between successive order statistics, and their functions are also important in statistics, in general, and in particular in the context of life testing and reliability models. The books by David and Nagaraja [3], and Arnold et al. [4]; and two volumes of papers on this topic by Balakrishnan and Rao [5, 6] are excellent sources of information on this topic. But most of this work has been confined to the case when the observations are independent and identically distributed (i.i.d.). In many practical situations, like in reliability theory, the observations are not necessarily i.i.d. Only during the last two decades or so this topic has got the attention of researchers. Some important early references for this case are Sen [7], David and Nagaraja [3], Shaked and Tong [8], Bapat and Beg [9], Boland et al. [10], Kochar [11], and Nappo and Spizzichino [12], Boland et al. [13] and Balakrishnan [14], among others.

Some interesting partial ordering results on order statistics and spacings from independent but nonidentically random variables have been obtained by Pledger and Proschan [15], Proschan and Sethuraman [16], Bapat and Kochar [17], Boland et al. [18], Kochar and Kirmani [19], Kochar and Korwar [20], Kochar and Rojo [21], Dykstra et al. [22], Kochar and Ma [23], Bon and Pǎltǎnea [24], Kochar [25], Khaledi and Kochar [26–30]. The book by Shaked and Shanthikumar [2] gives an excellent description of the various results on this topic till 2007.

In this review paper, we will discuss some latest developments on the topic of stochastic comparisons of order statistics and spacings. In Section 2, we introduce the required notation and definitions. Sections 3 and 4 are devoted to stochastic comparisons of order statistics in one-sample and two-sample problems, respectively. In Section 5, we discuss the topic of stochastic orderings among spacings in one-sample problem and two sample problems. Section 6 is devoted to some applications of these results. Throughout this chapter increasing means nondecreasing and decreasing means nonincreasing; and we will be assuming that all distributions under study are absolutely continuous.

2. Definitions and Some Preliminaries

In this section, we recall some basic definitions of stochastic orders and their properties. Let 𝑋 and 𝑌 be univariate random variables with distribution functions 𝐹 and 𝐺, survival functions 𝐹 and G, density functions 𝑓 and 𝑔; and hazard rates 𝑟_𝐹 (=𝑓/𝐹) and 𝑟_𝐺 (=𝑔/𝐺), respectively. Let 𝑙_𝑋(𝑙_𝑌) and 𝑢_𝑋(𝑢_𝑌) be the left and the right endpoints of the support of 𝑋(𝑌).

2.1. Magnitude Orders

First we give some definitions of stochastic orders to compare the magnitudes of two random variables.

Definition 2.1. 𝑋is said to be stochastically smaller than 𝑌 (denoted by 𝑋≤_st𝑌) if 𝐹(𝑥)≤𝐺(𝑥)∀𝑥. (2.1)

It is easy to see that 𝑋≤_st𝑌 if and only if 𝐹⁻¹(𝑢)≤𝐺⁻¹(𝑢)∀𝑢∈(0,1). (2.2) It can be shown that this is also equivalent to 𝐸[𝑔(𝑋)]≤𝐸[𝑔(𝑌)] for all increasing function 𝑔 for which expectations exist.

Definition 2.2. 𝑋 is said to be smaller than 𝑌 in hazard rate ordering (denoted by 𝑋≤_hr𝑌) if 𝐺(𝑥)/𝐹(𝑥) is increasing in 𝑥∈(−∞,max(𝑢_𝑋,𝑢_𝑌)).

It can be shown that 𝑋≤_hr𝑌 is equivalent to the inequalities 𝑃[𝑋−𝑡>𝑥∣𝑋>𝑡]≤𝑃[𝑌−𝑡>𝑥∣𝑌>𝑡],∀𝑥≥0,𝑡. (2.3) In other words, the conditional distributions, given that the random variables are at least of a certain size, are all stochastically ordered (in the standard sense) in the same direction. Thus, if 𝑋 and 𝑌 represent the survival times of different models of an appliance that satisfy this ordering, one model is better (in the sense of stochastic ordering) when the appliances are new, the same appliance is better when both are one month old, and in fact is better no matter how much time has elapsed. It is clearly useful to know when this strong type of stochastic ordering holds since qualitative judgements are then easy to make. In case the hazard rates exist, it is easy to see that 𝑋≤_hr𝑌, if and only if, 𝑟_𝐺(𝑥)≤𝑟_𝐹(𝑥) for every 𝑥. The hazard rate ordering is also known as uniform stochastic ordering in the literature.

Definition 2.3. 𝑋 is said to be smaller than 𝑌 in the reverse hazard rate order, denoted by 𝑋≤_rh𝑌, if 𝑃(𝑡−𝑋>𝑥∣𝑋≤𝑡)≥𝑃(𝑡−𝑌>𝑥∣𝑌≤𝑡),∀𝑥≥0andall𝑡. (2.4)

It is shown in Ross [31] that if 𝑋 and 𝑌 are two independent random variables, then 𝑋≤_st𝑌 may not imply 𝑋+(𝑋+𝑌)≤_st𝑌+(𝑋+𝑌). (2.5) However (2.5) will hold if 𝑋 and 𝑌 satisfy a stronger ordering called likelihood ratio ordering as defined below.

Definition 2.4. 𝑋 is said to be smaller than Y in likelihood ratio ordering (denoted by 𝑋≤_lr𝑌) if 𝑔(𝑥)/𝑓(𝑥) is increasing in 𝑥∈(𝑙_𝑋,𝑢_𝑋)∪(𝑙_𝑌,𝑢_𝑌).

The mean residual life function of a random variable 𝑋 is defined as 𝜇_𝑋(𝑡)=∫^∞_𝑥𝐹(𝑡)𝑑𝑡. An interesting order based on the mean residual life is defined as follows.

Definition 2.5. 𝑋 is said to be smaller than 𝑌 in the mean residual life order, denoted by 𝑋≤_mrl𝑌, if 𝜇_𝑋(𝑡)≤𝜇_𝑌(𝑡)∀𝑡. (2.6)

When the supports of 𝑋 and 𝑌 have a common left end-point, we have the following chain of implications among the above stochastic orders: 𝑋≤_lr𝑌⇒𝑋≤_hr𝑌⇒𝑋≤_mrl𝑌⇓⇓⇓𝑋≤_rh𝑌⇒𝑋≤_st𝑌⇒𝐸[𝑋]≤𝐸[𝑌]. (2.7)

Definition 2.6. 𝑋 is said to be smaller than 𝑌 in the increasing convex order (denoted by 𝑋≤_icx𝑌) if _𝑡^+∞𝐹(𝑥)𝑑𝑥≤_𝑡^+∞𝐺(𝑥)𝑑𝑥,∀𝑡. (2.8)

Note that (2.8) holds if and only if 𝐸𝜙(𝑋)≤𝐸𝜙(𝑌) (2.9) for every convex function 𝜙 for which the above expectations exist. Also note that 𝑋≤_st𝑌⇒𝑋≤_icx𝑌.

Another order closely related to the likelihood ratio order is the joint likelihood ordering as introduced by Shanthikumar and Yao [32].

Definition 2.7. For a bivariate random variable (𝑋,𝑌), 𝑋 is said to be smaller than 𝑌 according to joint likelihood ordering, denoted by 𝑋≤_{ℓ𝑟∶𝑗}, if and only if 𝐸𝜙(𝑋,𝑌)≥𝐸𝜙(𝑌,𝑋),𝜙∈𝒢_ℓ𝑟, (2.10) where 𝒢_ℓ𝑟:𝜙∶𝜙𝑦,𝑥≤𝜙𝑥,𝑦,𝑥≤𝑦. (2.11)

It can be seen that 𝑋₁≤_{ℓ𝑟∶𝑗}𝑋₂⟺𝑓∈𝒢_ℓ𝑟, (2.12) where 𝑓(⋅,⋅) denotes the joint density of (𝑋₁,𝑋₂).

As pointed out by Shanthikumar and Yao [32], joint likelihood ratio ordering between the components of a bivariate random vector may not imply likelihood ratio ordering between their marginal distributions unless the random variables are independent, but it does imply stochastic ordering between them, that is, 𝑋≤^{ℓ𝑟∶𝑗}𝑌⇒𝑋≤^st𝑌. (2.13)

A bivariate function 𝜙∈𝒢_ℓ𝑟 is called arrangement increasing (AI). Hollander et al. [33] have studied many interesting properties of such functions, though, apparently, they did not relate it to the notion of likelihood ratio ordering.

The above idea can be extended to compare the components of an 𝑛-dimensional vector 𝐗=(𝑋₁,…,𝑋_𝑛). We define 𝑋₁≤_{ℓ𝑟∶𝑗}⋯≤_{ℓ𝑟∶𝑗}𝑋_𝑛 if the joint density 𝑓(𝑥₁,…,𝑥_𝑛) of 𝐗 is an arrangement increasing function. (See, Marshall et al. [34] for the definition of an arrangement increasing function on ℛ^𝑛.)

In a different context, Robertson and Wright [35] studied a subclass of arrangement increasing functions on ℛ^𝑛, which they call as ISO^∗ functions, as described below. Let 𝐱 and 𝐲 be two vectors on ℛ^𝑛 such that ∑^𝑗_𝑖=1𝑦_𝑖≤∑^𝑗_𝑖=1𝑥_𝑖,𝑗=1,…,𝑛−1 and ∑^𝑛_𝑖=1𝑦_𝑖=∑^𝑛_𝑖=1𝑥_𝑖. We will denote this partial ordering between the vectors by 𝐱≪^⋆𝐲.

Definition 2.8. A real-valued function 𝜙 defined on a set 𝒜⊂ℛ^𝑛 is said to be ISO^∗ on 𝒜 if 𝜙(𝐱)≤𝜙(𝐲),forall𝐱≪^⋆𝐲.

As mentioned earlier, an ISO^∗ function is arrangement increasing but the converse is not true. It is easy to see that the joint density 𝑓(𝑥₁,𝑥₂) of a bivariate random vector (𝑋₁,𝑋₂) is ISO^∗ if and only if the conditional density of 𝑋₂ given 𝑋₁+𝑋₂=𝑡 is monotonically increasing for each fixed 𝑡.

The usual likelihood ratio order has the following multivariate version.

Definition 2.9. Let 𝐗=(𝑋₁,𝑋₂,…,𝑋_𝑛) and 𝐘=(𝑌₁,𝑌₂,…,𝑌_𝑛) be two 𝑛-dimensional random vectors with absolutely continuous (or discrete) distribution functions and let 𝑓 and 𝑔 denote their continuous (or discrete) density functions, respectively. Suppose that 𝑓(𝐱)𝑔(𝐲)≤𝑓(𝐱∧𝐲)𝑔(𝐱∨𝐲) (2.14) for every 𝐱 and 𝐲. Then 𝑋 is said to be smaller than 𝑌 in the multivariate likelihood ratio order (denoted as 𝐗≤_lr𝐘).

Definition 2.10. The random vector 𝐗=(𝑋₁,…,𝑋_𝑛) is smaller than the random vector 𝐘=(𝑌₁,…,𝑌_𝑛) in the multivariate stochastic order (denoted by 𝐗⪯^st𝐘) if ℎ(𝐗)≤_stℎ(𝐘) for all increasing functions ℎ.

It is known that multivariate likelihood ratio ordering implies multivariate stochastic ordering, but the converse is not true. Also if two random vectors are ordered according to multivariate likelihood ratio (stochastic) ordering, then their corresponding subsets of components are also ordered accordingly. For more details on stochastic orderings, see Chapters 1 and 4 of Shaked and Shanthikumar [2].

2.2. Variability Orders

One of the basic criteria for comparing variability in probability distributions is that of dispersive ordering. Let 𝐹⁻¹ and 𝐺⁻¹ be the right continuous inverses (quantile functions) of 𝐹 and 𝐺, respectively.

Definition 2.11. 𝑋 is less dispersed than 𝑌 (denoted by 𝑋≤_disp𝑌) if 𝐹⁻¹𝛽−𝐹⁻¹(𝛼)≤𝐺⁻¹𝛽−𝐺⁻¹(𝛼),∀0≤𝛼≤𝛽≤1. (2.15)

Note that (2.15) is equivalent to 𝐺⁻¹𝐹(𝑥)−𝑥increasingin𝑥. (2.16)

When (2.16) holds, Doksum [36] called this ordering as tail-ordering and used it to find bounds on powers and efficiencies of nonparametric tests. Deshpande and Kochar [37] pointed out that tail ordering is same as dispersive ordering and obtained some new results for this partial order.

A consequence of 𝑋≤_disp𝑌 is that |𝑋₁−𝑋₂|≤_st|𝑌₁−𝑌₂| and which in turn implies var(𝑋)≤var(𝑌) as well as 𝐸[|𝑋₁−𝑋₂|]≤𝐸[|𝑌₁−𝑌₂|], where 𝑋₁,𝑋₂(𝑌₁,𝑌₂) are two independent copies of 𝑋(𝑌). For details, see Saunders [38], Saunders and Moran [39], Lewis and Thompson [40], Deshpande and Kochar [37], Bagai and Kochar [41], Bartoszewicz [42, 43], and Section 2.B of Shaked and Shanthikumar [2].

Connection between Hazard Rate Order and Dispersive Order
Kochar [44] observed that 𝑋≤_hr𝑌⇒𝑃[𝑋𝑋𝑌𝑌]+𝑃[𝑌𝑋𝑋𝑌]≤𝑃[𝑌𝑌𝑋𝑋]+𝑃[𝑋𝑌𝑌𝑋], (2.17) where, for example, 𝑃[𝑋𝑋𝑌𝑌] means 𝑃[𝑋_1∶2≤𝑋_2∶2≤𝑌_1∶2≤𝑌_2∶2] with the notation that 𝑍_𝑖∶𝑛 denotes the 𝑖th order statistic of a random sample 𝑍₁,…,𝑍_𝑛 on 𝑍.

Equation (2.17) indicates that hazard rate ordering not only compares the magnitudes of two random variables, but it also has perhaps some connection with the variability between the random variables. On differentiating (2.16), one can easily see that 𝑋≤_disp𝑌⟺𝑟_𝐺𝐺⁻¹(𝑢)≤𝑟_𝐹𝐹⁻¹(𝑢)∀𝑢∈(0,1), (2.18) when the random variables 𝑋 and 𝑌 admit densities. This lead Bagai and Kochar [41] to prove the following connection between hazard rate ordering and dispersive ordering.

Theorem 2.12. (a) If 𝑋 or 𝑌 have increasing failure rate (IFR), then 𝑋≤_{𝑑𝑖𝑠𝑝}⇒𝑋≤_ℎ𝑟𝑌.
(b) If 𝑋 or 𝑌 have decreasing failure rate (DFR), then 𝑋≤_ℎ𝑟⇒𝑋≤_{𝑑𝑖𝑠𝑝}𝑌.

For a review on applications of dispersive ordering, see Joen et al. [45].

A weaker variability ordering is right spread order as introduced in Fernandez-Ponce et al. [46]. It was also independently studied by Shaked and Shanthikumar[47] who call it as excess wealth order.

Definition 2.13. 𝑋 is said to be smaller than 𝑌 in the excess wealth order (denoted by 𝑋≤_ew𝑌) if ^∞_𝐹−1𝑝𝐹(𝑥)𝑑𝑥≤^∞_𝐺−1𝑝𝐺(𝑥)𝑑𝑥,0≤𝑝≤1. (2.19)

Dispersive ordering implies excess wealth ordering which in turn implies that the commonly used measures of variability like variances are ordered. One may refer to Shaked and Shanthikumar [2] for a comprehensive discussion of this order.

2.3. Skewness Orderings

Skewness describes the departure of a distribution from symmetry, where one tail of the density is more “stretched out" than the other. Several partial orders have been introduced in the literature to compare the relative skewness of probability distributions. van Zwet [48] introduced the concept of convex transform order to compare two distributions according to skewness as defined below.

Definition 2.14. 𝑋 is said to be smaller than 𝑌 in the convex transform order, denoted by 𝑋≤_𝑐𝑌 if and only if, 𝐺⁻¹𝐹(𝑥) is convex in 𝑥 on the support of 𝑋.

If 𝑋≤_𝑐𝑌, then 𝑌 is more skewed than 𝑋 as explained in van Zwet [48] and Marshall et al. [34]. The convex transform order is also called more IFR (increasing failure rate) order in reliability theory, since when 𝑓 and 𝑔 exist, the convexity of 𝐺⁻¹𝐹(𝑥) means that 𝑓𝐹⁻¹(𝑢)𝑔𝐺⁻¹(𝑢)=𝑟_𝑋𝐹⁻¹(𝑢)𝑟_𝑌𝐺⁻¹(𝑢), (2.20) is increasing in 𝑢∈[0,1]. Thus 𝑋≤_𝑐𝑌 can be interpreted to mean that 𝑋 ages faster than 𝑌 in some sense. Gamma distributions are ordered according to the convex transform order in terms of their shape parameters.

Another well-known partial order to compare the skewness of two probability distributions is star order.

Definition 2.15. 𝑋 is said to be smaller than 𝑌 in the star order, denoted by 𝑋≤_⋆𝑌 (or 𝐹≤_⋆𝐺) if the function 𝐺⁻¹𝐹(𝑥) is star shaped in the sense that 𝐺⁻¹𝐹(𝑥)/𝑥 is increasing in 𝑥 on the support of 𝑋.

The star order is also called more IFRA (increasing failure rate in average) order in reliability theory, since the average failure of 𝐹 at 𝑥 is 𝑟_𝑋(𝑥)=1𝑥^𝑥₀𝑟_𝑋(𝑢)𝑑𝑢=−ln𝐹(𝑥)𝑥. (2.21) Thus 𝐹≤_⋆𝐺 can be interpreted in terms of average failure rates as 𝑟_𝑋𝐹⁻¹(𝑢)𝑟_𝑌𝐺⁻¹(𝑢) (2.22) is increasing in 𝑢∈(0,1]. Note that 𝑋 has an increasing failure rate if and only if 𝐹 is star-ordered with respect to exponential distribution.

The function 𝐿_𝑋(𝑢)=1𝐸(𝑋)^𝐹−1₀^(𝑢)𝑥𝑑𝐹(𝑥) (2.23) is known as Lorenz curve in the economics literature. It is often used to express inequality in incomes. Based on Lorenz curve, the Lorenz order has been proposed in Economics to compare income inequalities.

Definition 2.16. 𝑋 is said to be smaller than 𝑌 in the Lorenz order, denoted by 𝑋≤_Lorenz𝑌, if 1𝐸(𝑋)^𝐹−1₀^(𝑢)(𝑥)≥1𝑥𝑑𝐹𝐸(𝑌)^𝐺−1₀^(𝑢)𝑥𝑑𝐺(𝑥),∀𝑢∈(0,1]. (2.24)

It is known in the literature (Marshall et al. [34] ) that, 𝑋≤_𝑐𝑌⇒𝑋≤_⋆𝑌⇒𝑋≤_Lorenz𝑌⇒cv(𝑋)≤cv(𝑌), (2.25) where cv(𝑋)=Var(𝑥)/𝐸(𝑋) denote the coefficient of variation of 𝑋.

All the above partial orders are scale invariant. A good discussion of the star order and Lorenz order can be found in Barlow and Proschan [49], Marshall et al. [34], and Kochar [50].

2.4. Dependence Concepts and Orderings

Let (𝑋,𝑌) be a bivariate ramdom vector with joint distribution 𝐹(𝑥,𝑦) and with density function 𝑓(𝑥,𝑦). Lehmann [51] introduced many partial orderings for dependence.

Definition 2.17. (a) 𝑋 and 𝑌 are TP₂dependent or likelihood dependent if their joint density 𝑓(𝑥,𝑦) is totally positive of order 2 in 𝑥 and 𝑦, or more precisely if 𝑓𝑥₁,𝑦₁𝑓𝑥₁,𝑦₂𝑓𝑥₂,𝑦₁𝑓𝑥₂,𝑦₂≥0 (2.26) whenever 𝑥₁<𝑥₂, 𝑦₁<𝑦₂.
(b) 𝑌 is stochastically increasing in 𝑋, denoted by SI(𝑌∣𝑋), if 𝑃(𝑌>𝑦∣𝑋=𝑥) is increasing in 𝑥 for all 𝑦, or equivalently, 𝑃𝑌≤𝑦∣𝑋=𝑥≥𝑃𝑌≤𝑦∣𝑋=𝑥^∗,𝑥≤𝑥^∗. (2.27)
(c) 𝑌 is right tail increasing in 𝑋, denoted by RTI(𝑌∣𝑋), if 𝑃(𝑌>𝑦∣𝑋>𝑥) is increasing in 𝑥 for all 𝑦.
(d) 𝑌 is left tail decreasing in 𝑋, denoted by LTD(𝑌∣𝑋), if 𝑃(𝑌≤𝑦∣𝑋≤𝑥) is decreasing in 𝑥 for all 𝑦.

Finally, random variables 𝑋 and 𝑌 are associated (written 𝐴(𝑋,𝑌)) if cov[Γ(𝑋,𝑌),▵(𝑋,𝑌)]≥0 for all pairs of increasing binary functions Γ and ▵. As shown in Barlow and Proschan [49, page 143] the following chain of implications holds among the above notions of positive dependence: TP₂(𝑋,𝑌)⇒SI(𝑋∣𝑌)⇒RTI(𝑋∣𝑌)⇒𝐴(𝑆,𝑇)⇒Cov(𝑋,𝑌)≥0. (2.28)

These concepts of bivariate dependence can be easily extended to the multivariate case. A function 𝜓∶𝑅^𝑛→[0,∞) is said to be multivariate total positivity of order 2 (denoted by MTP₂) if 𝜓(𝐱)𝜓(𝐲)≤𝜓(𝐱∧𝐲)𝜓(𝐱∨𝐲)forevery𝐱and𝐲inℛ^𝑛, (2.29) where 𝐱∧𝐲=(min(𝑥₁,𝑦₁),…,min(𝑥_𝑛,𝑦_𝑛)) and 𝐱∨𝐲=(max(𝑥₁,𝑦₁),…,max(𝑥_𝑛,𝑦_𝑛)). Random variables 𝑋₁,…,𝑋_𝑛 are said to be MTP₂ dependent if their joint density function is MTP₂. It is shown in Kemperman [52] (see also Block and Ting, [53]) that if the support of a random vector 𝐗=(𝑋₁,…,𝑋_𝑛) is a lattice (i.e., if 𝐱 and 𝐲 are in the support of 𝐗 then so are 𝐱∧𝐲 and 𝐱∨𝐲) then 𝑋 is MTP₂ if and only if, its density function 𝑓 is TP₂ in each pair of its variables when the other (𝑛−2) variables are held fixed. See Karlin and Rinott [54] for more details on properties of MTP₂ functions. Also refer to Joe [55] and Nelsen [56] for a comprehensive discussion.

Observing that when 𝑋 and 𝑌 are continuous, inequality (2.27) can be written as 𝐻_[𝜉𝑞]∘𝐻⁻¹_[𝜉𝑞](𝑢)≤𝑢, (2.30) where 𝜉_𝑝=𝐹⁻¹(𝑝) stands for the 𝑝th quantile of the marginal distribution of 𝑋, and 𝐻_[𝑠] denotes the conditional distribution of 𝑌 given 𝑋=𝑠. Avérous et al. [57] proposed the following definition to measure the relative degree of monotone dependence between two pairs of bivariate random variables (𝑋₁,𝑌₁) and (𝑋₂,𝑌₂).

Definition 2.18. 𝑌₁ is said to be less stochastic increasing in 𝑋₁ than 𝑌₂ is in 𝑋₂, denoted by (𝑌₁∣𝑋₁)⪯_SI(𝑌₂∣𝑋₂), if and only if, for 0≤𝑢≤1, and 0<𝑝≤𝑞<1, 𝐻_2[𝜉2𝑞]∘H⁻¹_2[𝜉2𝑞](𝑢)≤𝐻_1[𝜉1𝑞]∘𝐻⁻¹_1[𝜉1𝑞](𝑢), (2.31) where 𝜉_𝑖𝑝=𝐹_𝑖⁻¹(𝑝) stands for the 𝑝th quantile of the marginal distribution of 𝑋_𝑖, and 𝐻_𝑖[𝑠] denotes the conditional distribution of 𝑌_𝑖 given 𝑋_𝑖=𝑠, for 𝑖=1,2.

Dolati et al. [58] proposed the following weaker dependence order based on RTI criteria, called more RTI order.

Definition 2.19. 𝑌₁ is said to be less right-tail increasing (RTI) in 𝑋₁ than 𝑌₂ is in 𝑋₂, denoted by (𝑌₁∣𝑋₁)⪯_RTI(𝑌₂∣𝑋₂), if and only if, for 0≤𝑢≤1, and 0<𝑝≤𝑞<1, 𝐻^∗_2[𝜉2𝑞]∘𝐻^∗−1_2[𝜉2𝑞](𝑢)≤𝐻^∗_1[𝜉1𝑞]∘𝐻^∗−1_1[𝜉1𝑞](𝑢), (2.32) where 𝜉_𝑖𝑝=𝐹_𝑖⁻¹(𝑝) stands for the 𝑝th quantile of the marginal distribution of 𝑋_𝑖, and 𝐻^∗_𝑖[𝑠] denotes the conditional distribution of 𝑌_𝑖 given 𝑋_𝑖>𝑠, for 𝑖=1,2.

It is easy to see that both more SI order and more RTI order are copula-based orders, and more SI order implies more RTI order which in turn implies more concordance ordering (i.e., the two copulas are ordered). For the concept of copula, please refer to Nelsen [59] for more details.

As observed in Avérous et al. [57] and Genest et al. [60], there is a close connection between the above concepts of more dependence and the notion of dispersive ordering.

2.5. Notions of Majorization and Related Orderings

One of the basic tools in establishing various inequalities in statistics and probability is the notion of majorization.

Let {𝑥₍₁₎≤𝑥₍₂₎≤⋯≤𝑥_(𝑛)} denote the increasing arrangement of the components of the vector 𝐱=(𝑥₁,𝑥₂,…,𝑥_𝑛).

Definition 2.20. The vector 𝐱 is said to majorize the vector 𝐲 (written 𝐱≽^𝑚𝐲) if ∑^𝑗_𝑖=1𝑥_(𝑖)≤∑^𝑗_𝑖=1𝑦_(𝑖) for 𝑗=1,…,𝑛−1 and ∑^𝑛_𝑖=1𝑥_(𝑖)=∑^𝑛_𝑖=1𝑦_(𝑖).

Functions that preserve the majorization ordering are called Schur-convex functions. The vector 𝐱 is said to majorize the vector 𝐲 weakly (written 𝐱≽^𝑤𝐲) if ∑^𝑗_𝑖=1𝑥_(𝑖)≤∑^𝑗_𝑖=1𝑦_(𝑖) for 𝑗=1,…,𝑛. Marshall et al. [34] provides extensive and comprehensive details on the theory of majorization and its applications in statistics.

Bon and Pǎltǎnea [24] have considered a preorder on ℝ^+𝑛, which they call as a𝑝-larger order.

Definition 2.21. A vector 𝐱 in ℝ^+𝑛 is said to be 𝑝-larger than another vector 𝐲 also in ℝ^+𝑛 (written 𝐱≽^𝑝𝐲) if ∏^𝑗_𝑖=1𝑥_(𝑖)≤∏^𝑗_𝑖=1𝑦_(𝑖),𝑗=1,…,𝑛.

Let log(𝐱) denote the vector of logarithms of the coordinates of 𝐱. It is easy to verify that 𝐱≽^𝑝𝐲⟺log(𝐱)≽^𝑤log(𝐲). (2.33) It is known that 𝐱≽^𝑚𝐲⇒(𝑔(𝑥₁),…,𝑔(𝑥_𝑛))≽^𝑤(𝑔(𝑦₁),…,𝑔(𝑦_𝑛)) for all concave functions 𝑔 (cf. [34]). From this and (2.33), it follows that when 𝐱,𝐲∈ℝ^+𝑛𝐱≽^𝑚𝐲⇒𝐱≽^𝑝𝐲. (2.34) The converse is, however, not true. For example, the vectors (0.2,1,5)≽^𝑝(1,2,3) but majorization does not hold between these two vectors.

3. Stochastic Comparisons of Order Statistics in the One-Sample Problem

In this section, we compare order statistics from a single sample according to various stochastic orders.

3.1. Magnitude Orderings between Order Statistics in the One-Sample Problem

Let 𝑋₁,…,𝑋_𝑛 be a set of independent and identically distributed random variables. It is easy to see that 𝑋_𝑖∶𝑛≤_st𝑋_𝑗∶𝑛, for all 1≤𝑖<𝑗≤𝑛. Boland et al. [18] extended this result from usual stochastic order to hazard rate order. Using the definition of likelihood ratio ordering, it is not hard to prove that 𝑋_𝑖∶𝑛≤_lr𝑋_𝑗∶𝑛 for 1≤𝑖<𝑗≤𝑛. Shaked and Shanthikumar [2] considered the problem of comparing order statistics from samples with possibly unequal sample sizes. They showed that if random variables 𝑋_𝑖’s are iid, then 𝑋_𝑛∶𝑛≤_lr𝑋_{𝑛+1∶𝑛+1} and 𝑋_1∶𝑛≥_lr𝑋_1∶𝑛+1. Raqab and Amin [61] strengthened this result and proved that 𝑋_𝑖∶𝑛≤_lr𝑋_𝑗∶𝑚, whenever 𝑖≤𝑗 and 𝑛−𝑖≥𝑚−𝑗.

It is interesting to investigate the above stochastic relations among order statistics when the random variables are independent but not identically distributed. Boland et al. [18] showed that if random variables are independent and 𝑋_𝑘≤_hr𝑋_𝑛+1, 𝑘=1,…,𝑛, then 𝑋_{𝑖−1∶𝑛}≤_hr𝑋_{𝑖∶𝑛+1}, 𝑖=1,…,𝑛+1. They also proved that if 𝑋_𝑖’s are independent and 𝑋_𝑛+1≤_hr𝑋_𝑘, 𝑘=2,…,𝑛, then 𝑋_𝑖∶𝑛≥_hr𝑋_{𝑖∶𝑛+1}, 𝑖=1,…,𝑛. Assuming 𝑋₁≤_lr𝑋₂≤_lr⋯≤_lr𝑋_𝑛, Bapat an Kochar [17] proved that 𝑋_𝑖∶𝑛≤_lr𝑋_j∶𝑛, 𝑖<𝑗.

3.2. Variability Orderings between Order Statistics in the One-Sample Problem

David and Groeneveld [62] proved that if 𝑋_𝑖’s are iid random variables with a common decreasing failure rate (DFR) distribution, then var(𝑋_𝑖∶𝑛)≤var(𝑋_𝑗∶𝑛), for 1≤𝑖<𝑗≤𝑛. Kochar [11] strengthened this result to prove that under the same conditions, 𝑋_𝑖∶𝑛≤_disp𝑋_𝑗∶𝑛, 1≤𝑖<𝑗≤𝑛. Khaledi and Kochar [27] further proved that if 𝑋_𝑖’s are iid with a DFR distribution, then 𝑋_𝑖∶𝑛≤_disp𝑋_𝑗∶𝑚, whenever 𝑖≤𝑗 and 𝑛−𝑖≥𝑚−𝑗.

3.3. Skewness Orderings between Order Statistics in the One-Sample Problem

Arnold and Villasenor [63], Arnold and Nagaraja [64], Wilfling [65], and Kleiber [66], among others, studied Lorenz order relations between order statistics from uniform and other distributions. In particular, Arnold and Villasenor [63] proved the following result on Lorenz ordering between the order statistics from uniform distributions.

Theorem 3.1. Let 𝑈_𝑖∶𝑛 denote the 𝑖th order statistic of a random sample of size 𝑛 from a uniform distribution over (0,1), 𝑖=1,…,𝑛. Then (a)𝑈_{𝑖+1∶𝑛}≤_{𝐿𝑜𝑟𝑒𝑛𝑧}𝑈_𝑖∶𝑛,𝑓𝑜𝑟𝑎𝑙𝑙𝑖≤𝑛−1, (b)𝑈_𝑖∶𝑛≤_{𝐿𝑜𝑟𝑒𝑛𝑧}𝑈_{𝑖∶𝑛+1},𝑓𝑜𝑟𝑎𝑙𝑙𝑖≤𝑛+1, (c)𝑈_{𝑛−𝑖+1∶𝑛+1}≤_{𝐿𝑜𝑟𝑒𝑛𝑧}𝑈_{𝑛−𝑖∶𝑛},𝑓𝑜𝑟𝑎𝑙𝑙𝑖≤𝑛, (d)𝑈_{𝑛+2∶2𝑛+3}≤_{𝐿𝑜𝑟𝑒𝑛𝑧}𝑈_{𝑛+1∶2𝑛+1},𝑓𝑜𝑟𝑎𝑙𝑙𝑛.

The last inequality may be described as “sample medians exhibit less variability as sample size increases.” Arnold and Villasenor [63] wonder about the conditions on 𝑖, 𝑗, 𝑚, and 𝑛 under which 𝑈_𝑗∶𝑚≤_Lorenz𝑈_𝑖∶𝑛 (3.1) holds.

Kochar [67] answered this question in Theorem 3.2 below in which sufficient conditions on the parent distribution 𝐹 are obtained under which 𝑋_𝑗∶𝑚≤_⋆𝑋_𝑖∶𝑛 holds. Many of the previously known results follow from this general result as particular cases as star ordering implies Lorenz ordering.

Theorem 3.2. For 𝑖=1,…,𝑛, let 𝑋_𝑖∶𝑛 denote the 𝑖th order statistic of a random sample of size 𝑛 from a distribution with reverse hazard rate 𝑟_𝐹̃. If 𝑥𝑟_𝐹̃(𝑥)𝑖𝑠𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑖𝑛𝑥, (3.2) then for 𝑖≤𝑗 and 𝑛−𝑖≥𝑚−𝑗, 𝑋_𝑗∶𝑚≤_⋆𝑋_𝑖∶𝑛. (3.3)

Remark 3.3. Arnold and Villasenor [63] mention (3.2) as a sufficient condition for the relation 𝑋_{𝑖+1∶𝑛}≤_Lorenz𝑋_𝑖∶𝑛 (3.4) to hold. We have a more general result.

The following theorem is also proved in in Kochar [67].

Theorem 3.4. For 𝑖=1,…,𝑛, let 𝑋_𝑖∶𝑛 denote the 𝑖th order statistic of a random sample of size 𝑛 from a distribution with its hazard rate 𝑟_𝐹(𝑥)=𝑓(𝑥)/𝐹(𝑥) satisfying the condition, 𝑥𝑟_𝐹(𝑥)𝑖𝑠𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑖𝑛𝑥. (3.5)
Then 𝑋_𝑖∶𝑛≤_⋆𝑋_𝑗∶𝑚𝑓𝑜𝑟𝑖≤𝑗,𝑛−𝑖≥𝑚−𝑗. (3.6)

The above theorems immediately lead to the following result because of the relation between star ordering and Lorenz ordering.

Corollary 3.5. If for 𝑖=1,…,𝑛,  𝑋_𝑖∶𝑛 denotes the 𝑖th order statistic of a random sample of size 𝑛 from a distribution satisfying (a)condition (3.2), then 𝑋_𝑗∶𝑚≤_{𝐿𝑜𝑟𝑒𝑛𝑧}𝑋_𝑖∶𝑛 (3.7) for 𝑖≤𝑗 and 𝑛−𝑖≥𝑚−𝑗. (b)condition (3.5), then 𝑋_𝑗∶𝑚≥_{𝐿𝑜𝑟𝑒𝑛𝑧}𝑋_𝑖∶𝑛 (3.8) for 𝑖≤𝑗 and 𝑛−𝑖≥𝑚−𝑗.

Example 3.6. The condition (3.2) is satisfied by the power function distribution with distribution function, 𝐹_𝑋(𝑥)=[𝑥/𝑐]^𝛾, 𝑜≤𝑥≤𝑐, 𝛾>0. Therefore, the conclusions of Theorem 3.2 and Corollary 3.5(a) hold for this distribution. Arnold and Villasenor [63] also conjectured that for this distribution, 𝑋_{𝑛−𝑗+1∶𝑛+1}≤_Lorenz𝑋_{𝑛−𝑗∶𝑛},forevery1≤𝑗≤𝑛. (3.9) Its proof immediately follows from Theorem 3.2 and Corollary 3.5.

Example 3.7. Let 𝑋 has Pareto distribution with 𝐹(𝑥)=1−(𝑥/𝑐)^−𝛼, 𝑥>𝑐, 𝛼>1. Condition (3.5) is satisfied by this distribution. Hence the conclusions of Theorem 3.4 and Corollary 3.5(b) hold for this distribution.

3.4. Dependence among Order Statistics in the One-Sample Problem

Assuming that 𝑋₁,…,𝑋_𝑛 are independent and identically random variables, Bickel [68] showed that Cov𝑋_𝑖∶𝑛,𝑋_𝑗∶𝑛≥0. (3.10) When the parent distribution has an increasing hazard rate and a decreasing reverse hazard rate, Tukey [69] showed that Cov𝑋_{𝑖′∶𝑛},𝑋_{𝑗′∶𝑛}≤Cov𝑋_𝑖∶𝑛,𝑋_𝑗∶𝑛 (3.11) for either 𝑖=𝑖^′ and 𝑗≤𝑗^′; or 𝑗=𝑗^′ and 𝑖^′≤𝑖.

Kim and David [70] proved that if both the hazard and the reverse hazard rates of the 𝑋_𝑖’s are increasing, then inequality (3.11) remains valid when 𝑖=𝑖^′ and 𝑗≤𝑗^′; However, the inequality (3.11) is reversed when 𝑗=𝑗^′ and 𝑖^′≤𝑖.

Avérous et al. [57] used the more SI concept to study the relative degree of dependence between two pairs of random variables. They proved the following result.

Theorem 3.8. Let 𝑋_1∶𝑛≤⋯𝑋_𝑛∶𝑛 and 𝑋_1∶𝑛′≤⋯𝑋_{𝑛′∶𝑛′} be the order statistics associated with two independent random samples of sizes 𝑛 and 𝑛^′ from the same continuous distribution. Then, for 1≤𝑖≤𝑗≤𝑛, 1≤𝑖^′≤𝑗^′≤𝑛^′, and 𝑖^′≤𝑖, 𝑗−𝑖≤𝑗^′−𝑖^′, 𝑛−𝑖≤𝑛^′−𝑖^′, 𝑛^′−𝑗^′≤𝑛−𝑗, one hold that 𝑋_{𝑗′∶𝑛′}∣𝑋_{𝑖′∶𝑛′}⪯_𝑆𝐼𝑋_𝑗∶𝑛∣𝑋_𝑖∶𝑛. (3.12)

As a direct consequence, we have the following result.

Corollary 3.9. Let X_1∶𝑛≤⋯𝑋_𝑛∶𝑛 be order statistics from the same continuous distribution. Then, (1)(𝑋_𝑘∶𝑛∣𝑋_𝑖∶𝑛)⪯_𝑆𝐼(𝑋_𝑗∶𝑛∣𝑋_𝑖∶𝑛)for 1≤𝑖<𝑗<𝑘≤𝑛; (2)(𝑋_𝑗∶𝑛∣𝑋_𝑖∶𝑛)⪯_𝑆𝐼(𝑋_{𝑗+1∶𝑛+1}∣𝑋_{𝑖+1∶𝑛+1})for 1≤𝑖<𝑗≤𝑛; (3)(𝑋_𝑛+1∶n+1∣𝑋_1∶𝑛+1)⪯_𝑆I(𝑋_𝑛∶𝑛∣𝑋_1∶𝑛)for 𝑛≥2.

It can be seen from the above result that the dependence between the components of a pair (𝑋_𝑗∶𝑛,𝑋_𝑖∶𝑛) of order statistics decreases in the sense of SI ordering as 𝑖 and 𝑗 get further apart.

Remark 3.10. Since the copula of a pair of order statistics of a random sample is independent of the parent distribution and since the concept of more SI is copula based, it follows that in Theorem 3.8 and Corollary 3.9, the two samples could be from different distributions.

As explained in Avérous et al. [57], the following result follows immediately from Theorem 3.8.

Corollary 3.11. Under the assumptions of Theorem 3.8, one has 𝜅𝑋_{𝑗′∶𝑛′},𝑋_{𝑖′∶𝑛′}≤𝜅𝑋_𝑗∶𝑛,𝑋_𝑖∶𝑛, (3.13) where 𝜅(𝑋,𝑌) represents Spearman’s rho, Kendall’s tau, Gini’s coefficient, or indeed any other copula-based measure of concordance satisfying the axioms of Scarsini [71].

3.4.1. Kendall’s 𝜏 and Spearman’s 𝜌 for Order Statistics

Two popular nonparametric measures of association for bivariate random variables are Kendall’s 𝜏 and Spearman’s 𝜌, which measure different aspects of the dependence structure. In terms of dependence properties, Spearman’s 𝜌 is a measure of average quadrant dependence, while Kendall’s 𝜏 is a measure of average likelihood ratio dependence (cf. [59, 72]).

Avérous et al. [57] made an important observation that in the case of a random sample from a continuous distribution with cdf 𝐹, the copula of a pair of order statistics is independent of the parent distribution 𝐹. As a result the value of any copula-based measure of dependence like Kendall’s tau or Spearman’s coefficient for any pair of order statistics (𝑋_𝑖∶𝑛,𝑋_𝑗∶𝑛) will be the same for all continuous distributions Schmitz [73] derived the following formulas: 𝜏𝑋_1∶𝑛,𝑋_𝑛∶𝑛=1,𝜌𝑋2𝑛−1_1∶𝑛,𝑋_𝑛∶𝑛=3−12𝑛_𝑛2𝑛^𝑛_𝑘=0(−1)^𝑘()2𝑛−𝑘2𝑛𝑛+𝑘+12𝑛!³()!()3𝑛−1^𝑛. (3.14) X. Li and Z. Li [74] proved a conjecture in Schmitz [73] that 𝜌𝑋_1∶𝑛,𝑋_𝑛∶𝑛𝜏𝑋_1∶𝑛,𝑋_𝑛∶𝑛→32,𝑛→∞. (3.15)

Avérous et al. [57] used a combinatorial approach to prove the following formula of Kendall’s 𝜏 for any pair of order statistics from the same continuous distribution: 𝜏𝑋_𝑖∶𝑛,𝑋_𝑗∶𝑛2()=1−𝑛−12𝑛−1𝑛−2𝑖−1𝑛−𝑖−1𝑗−𝑖−1^{𝑛−𝑗𝑠=0𝑖−1𝑟=0}(^𝑛_𝑟)(_𝑠^𝑛−𝑟)_{2𝑛−2𝑛−𝑗+𝑠,𝑟+𝑖−1}. (3.16)

Subsequently, Chen [75] developed three new formulas to compute 𝜌(𝑋_1∶𝑛,𝑋_𝑛∶𝑛): 𝜌𝑋_1∶𝑛,𝑋_𝑛∶𝑛=31−4𝑎_𝑛, (3.17) where 𝑎_𝑛 can be computed by any one of the following formulas. (1)Formula 1: 𝑎_𝑛=𝑛(𝑛−1)¹₀^𝑡₀(1−𝑠)^𝑛𝑡^𝑛(𝑡−𝑠)^𝑛−2𝑑𝑠𝑑𝑡; (3.18)(2)Formula 2: 𝑎_𝑛=^𝑛_𝑘=0(−1)^𝑘(^𝑛_𝑘)_{𝑘𝑛+𝑘}𝑛+𝑘;2𝑛+𝑘 (3.19)(3)Formula 3: 𝑎_𝑛=𝑛(𝑛−1)(3𝑛)!^𝑛_𝑗=0𝑛!𝑗!^𝑛_𝑘=0𝑛!𝑘!𝑛−2+𝑗+𝑘!. (3.20)He further showed the following compound inequality: 3(2𝑛−1)14𝑛²+15𝑛+356𝑛³+86𝑛²≤𝜌𝑋+43𝑛+7^1∶𝑛,𝑋_𝑛∶𝑛𝜏𝑋_1∶𝑛,𝑋_𝑛∶𝑛≤3(2𝑛−1)14𝑛²−13𝑛+256𝑛³−82𝑛²,+39𝑛−6 (3.21) from which, (3.15) follows immediately.

Recently, Navarro and Balakrishnan [76] have also studied this problem and have obtained alternate expressions for computing these measures of dependence.

3.4.2. Dependence Orderings among Order Statistics When the Observations Are Not Identically Distributed

The proportional hazard rates (PHRs) model is a popular model in survival analysis, reliability theory, and other branches of statistics. Let 𝑋_𝑖 denote the lifetime of the 𝑖𝑡ℎ component of a reliability system with survival function 𝐹_𝑖(𝑡), 𝑖=1,…,𝑛. Then they have proportional hazard rates if there exist constants 𝜆₁,…,𝜆_𝑛 and a (cumulative hazard) function 𝑅(𝑡)≥0 such that 𝐹_𝑖(𝑡)=𝑒^{−𝜆𝑖𝑅(𝑡)} for 𝑖=1,…,𝑛. Clearly then the hazard rate of 𝑋_𝑖 is 𝑟_𝑖(𝑡)=𝜆_𝑖𝑅^′(𝑡) (assuming it exists). An example of such a situation is when the components have independent exponential lifetimes with respective hazard rates 𝜆₁,…,𝜆_𝑛.

Boland et al. [10] studied in detail the dependence properties of order statistics when the observations are not necessarily identically distributed. They proved the following dependence result for the PHR model.

Theorem 3.12. Let 𝑋₁,…,𝑋_𝑛 be independent random variables with differentiable densities and follow the PHR model on an interval. Then for 1<𝑖≤𝑛, 𝑋_𝑖∶𝑛 is SI in 𝑋_1∶𝑛.

They also gave a counterexample to illustrate that, in general, 𝑋_𝑖∶𝑛 is not SI in 𝑋_1∶𝑛. However, they showed that for 1≤𝑖<𝑗≤𝑛, 𝑋_𝑗∶𝑛 is RTI in 𝑋_𝑖∶𝑛.

Theorem 3.13. Let 𝑋₁,…,𝑋_𝑛 be independent random variables. Then for any 𝑖≤𝑗, RTI(𝑋_𝑗∶𝑛∣𝑋_𝑖∶𝑛) and LTD(𝑋_𝑖∶𝑛∣𝑋_𝑗∶𝑛).

They showed with the help of a counter example that, in general, the relation RTI(𝑋_𝑖∶𝑛∣𝑋_𝑗∶𝑛) may not hold for 𝑖<𝑗.

This topic has been further developed by Hu and Xie [77], where they exploited the negative dependence of occupancy numbers in the balls and bins experiment. They proved the following result.

Theorem 3.14. Let 𝑋₁,…,𝑋_𝑛 be independent random variables. For 1≤𝑖≤𝑗₁≤𝑗₂≤⋯≤𝑗_𝑟≤𝑛, and (𝑥₁,…,𝑥_𝑛)∈ℝ^𝑛,(1)if 𝐴_{𝑖,𝑛,𝑦}={𝑋_𝑖∶𝑛>𝑦}, then 𝑃𝑋_𝑗1>𝑥₁,…,𝑋_𝑗𝑟>𝑥_𝑟∣𝐴_{𝑖,𝑛,𝑦} (3.22) is increasing in 𝑦; (2)if event 𝐴_{𝑖,𝑛,𝑦} is either {𝑋_𝑖∶𝑛>𝑦} or {𝑋_𝑖∶𝑛≤𝑦}, then the conditional probability in (3.22) is decreasing in 𝑖 for each 𝑦; and (3)if each 𝑋_𝑘 has a continuous distribution function, and if 𝐴_{𝑖,𝑛,𝑦} is either {𝑋_𝑖∶𝑛=𝑦} or {𝑋_{𝑖−1∶𝑛}<𝑦<𝑋_𝑖∶𝑛}, then (3.22) is decreasing in 𝑖 for each 𝑦, where 𝑋_0∶𝑛≡−∞.

Dubhashi and Häggström [78] further extended the above result to multivariate stochastic comparisons.

Theorem 3.15. Let 𝑋₁,…,𝑋_𝑛 be independent random variables. Then 𝑋_𝑖∶𝑛,…,𝑋_𝑛∶𝑛∣𝑋_𝑖∶𝑛>𝑦⪯_𝑠𝑡𝑋_𝑖∶𝑛,…,𝑋_𝑛∶𝑛∣𝑋_𝑖∶𝑛>𝑦^′,𝑦≤𝑦^′. (3.23)

Subsequently, Theorem 3.14 was further extended by Hu and Chen [79] as follows.

Theorem 3.16. Let 𝑋₁,…,𝑋_𝑛 be independent random variables. (1)If 𝑗−𝑖≥max{𝑛−𝑚,0}, then 𝑃𝑋_𝑗∶𝑛>𝑥₁,𝑋_{𝑗+1∶𝑛}>𝑥₂,…,𝑋_𝑛∶𝑛>𝑥_{𝑛−𝑗+1}∣𝑋_𝑖∶𝑚>𝑦 (3.24) is increasing in 𝑦 for all (𝑥₁,…,𝑥_{𝑛−𝑗+1})∈ℝ^{𝑛−𝑗+1}; (2)If 𝑗−𝑖≤min{𝑛−𝑚,0}, then 𝑃𝑋_1∶𝑛≤𝑥₁,𝑋_2∶𝑛≤𝑥₂,…,𝑋_𝑗∶𝑛≤𝑥_{𝑛−𝑗+1}∣𝑋_𝑖∶𝑚≤𝑦 (3.25) is decreasing in 𝑦 for all (𝑥₁,…,𝑥_𝑗)∈ℝ^𝑗.

Recently, Zhuang et al. [80] discussed the dependence among order statistics in the sense of multivariate stochastic comparisons, which extends the results in Hu and Chen [79] and Dubhashi and Häggström [78].

Theorem 3.17. Let 𝑋₁,…,𝑋_𝑛 be independent random variables. (1)If 𝑗−𝑖≥max{𝑛−𝑚,0}, then 𝑋_𝑗∶𝑛,…,𝑋_𝑛∶𝑛∣𝑋_𝑖∶𝑚>𝑦⪯_𝑠𝑡𝑋_𝑗∶𝑛,…,𝑋_𝑛∶𝑛∣𝑋_𝑖∶𝑚>𝑦^′,𝑦≤𝑦^′. (3.26)(2)If 𝑗−𝑖≤min{𝑛−𝑚,0}, then 𝑋_1∶𝑛,…,𝑋_𝑗∶𝑛∣𝑋_𝑖∶𝑚≤𝑦⪯_𝑠𝑡𝑋_1∶𝑛,…,𝑋_𝑗∶𝑛∣𝑋_𝑖∶𝑛≤𝑦^′,𝑦≤𝑦^′. (3.27)

4. Stochastic Comparisons of Order Statistics in the Two-Sample Problem

Let 𝑋₁,…,𝑋_𝑛 be a set of independent random variables and 𝑌₁,…,𝑌_𝑛 be another set of independent random variables. Ross [31] proved that if 𝑋_𝑖≤_st𝑌_𝑖, 𝑖=1,…,𝑛, then (𝑋₁,…,𝑋_𝑛)≤_st(𝑌₁,…,𝑌_𝑛). A consequence of this result is that 𝑋_𝑖∶𝑛≤_st𝑌_𝑖∶𝑛for𝑖=1,…,𝑛. Lynch et al. [81] generalized this result from stochastic ordering to hazard rate ordering though under a stronger condition. They proved that if 𝑋_𝑖≤_hr𝑌_𝑗, 𝑖,𝑗∈{1,…,𝑛}, then 𝑋_𝑖∶𝑛≤_hr𝑌_𝑖∶𝑛, 𝑖=1,…,𝑛. A similar result for likelihood ratio ordering has been proved by Chan et al. [82]. They proved that if 𝑋_𝑖≤_lr𝑌_𝑗,  𝑖,𝑗∈{1,…,𝑛}, then 𝑋_𝑖∶𝑛≤_lr𝑌_𝑖∶𝑛, 𝑖=1,…,𝑛. Lillo et al. [83] in the following theorem strengthened this result to the case when the number of 𝑋_𝑖’s and 𝑌_𝑖’s is not necessarily equal.

Theorem 4.1. Let 𝑋₁,…,𝑋_𝑛 be independent random variables and let 𝑌₁,…,𝑌_𝑚 be another set of independent random variables, all having absolutely continuous distributions. Then 𝑋_𝑖≤_𝑙𝑟𝑌_𝑗 for all 𝑖,𝑗 implies 𝑋_𝑖∶𝑛≤_𝑙𝑟𝑌_𝑗∶𝑚 whenever 𝑖≤𝑗 and 𝑛−𝑖≥𝑚−𝑗.

In the next theorem we establish dispersive ordering between order statistics when the random samples are drawn from different distributions.

Theorem 4.2. Let 𝑋₁,…,𝑋_𝑛 be a random sample of size 𝑛 from a continuous distribution 𝐹 and let 𝑌₁…,𝑌_𝑚 be a random sample of size 𝑚 from another continuous distribution 𝐺. If either 𝐹 or 𝐺 is 𝐷𝐹𝑅, then 𝑋≤_{𝑑𝑖𝑠𝑝}𝑌⇒𝑋_𝑖∶𝑛≤_{𝑑𝑖𝑠𝑝}𝑌_𝑗∶𝑚𝑓𝑜𝑟𝑖≤𝑗,𝑛−𝑖≥𝑚−𝑗. (4.1)

Proof. Let 𝐹 be a DFR distribution. The proof for the case when 𝐺 is DFR is similar. Khaledi and Kochar [27] proved that 𝑋_𝑖∶𝑛≤_disp𝑋_𝑗∶𝑚 for 𝑖≤𝑗 and 𝑛−𝑖≥𝑚−𝑗. Bartoszewicz [42] proved that if 𝑋≤_disp𝑌 then 𝑋_𝑗∶𝑚≤_disp𝑌_𝑗∶𝑚. Combining these we get the required result.

Since the property 𝑋≤_hr𝑌 together with the condition that either 𝐹 or 𝐺 is DFR implies that 𝑋≤_disp𝑌, we get the following result from the above theorem.

Corollary 4.3. Let 𝑋₁,…,𝑋_𝑛 be a random sample of size 𝑛 from a continuous distribution 𝐹 and 𝑌₁…,𝑌_𝑚 be a random sample of size 𝑚 from another continuous distribution 𝐺. If either 𝐹 or 𝐺 is 𝐷𝐹𝑅, then 𝑋≤_ℎ𝑟𝑌⇒𝑋_𝑖∶𝑛≤_{𝑑𝑖𝑠𝑝}𝑌_𝑗∶𝑚𝑓𝑜𝑟𝑖≤𝑗,𝑛−𝑖≥𝑚−𝑗. (4.2)

4.1. Stochastic Comparisons of Order Statistics When Observations Are Heterogeneous

Many researchers have investigated the effect on the survival function, the hazard rate function and other characteristics of the time to failure of this system when we switch the vector (𝜆₁,…,𝜆_𝑛) to another vector say (𝜆^∗₁,…,𝜆^∗_𝑛) in the PHR model. Pledger and Proschan [15], for the first time, studied this problem and proved the following interesting result among many other results.

Theorem 4.4. Let (𝑋₁,…,X_𝑛) and (𝑋^∗₁,…,𝑋^∗_𝑛) be two random vectors of independent lifetimes with proportional hazards with 𝜆₁,…,𝜆_𝑛 and 𝜆^∗₁,…𝜆^∗_𝑛 as the constants of proportionality. Then for 𝑖=1,…,𝑛.𝜆₁,𝜆₂,…,𝜆≽_𝑛^𝑚𝜆^∗₁,𝜆^∗₂,…,𝜆^∗_𝑛⇒𝑋_𝑘∶𝑛≥_𝑠𝑡𝑋^∗_𝑘∶𝑛for𝑘=1,…,𝑛. (4.3)

Proschan and Sethuraman [16] strengthened this result from component wise stochastic ordering to multivariate stochastic ordering. That is, under the assumptions of Theorem 4.4, they proved that 𝑋_1∶𝑛,…,𝑋_𝑛∶𝑛≥_st𝑋^∗_1∶𝑛,…,𝑋^∗_𝑛∶𝑛. (4.4)

These results show that if the parent observations have proportional hazard rates, then their order statistics are stochastically larger when the proportionality parameters are dispersed in the sense of majorization.

Boland et al. [18] showed with the help of the following counterexample that (4.3) can not be strengthened from stochastic ordering to hazard rate ordering for 𝑘=𝑛 when 𝑛≥3.

Example 4.5. Let (𝑋₁,𝑋₂,𝑋₃) be independent exponential random vector with hazard rate vector (𝜆₁,𝜆₂,𝜆₃)=(0.1,1,9) and (𝑋^∗₁,𝑋^∗₂,𝑋^∗₃) be independent exponential random vector with hazard rate vector (𝜆^∗₁,𝜆^∗₂,𝜆^∗₃)=(0.1,4,6). It is easily seen that 𝜆₁,𝜆₂,𝜆₃^𝑚≽𝜆^∗₁,𝜆^∗₂,𝜆^∗₃. (4.5) However, 𝑟_𝑋3∶3(2)≈0.113>𝑟_𝑋^∗_3∶3(2)≈0.100. (4.6) Hence, 𝑋_3∶3≱_hr𝑋^∗_3∶3. (4.7)

Khaledi and Kochar [28] proved the following result for the largest order statistics under p-larger ordering.

Theorem 4.6. Let (𝑋₁,…,𝑋_𝑛) and (𝑋^∗₁,…,𝑋^∗_𝑛) be two random vectors of independent lifetimes with proportional hazards with 𝜆₁,…,𝜆_𝑛 and 𝜆^∗₁,…𝜆^∗_𝑛 as the constants of proportionality. Then 𝜆₁,𝜆₂,…,𝜆≽_𝑛^𝑝𝜆^∗₁,𝜆^∗₂,…,𝜆^∗_𝑛⇒𝑋_𝑛∶𝑛≥_𝑠𝑡𝑋^∗_𝑛∶𝑛. (4.8)

The following example due to Khaledi and Kochar [28] shows that Theorem 4.6 may not hold for other order statistics.

Example 4.7. Let (𝑋₁,𝑋₂,𝑋₃) be independent exponential random vector with hazard rate vector (𝜆₁,𝜆₂,𝜆₃)=(0.1,1,7.9) and (𝑋^∗₁,𝑋^∗₂,𝑋^∗₃) be independent exponential random vector with hazard rate vector (𝜆^∗₁,𝜆^∗₂,𝜆^∗₃)=(1,2,5). It is easily seen that 𝜆₁,𝜆₂,𝜆≽₃^𝑝𝜆^∗₁,𝜆^∗₂,𝜆^∗₃. (4.9) However, 𝑟_𝑋1∶3(𝑥)=9>𝑟_𝑋^∗_1∶3(𝑥)=8, (4.10) which implies 𝑋_1∶3≤_st𝑋^∗_1∶3. (4.11)

Khaledi and Kochar [28] also showed that in, general, 𝜆₁,𝜆₂,…,𝜆≽_𝑛^𝑝𝜆^∗₁,𝜆^∗₂,…,𝜆⇒^∗_𝑛¨^¨𝑋^∗_𝑛∶𝑛≤_rh𝑋_𝑛∶𝑛. (4.12)

Torrado and Veerman [84] formulated a simple criteria in terms of the 𝜆_𝑖’s to see whether two 𝑘-out-of-𝑛 systems are stochastically ordered.

4.1.1. Comparison of Order Statistics from Heterogenous and Homogeneous Samples

Let 𝑋₁,…,𝑋_𝑛 be continuous independent random variables such that the c.d.f. of 𝑋_𝑖 is 𝐹_𝑖, 𝑖=1,…,𝑛, where 𝐹_𝑖’s are not necessarily all equal. Let 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from a distribution with c.d.f 𝐺=(1/𝑛)∑^𝑛_𝑖=1𝐹_𝑖. Sen [7] proved a very general result that whereas for the first order statistic, 𝑋_1∶𝑛≤_st𝑌_1∶𝑛, the inequality is reversed for the largest order statistic, that is, 𝑋_𝑛∶𝑛≥_st𝑌_𝑛∶𝑛.

Dykstra et al. [22] studied the above problem for the exponential distributions. They proved the following stronger result on hazard rate ordering for largest order statistics.

Theorem 4.8. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛; and let 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆. Then for 𝜆=∑𝜆=^𝑛𝑖=1𝜆_𝑖𝑛,𝑌_𝑛∶𝑛≤_ℎ𝑟𝑋_𝑛∶𝑛 (4.13)(4.14) as well as 𝑌_𝑛∶𝑛≤_{𝑑𝑖𝑠𝑝}𝑋_𝑛∶𝑛. (4.15)

Khaledi and Kochar [28] proved that (4.14) and (4.15) continue to hold with 𝜆=𝜆=(∏^𝑛_𝑖=1𝜆_𝑖)^1/𝑛, the geometric mean of the 𝜆^′s. This result gives better bounds on the hazard rate and measures of variability of a parallel system with heterogeneous exponential components.

Khaledi and Kochar [28] also extended the results (4.14) and (4.15) from the exponential case to the PHR model as stated in the next theorem.

Theorem 4.9. Let 𝑋₁,…,𝑋_𝑛 be independent random variables with 𝑋_𝑖 having survival function 𝐹^𝜆𝑖, 𝑖=1,…,𝑛. Let 𝑌₁,…,𝑌_𝑛 be a random sample with common population survival distribution 𝐹𝜆, where 𝜆=(∏^𝑛_𝑖=1𝜆_𝑖)^1/𝑛, then (i)𝑌_𝑛∶𝑛≤_ℎ𝑟𝑋_𝑛∶𝑛; (ii)𝑌_𝑛∶𝑛≤_{𝑑𝑖𝑠𝑝}𝑋_𝑛∶𝑛 if 𝐹 has decreasing failure rate (DFR).

These results give nice bounds on the hazard rate and the variance of a parallel system with components which are independent following the PHR model in terms of the case when they are independent and identically distributed.

Kochar and Xu [85] further strengthened the conclusion (4.14) of Theorem 4.8 of Dykstra et al. [22] from hazard rate ordering to likelihood ratio order for the PHR model, which includes exponential as a special case.

Theorem 4.10. Let 𝑋₁,…,𝑋_𝑛 be independent random variables with 𝑋_𝑖 having survival function 𝐹^𝜆𝑖, 𝑖=1,…,𝑛. Let 𝑌₁,…,𝑌_𝑛 be a random sample with common population survival distribution 𝐹^𝜆, where 𝜆=∑^𝑛_𝑖=1𝜆_𝑖/𝑛, then 𝑌_𝑛∶𝑛≤_𝑙𝑟𝑋_𝑛∶𝑛. (4.16)

The following example due to Kochar and Xu [85] shows that the result of Khaledi and Kochar [27] above cannot be strengthened from the hazard rate order to the likelihood ratio order under p-larger ordering.

Example 4.11. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛, and 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆=(∏^𝑛_𝑖=1𝜆_𝑖)^1/𝑛. Then, the reversed hazard rate of 𝑋_𝑛∶𝑛 is 𝑓_𝑛∶𝑛(𝑥)𝐹_𝑛∶𝑛(𝑥)=^𝑛_𝑖=1𝜆_𝑖𝑒^{−𝜆𝑖𝑥}1−𝑒^{−𝜆𝑖𝑥}. (4.17) Similarly, the reversed hazard rate of 𝑌_𝑛∶𝑛 is 𝑔_𝑛∶𝑛(𝑥)𝐺_𝑛∶𝑛(𝑥)𝜆𝑒=𝑛⁻𝜆_𝑥1−𝑒⁻𝜆_𝑥. (4.18) Let 𝜆₁=𝜆₂=1, 𝜆₃=3, and 𝑛=3, then, 𝑓_𝑛∶𝑛(1)𝐹_𝑛∶𝑛(1)𝑔≈1.321≤1.339≈^𝑛∶𝑛(1)𝐺_𝑛∶𝑛(1). (4.19) Thus, 𝑋_𝑛∶𝑛≱_rh𝑌_𝑛∶𝑛, (4.20) which implies that 𝑋_𝑛∶𝑛≱_lr𝑌_𝑛∶𝑛. (4.21)

Bon and Pǎltǎnea [86] proved the following result on stochastic ordering between 𝑘th order statistics from heterogeneous and homogeneous samples from exponential distributions.

Theorem 4.12. Let 𝑋₁,…,𝑋_𝑝 be independent exponential random variables with hazard rates 𝜆_𝑖, 𝑖=1,…,𝑛. Let 𝑌₁,…,𝑌_𝑛 be independent exponential random variables with a common hazard rate 𝜆. Then, 𝜆≥𝜆⟺𝑋_𝑘∶𝑛≥_𝑠𝑡𝑌_𝑘∶𝑛, (4.22) where 𝑛𝑘𝜆=⁻¹_{1≤𝑖1<⋯<𝑖𝑘≤𝑛}𝜆_𝑖1⋯𝜆_𝑖𝑘^1/𝑘, (4.23) a result which also follows from Khaledi and Kochar [27] for the case when 𝑘=𝑛.

Kochar and Xu [87] proved the following results on excess wealth ordering between largest order statistics from heterogeneous and homogeneous samples.

Theorem 4.13. 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. Then, 𝜆^∗≤𝜆⟺𝑌_𝑛∶𝑛≤_𝑅𝑆𝑋_𝑛∶𝑛, (4.24) where 𝜆^∗=^𝑛_𝑖=11𝑖^𝑛_𝑘=1(−1)^𝑘+1_{1≤𝑖1≤⋯≤𝑖𝑘≤𝑛}1∑^𝑘_𝑗=1𝜆_𝑖𝑗⁻¹. (4.25)

Theorem 4.14. Let 𝑋₁,…,𝑋_𝑛 be independent random variables with 𝑋_𝑖 having survival function 𝐹^𝜆𝑖, 𝑖=1,…,𝑛, and let 𝑌₁,…,𝑌_𝑛 be another random sample with the common survival function 𝐹^𝜆. If 𝐹 is DFR, then 𝜆^∗≤𝜆⇒𝑌_𝑛∶𝑛≤_𝑅𝑆𝑋_𝑛∶𝑛, (4.26) where 𝜆^∗ is given in (4.24).

As a consequence, we have the following result.

Corollary 4.15. Let 𝑋₁,…,𝑋_𝑛 be independent random variables with 𝑋_𝑖 having survival function 𝐹^𝜆𝑖, 𝑖=1,…,𝑛, and let 𝑌₁,…,𝑌_𝑛 be another random sample with the common survival function 𝐹^𝜆. If 𝐹 is DFR, then 𝜆^∗≤𝜆⇒Var𝑌_𝑛∶𝑛≤Var𝑋_𝑛∶𝑛, (4.27) where 𝜆^∗ is given in (4.24).

Remark 4.16. In the special case of exponential distribution, 𝐹(𝑥)=𝑒^−𝑥, Khaledi and Kochar [28] gave a lower bound on the variance of 𝑋_𝑛∶𝑛 in terms of the geometric mean, Var𝑋_𝑛∶𝑛≥1𝜆^𝑛_𝑖=11𝑖². (4.28) Since the right spread order implies the variance being ordered (cf. Shaked and Shanthikumar [2]), Corollary 4.15 provides a better lower bound for the variance, Var𝑋_𝑛∶𝑛≥1𝜆^∗𝑛_𝑖=11𝑖², (4.29) as 𝜆^∗≤𝜆.

4.1.2. Skewness Comparisons of Order Statistics from Two Samples

Kochar and Xu [87] for the first time studied the problem of comparing order statistics according to shape orderings like convex ordering and star ordering. They proved that a parallel system with heterogenous exponential component lifetimes is more skewed (according to convex transform order) than the one with i.i.d. exponential components. This is stated in the following theorem.

Theorem 4.17. 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. Then, 𝑌_𝑛∶𝑛≤_𝑐𝑋_𝑛∶𝑛. (4.30)

Remark 4.18. Theorem 4.17 means that a parallel system with homogeneous exponential components ages faster than than a system with heterogenous exponential components in the sense of “more IFR” property. Note that a parallel system with homogeneous exponential components is IFR (Barlow and Proschan [49]). However, a parallel system with heterogenous exponential components may not be IFR.

Remark 4.19. It is interesting to note that, unlike the magnitude and variability orders, no restriction on the parameters is needed for Theorem 4.17 to hold as the convex transform order is scale invariant. Intuitively, due to the heterogeneity, the largest order statistic from a heterogenous sample will be more skewed than that from a homogeneous sample. Theorem 4.17 confirms this fact for exponential samples.

The following result, which is of independent interest in economics, is a direct consequence of Theorem 4.17

Corollary 4.20. 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. Then, 𝑌_𝑛∶𝑛≤_{𝐿𝑜𝑟𝑒𝑛𝑧}𝑋_𝑛∶𝑛. (4.31)

From Barlow and Proschan [49, page 60], it follows that, 𝐸𝑌_𝑛∶𝑛=1𝜆^𝑛_𝑖=11𝑖,Var𝑌_𝑛∶𝑛=1𝜆^2𝑛_𝑖=11𝑖². (4.32) Using (2.25), one get the following lower bound on the coefficient of variation of 𝑋_𝑛∶𝑛, cv𝑋_𝑛∶𝑛≥∑^𝑛_𝑖=11/𝑖²∑^𝑛_𝑖=1().1/𝑖 (4.33)

Kochar and Xu [88] extended Theorem 4.17 to the PHR family as follows.

Theorem 4.21. Let 𝑋₁,…,𝑋_𝑛 be independent random variables with 𝑋_𝑖 having survival function 𝐹^𝜆𝑖, 𝑖=1,…,𝑛, and let 𝑌₁,…,𝑌_𝑛 be a random sample from a distribution with the common survival distribution 𝐹^𝜆 where 𝜆≥𝜆=^𝑛∏^𝑛_𝑖=1𝜆_𝑖, the geometric mean of 𝜆_𝑖’s. If 𝑅(𝑥)𝑥𝑟(𝑥)𝑖𝑠𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑖𝑛𝑥≥0, (4.34) then 𝑋_𝑛∶𝑛≥_⋆𝑌_𝑛∶𝑛, (4.35) where 𝑅(𝑥)=−log𝐹(𝑥) is the cumulative hazard rate function, and 𝑟(𝑥)=𝑓(𝑥)/𝐹(𝑥) is the hazard rate function of 𝐹.

It can be seen that distributions like Weibull, Lomax, and Pareto satisfy the condition (4.34). Kochar and Xu [88] proved the following result on star ordering for 𝑘th order statistics.

Theorem 4.22. Let 𝑋₁,…,𝑋_𝑝 be i.i.d. random variables with the common survival distribution 𝐹^𝜆1, and let 𝑋_𝑝+1,…,𝑋_𝑛 be another set of i.i.d. random variables with common survival distribution 𝐹^𝜆2, and let 𝑌₁,…,𝑌_𝑛 be a random sample from a distribution with common survival distribution 𝐹^𝜆 where 𝜆≥𝜆. If 𝑅(𝑥)𝑥𝑟(𝑥)𝑖𝑠𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑖𝑛𝑥≥0, (4.36) then 𝑋_𝑘∶𝑛≥_⋆𝑌_𝑘∶𝑛, (4.37) and hence, 𝑋_𝑘∶𝑛≥_{𝐿𝑜𝑟𝑒𝑛𝑧}𝑌_𝑘∶𝑛,𝑘=1,…,𝑛, (4.38) where 𝑛𝑘𝜆=⁻¹_𝑙∈𝕃𝑝𝑙𝜆𝑛−𝑝𝑘−𝑙^𝑙1𝜆₂^{𝑘−𝑙1/𝑘} (4.39) and 𝕃={𝑙∶max{𝑘−𝑛+𝑝,0}≤𝑙≤min{𝑝,𝑘}}.

4.1.3. Stochastic Comparisons of Second Order Statistics

In the next theorem, we give a collection of results on various types of stochastic orders for the second order statistics.

Theorem 4.23. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with parameters 𝜆₁,…,𝜆_𝑛, respectively; and let 𝑌₁,…,𝑌_𝑛 be independent random variables with common hazard rate 𝜆. (a)(Pǎltǎnea [89]) 𝑌_2∶𝑛≤_ℎ𝑟𝑋_2∶𝑛⟺𝜆≥𝜆=𝑛2^{−11≤𝑖<𝑗≤𝑛}𝜆_𝑖𝜆_𝑗. (4.40)(b) (Zhao and Balakrishnan [90]) 𝑌_2∶𝑛≤_disp𝑋_2∶𝑛⟺𝜆≥𝜆. (4.41)(c) (Zhao et al. [91]) 𝑌_2∶𝑛≤_𝑙𝑟𝑋_2∶𝑛⟺𝜆≥𝜆^∗∗=12𝑛−12Λ₁+Λ₃−Λ₁Λ₂Λ²₁−Λ₂, (4.42) where Λ_𝑘=^𝑘_𝑖=1𝜆^𝑘_𝑖,𝑘=1,2,3. (4.43)(d) (Zhao and Balakrishnan [92]) 𝑌_2∶𝑛≤_𝑚𝑟𝑙𝑋_2∶𝑛⟺𝜆≥𝜆^∗∗, (4.44) where 𝜆^∗∗=2𝑛−1𝑛(𝑛−1)∑^𝑛_𝑖=11/Λ₁−𝜆_𝑖−(𝑛−1)/Λ₁. (4.45)

4.2. The Scale Family

Independent random variables 𝑋_𝜆1,…,𝑋_𝜆𝑛 are said to belong to the scale family of distributions if 𝑋_𝜆𝑖∼𝐹(𝜆_𝑖𝑥), for 𝑖=1,…,𝑛, where 𝐹 is an absolutely continuous distribution function with density function 𝑓. It means that the random variables 𝜆₁𝑋_𝜆1,…,𝜆_𝑛𝑋_𝜆𝑛 are independent and identically distributed with common cdf 𝐹. 𝐹 is called the baseline distribution and the 𝜆_𝑖’s are the scale parameters. It includes many important distributions like normal, exponential, Weibull, and gamma as special cases. The scale model is of theoretical as well as practical importance in various fields of probability and statistics. As we have seen above, there is an extensive literature on stochastic orderings among order statistics when the observations follow the exponential distribution with different scale parameters.

Pledger and Proschan [15] proved the following general result for the scale model.

Theorem 4.24. Let X₁,…,𝑋_𝑛; 𝑌₁,…,𝑌_𝑛 be independent nonnegative random variables with 𝑋_𝑖∼𝐹(𝜆_𝑖𝑥), 𝑌_𝑖∼𝐹(𝜇_𝑖𝑥), 𝜆_𝑖>0, 𝜇_𝑖>0, 𝑖=1,…,𝑛 where 𝐹 is an absolutely continuous distribution. If 𝑟(𝑥), the hazard rate of 𝐹, is decreasing, then 𝜆₁,…,𝜆≽_𝑛^𝑚𝜇₁,…,𝜇_𝑛⇒𝑋_𝑘∶𝑛≥_𝑠𝑡𝑌_𝑘∶𝑛, (4.46) for 𝑘=1,…,𝑛.

Bon and Pǎltǎnea [86] proved the following result for the scale model when one set of variables are iid.

Theorem 4.25. Let 𝑋₁,…,𝑋_𝑛 be independent nonnegative random variables with 𝑋_𝑖∼𝐹(𝜆_𝑖𝑥) and let 𝑌₁,…,𝑌_𝑛 be a random sample from a distribution with cdf 𝐹(𝑚_𝑘(𝜆)𝑥), where 𝐹 is the cdf of a nonnegative random variable with nonincreasing failure rate and 𝑚_𝑘(𝜆)=𝑛𝑘^−11≤𝑖₁^<⋯<𝑖_𝑘^≤𝑛𝜆_𝑖1⋯𝜆_𝑖𝑘^1/𝑘. (4.47)
If 𝐹(𝑥)/[𝑥𝐹(𝑥)] is an increasing function on [0,∞), then for 𝑘∈{1,…,𝑛}, 𝑋_𝑘∶𝑛≥_𝑠𝑡𝑌_𝑘∶𝑛. (4.48)

Hu [93] also considered the general scale problem and proved the following result.

Theorem 4.26. Let 𝑋₁,…,𝑋_𝑛; 𝑌₁,…,𝑌_𝑛 be independent nonnegative random variables with 𝑋_𝑖∼𝐹(𝜆_𝑖𝑥), 𝑌_𝑖∼𝐹(𝜇_𝑖𝑥), where 𝜆_𝑖>0, 𝜇_𝑖>0, 𝑖=1,…,𝑛 are such that 𝜆₁,…,𝜆≽_𝑛^𝑚𝜇₁,…,𝜇_𝑛. (4.49) Assume that the failure rate of 𝐹, 𝑟(𝑥) is decreasing and 𝑥𝑟(𝑥) is increasing. Then on some probability space (Ω,ℱ,𝒫), there exist random variables 𝑋^′₁,…,𝑋^′_𝑛; 𝑌^′₁,…,𝑌^′_𝑛 such that 𝑋₁,…,𝑋=_𝑛^𝑠𝑡𝑋^′₁,…,𝑋^′_𝑛,𝑌₁,…,𝑌=_𝑛^𝑠𝑡𝑌^′₁,…,𝑌^′_𝑛,𝑋_𝑘∶𝑛≥𝑌_𝑘∶𝑛,𝑎.𝑠.𝑓𝑜𝑟𝑘=1,…,𝑛. (4.50) This implies in particular that 𝑋_1∶𝑛,…,𝑋≽_𝑛∶𝑛^𝑠𝑡𝑌_1∶𝑛,…,𝑋_𝑛∶𝑛. (4.51)

Let 𝐺(𝛼,𝜆) and 𝑊(𝛼,𝜆) denote gamma and Weibull random variables with shape parameter 𝛼 and scale parameter 𝜆. For these scale models, Sun and Zhang [94] and Khaledi and Kochar [95], respectively, proved the following theorem.

Theorem 4.27. Let (𝑋₁,…,𝑋_𝑛) and (𝑌₁,…,𝑌_𝑛) be two sets of independent random variables with (a)𝑋_𝑖∼𝑊(𝛼,𝜆_𝑖) and 𝑌_𝑖∼𝑊(𝛼,𝜇_𝑖), 𝑖=1,…,𝑛 or (b)𝑋_𝑖∼𝐺(𝛼,𝜆_𝑖) and 𝑌_𝑖∼𝐺(𝛼,𝜇_𝑖), 𝑖=1,…,𝑛. Then for 0<𝛼≤1,𝝀≽^𝑚𝝁⇒𝑋_1∶𝑛,…,𝑋≽_𝑛∶𝑛^𝑠𝑡𝑌_1∶𝑛,…,𝑌_𝑛∶𝑛. (4.52)

4.2.1. Generalized Gamma Distribution

A random variable 𝑋 is said to have a generalized gamma distribution, denoted by 𝑋∼𝐺𝐺(𝑝,𝑞), if it admits the following density function: 𝑔_𝑝,𝑞(𝑥)=𝑝Γ𝑥𝑞/𝑝^𝑞−1𝑒^−𝑥𝑝𝑥>0, (4.53) where 𝑝,𝑞(>0) are the shapes parameters. It was introduced by Stacy [96] and includes the wildly used exponential (𝑝=1, 𝑞=1), Weibull (𝑞=𝑝), and gamma (𝑝=1) distributions as special cases. It is a flexible family of distributions, having an increasing failure rate when 𝑞≥1,𝑝≥1, a bathtub failure rate when 𝑞<1,𝑝>1, an upside down bathtub (or unimodal) failure rate when 𝑞>1,𝑝<1.

It is easy to show that for 𝑝≤1,𝑞≤1, the hazard rate of generalized gamma distribution is decreasing. It is shown in Khaledi et al. [97] that 𝑥𝑟(𝑥) is an increasing function of 𝑥 for all 𝑝,𝑞>0. Thus the conditions of Theorem 4.26 are satisfied when the baseline distribution is generalized gamma distribution with 𝑝≤1,𝑞≤1 and thus extending the result of Theorem 1.1 in Sun and Zhang [94] from standard gamma distribution to generalized gamma distribution.

4.2.2. Power-Generalized Weibull Distribution

A random variable 𝑋 is said to have power-generalized Weibull distribution, denoted by 𝑋∼PGW(𝜈,𝛾), if its density function is 𝑓=𝜈𝑡,𝜈,𝛾𝛾𝑡^𝜈−11+𝑡^{𝜈1/𝛾−1}𝑒^{1−(1+𝑡𝜈)1/𝛾},𝑡>0;𝜈,𝛾>0, (4.54) and its survival function is 𝐹𝑡,𝜈,𝛾=𝑒^{1−(1+𝑡𝜈)1/𝛾},𝑡>0. (4.55) It has a decreasing failure rate when 𝜈≤𝛾,𝜈≤1, an increasing failure rate when 𝜈≥𝛾,𝜈≥1, a bathtub failure rate when 0<𝛾<𝜈<1, and an upside down bathtub (or unimodal) failure rate when 𝛾>𝜈>1. It includes Weibull and exponential distributions as special cases.

For more details on this family and its applications in probability and statistics, the reader is referred to Bagdonavicius and Nikulin [98]. Let 𝑟(𝑥) be the hazard rate function of power-generalized Weibull distribution. It is known that for 𝜈≤𝛾,0<𝜈≤1, 𝑟(𝑥) is a decreasing function (cf. [98]).

For all values of 𝜈 and 𝛾, the function (𝑥)=𝜈𝑥𝑟𝛾𝑥^𝜈1+𝑥^{𝜈1/𝛾−1} (4.56) is increasing in 𝑥. That is, the conditions of Theorem 4.26 are satisfied by the power-generalized Weibull distribution with 𝜈≤𝛾,0<𝜈≤1, thus extending the result of Theorem 2.2 in Khaledi and Kochar [95] from standard Weibull distribution to power-generalized Weibull distribution.

Khaledi et al. [97] proved that under some additional conditions the results of Theorem 4.24 can be extended from stochastic ordering to the hazard rate order when we compare two series systems.

Theorem 4.28. Let 𝑋_𝜆1,…,𝑋_𝜆𝑛 be independent nonnegative random variables with 𝑋_𝜆𝑖∼𝐹(𝜆_𝑖𝑥), 𝑖=1,…,𝑛, where 𝜆_𝑖>0, 𝑖=1,…,𝑛 and 𝐹 is an absolutely continuous distribution. Let 𝑟 be the hazard rate functions of 𝐹. If 𝑥²𝑟^′(𝑥) is decreasing and (𝜆₁,…,𝜆_𝑛)≽^𝑚(𝜇₁,…,𝜇_𝑛), then (i)𝑋_1∶𝑛≥_ℎ𝑟𝑌_1∶𝑛, and (ii)if 𝑟(𝑥) is decreasing then 𝑋_1∶𝑛≥_{𝑑𝑖𝑠𝑝}𝑌_1∶𝑛. The inequalities in (𝑖) and (𝑖𝑖) are reversed if 𝑥²𝑟^′(𝑥) is increasing.

It is shown in Khaledi et al. [97] that 𝑥²𝑟^′(𝑥) is decreasing for the PGW(𝜈,𝛾) distribution when 0<𝜈≤𝛾 and 𝜈<1. This is also true for the 𝐺𝐺(𝑝,𝑞) distribution when 𝑝,𝑞<1. We know that for 𝑝,𝑞≤1, the hazard rate function 𝑟(𝑥) is decreasing. That is, Theorem 4.28 (ii) can be applied to this case as well.

Khaledi et al. [97] also established a stochastic ordering result for the largest order statistics under p-larger ordering.

Theorem 4.29. Let 𝑋₁,…,𝑋_𝑛 be a set of independent nonnegative random variables with 𝑋_𝑖∼𝐹(𝜆_𝑖𝑥), 𝑖=1,…,𝑛, where 𝐹 is an absolutely continuous distribution function with density function 𝑓. Let 𝑌₁,…,𝑌_𝑛 be another set of independent nonnegative random variables with 𝑌_𝑖∼𝐹(𝜇_𝑖x), 𝑖=1,…,𝑛. Let 𝑟(̃𝑥) denote the reverse hazard rate of 𝐹. If 𝑥𝑟(̃𝑥) is decreasing in 𝑥, then 𝜆₁,…,𝜆≽_𝑛^𝑝𝜇₁,…,𝜇_𝑛⇒𝑋_𝑛∶𝑛≥_𝑠𝑡𝑌_𝑛∶𝑛. (4.57)

The above theorem immediately leads to the following corollary.

Corollary 4.30. Let 𝑋₁,…,𝑋_𝑛 be a set of independent nonnegative random variables with 𝑋_𝑖∼𝐹(𝜆_𝑖𝑥), 𝑖=1,…,𝑛, where 𝐹 is an absolutely continuous distribution function with density function 𝑓. Let 𝑌₁,…,𝑌_𝑛 be i.i.d. random variables with common c.d.f 𝐹(𝜆𝑥), where 𝜆 is the geometric mean of the 𝜆_𝑖’s. If 𝑥𝑟(̃𝑥) is decreasing in 𝑥, then 𝑋_𝑛∶𝑛≥_𝑠𝑡𝑌_𝑛∶𝑛.

The above corollary gives a lower bound on the survival function of a parallel system with non-identical components in terms of the one with i.i.d. components when the common scale parameter is the geometric mean of the scale parameters. The new bound is better than the one that follows from Hu [93] which is in terms of the arithmetic mean of the scale parameters since 𝐹_{𝑋𝑛∶𝑛}(𝑥) is a nonincreasing function of 𝜆 and the fact that the geometric mean of the 𝜆_𝑖’s is smaller than their arithmetic mean.

Under the conditions of Theorems 4.28 and 4.29 the improvements on the bounds are relatively more if the 𝜆_𝑖’s are more dispersed in the sense of majorization. This fact follows from the fact that the geometric mean is Schur concave whereas the arithmetic mean is Schur constant and the survival function of a parallel system of i.i.d components with baseline distribution 𝐹(𝑥) and common parameter 𝜆 is decreasing in 𝜆.

It is shown in Khaledi et al. [97] that the conditions of Theorem 4.29 are satisfied when the baseline distributions in the scale model are generalized gamma distribution and power-generalized Weibull distribution with arbitrary parameters.

The next theorem again by Khaledi et al. [97] extends Theorem 4.26 from stochastic ordering to the reverse hazard rate order when we compare two parallel systems, a result which is similar to Theorem 4.28.

Theorem 4.31. Let 𝑋_𝜆1,…,𝑋_𝜆𝑛 be independent nonnegative random variables with 𝑋_𝜆𝑖∼𝐹(𝜆_𝑖𝑥), 𝑖=1,…,𝑛, where 𝜆_𝑖>0, 𝑖=1,…,𝑛 and 𝐹 is an absolutely continuous distribution. Let 𝑌₁,…,𝑌_𝑛 be another set of independent nonnegative random variables with 𝑌_𝑖∼𝐹(𝜇_𝑖𝑥), 𝑖=1,…,𝑛. Let 𝑓 and 𝑟̃ be the density and the reverse hazard rate functions of 𝐹, respectively. If 𝑥²𝑟^′̃(𝑥) is increasing, then 𝜆₁,…,𝜆≽_𝑛^𝑚𝜇₁,…,𝜇_𝑛⇒𝑋≽_𝑛∶𝑛^𝑟ℎ𝑌_𝑛∶𝑛. (4.58)

It is proved in Khaledi et al. [97] that the conditions of Theorem 4.31 are satisfied by generalized gamma distribution with the parameters 𝑝<1 and 𝑞>0. That is, the reverse hazard rate of a parallel system consisting of independent components with 𝐺𝐺(𝑝,𝑞) lifetimes is Schur convex in the vector of scales parameters, when either 𝑝<1 and 𝑞<1 (i.e., 𝐹 is DFR,) or 𝑝<1 and 𝑞>1 (i.e., 𝐹 has an upside down bathtub failure rate).

Balakrishnan and Zhao [99] proved the following result for gamma distribution.

Theorem 4.32. Let 𝑋₁,…,𝑋_𝑛 be independent gamma random variables with 𝑋_𝑖 having shape parameter 0<𝑟≤1 and scale parameter 𝜆_𝑖, 𝑖=1,…,𝑛. Let 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from a gamma distribution with shape parameter 𝑟 and scale parameter 𝜆=𝜆≥𝜆=^𝑛∏^𝑛_𝑖=1𝜆_𝑖, the geometric mean of 𝜆_𝑖’s. Then 𝑋_𝑛∶𝑛≥_ℎ𝑟𝑌_𝑛∶𝑛. (4.59)

This result generalizes the results of Khaledi and Kochar [27] from exponential distribution to gamma distribution when the scale parameter is at most one.

4.3. Dependence Orderings among Order Statistics from Independent Random Variables with Proportional Hazard Rates

Sathe [100] proved that if 𝑋₁,…,𝑋_𝑛 are independent exponential random variables with distinct parameters 𝜆₁,𝜆₂,…,𝜆_𝑛, then for any 𝑘=2,…,𝑛, the Peasrson coefficient of correlation between 𝑋_𝑘∶𝑛 and 𝑋_1∶𝑛 is maximum when the 𝜆_𝑖’s are equal. The natural question is to see if we can extend this result to positive dependence orderings. Dolati et al. [58] studied this topic when the observations follow the proportional hazards rate model. They proved the following result for the extreme order statistics.

Theorem 4.33. Let 𝑋₁,…,𝑋_𝑛 be independent continuous random variables following the PHR model. Let 𝑌₁,…,𝑌_𝑛 be i.i.d. continuous random variables, then 𝑋_𝑛∶𝑛∣𝑋_1∶𝑛≺_𝑅𝑇𝐼𝑌_𝑛∶𝑛∣𝑌_1∶𝑛. (4.60)

They also wondered whether this result can be strengthened to more SI ordering. Genest et al. [60] gave a positive answer to this question by showing the following result.

Theorem 4.34. Let 𝑋₁,…,𝑋_𝑛 be independent continuous random variables following the PHR model. Let 𝑌₁,…,𝑌_𝑛 be i.i.d. continuous random variables, then 𝑋_𝑛∶𝑛∣𝑋_1∶𝑛≺_𝑆𝐼𝑌_𝑛∶𝑛∣𝑌_1∶𝑛. (4.61)

It will be of interest to know whether Theorem 4.34 can be extended to other order statistics, that is, for 2≤𝑗≤𝑛−1, 𝑋_𝑗∶𝑛∣𝑋_1∶𝑛⪯_SI𝑌_𝑗∶𝑛∣𝑌_1∶𝑛. (4.62)

It is true for 𝑗=2. To prove our conjecture one needs to prove that 𝑋_𝑗∶𝑛−𝑋_1∶𝑛≤_disp𝑌_𝑗∶𝑛−𝑌_1∶𝑛 (4.63) whose proof is still elusive for 3≤𝑗≤𝑛−1.

It is also worth noting that Dolati et al. [58] got a nice bound for Kendall’s tau of (𝑋_𝑛∶𝑛,𝑋_1∶𝑛) by using Theorem 4.33, 𝜏𝑋_𝑛∶𝑛,𝑋_1∶𝑛≤1.2𝑛−1 (4.64)

5. Stochastic Comparisons of Sample Spacings

Spacings are of great interest in many areas of statistics, in particular, in the characterizations of distributions, goodness-of-fit tests, auction theory, life testing, and reliability models. A large number of goodness-of-fit tests are based on functions of sample spacings, see Balakrishnan and Rao [5, 6], Jammalamadaka and Taufer [101], and Jammalamadaka and Goria [102] for more references.

Let 𝑋₁,…,𝑋_𝑛 be 𝑛 random variables. The random variables 𝐷_𝑖∶𝑛=𝑋_𝑖∶𝑛−𝑋_{𝑖−1∶𝑛} and 𝐷^∗_𝑖∶𝑛=(𝑛−𝑖+1)𝐷_𝑖∶𝑛, 𝑖=1,…,𝑛, with 𝑋_0∶𝑛≡0, are respectively called spacings and normalized spacings. They are of great interest in various areas of statistics, in particular, in characterizations of distributions, goodness-of-fit tests, life testing, and reliability models. In the reliability context they correspond to times elapsed between successive failures of components in a system. In stochastic auction theory, 𝐷_𝑛∶𝑛 and 𝐷_2∶𝑛 are of particular interest, which represent auction rents in buyer’s auction and reverse auction in the second-price business auction (see [103]). In this section we consider the stochastic properties of spacings. In Section 5.1 we investigate stochastic order relations between successive spacings. In Section 5.2 we study the effect of changes in the scale parameters of the observations on stochastic properties of spacings when the observations are independent exponentials.

5.1. Stochastic Orders between Successive Spacings in a Sample

It is well known that the normalized spacings of a random sample from an exponential distribution are i.i.d. random variables having the same exponential distribution. Many authors have studied the stochastic properties of spacings from restricted families of distributions. Barlow and Proschan [104] proved the following result on stochastic ordering between the successive normalized spacings from DFR distributions.

Theorem 5.1. Let 𝑋₁,…,𝑋_𝑛 be a random sample of size 𝑛 from a DFR distribution. Then
(a) 𝐷^∗_𝑖∶𝑛≤_𝑠𝑡𝐷^∗_{𝑖+1∶𝑛},𝑖=1,…,𝑛−1, (5.1)
(b) 𝐷^∗_{𝑖∶𝑛+1}≤_𝑠𝑡𝐷^∗_𝑖∶𝑛,𝑛≥𝑖𝑓𝑜𝑟𝑓𝑖𝑥𝑒𝑑𝑖. (5.2)

Similar results hold for the IFR case with the inequalities reversed in (a) and (b) above. Later Pledger and Proschan [15] partially extended this result to the case when the random variables are independent with proportional decreasing failure rates. Kim and David [70] have also obtained some results on spacings from IFR (DFR) distributions.

Kochar and Kirmani [19] strengthened this result from stochastic ordering to hazard rate ordering. They proved the following result.

Theorem 5.2. Let 𝑋₁,…,𝑋_𝑛 be a random sample of size 𝑛 from a DFR distribution. Then
(a) 𝐷^∗_𝑖∶𝑛≤_ℎ𝑟𝐷^∗_{𝑖+1∶𝑛}𝑓𝑜𝑟𝑖=1,…,𝑛−1. (5.3)
(b) 𝐷^∗_{𝑖∶𝑛+1}≤_ℎ𝑟𝐷^∗_𝑖∶𝑛,𝑛≥𝑖𝑓𝑜𝑟𝑓𝑖𝑥𝑒𝑑𝑖. (5.4)

Barlow and Proschan [104] have also shown that spacings of i.i.d. DFR random variables have also DFR distributions. Using this and the connection between hazard rate ordering and dispersive ordering, the proof of the next result follows from Theorem 2.12.

Theorem 5.3. If 𝑋₁,…,𝑋_𝑛 is a random sample from a DFR distribution, then for 𝑖=1,…,𝑛−1,(a)𝐷^∗_𝑖∶𝑛≤_{𝑑𝑖𝑠𝑝}𝐷^∗_{𝑖+1∶𝑛}, (b)𝐷^∗_{𝑖∶𝑛+1}≤_{𝑑𝑖𝑠𝑝}𝐷^∗_𝑖∶𝑛.

The corresponding problem when the random variables are not identically distributed or not independent has also been studied by many researchers. Kochar and Kirmani [19] proved that when the joint density of the parent observations is convex, the joint density of the normalized spacings is ISO^∗. This will hold, in particular, when the 𝑋^′_𝑖s are independent (but not necessarily identical) with log-convex densities.

Theorem 5.4. Let the joint density 𝑓_𝐗(𝑥₁,…,𝑥_𝑛) of 𝐗=(𝑋₁,…,𝑋_𝑛) be convex. Then the joint density of 𝐃=(𝐷_1∶𝑛,…,𝐷_𝑛∶𝑛) is ISO^∗.

This result immediately leads to the following interesting result.

Theorem 5.5. Let 𝑋₁,𝑋₂,…,𝑋_𝑛 be independent random variables with log-convex densities. Then 𝐷^∗_1∶𝑛≤_{ℓ𝑟∶𝑗}𝐷^∗_2∶𝑛≤_{ℓ𝑟∶𝑗}⋯≤_{ℓ𝑟∶𝑗}𝐷^∗_𝑛∶𝑛. (5.5)

If a density is log-convex, it is DFR, but the converse is not true. Hence, Theorem 5.5 establishes a stronger ordering between the normalized spacings than does Theorem 5.2 under a stronger condition on the parent distributions. A related paper on this topic is by Nappo and Spizzichino [12].

One may wonder whether under the condition of Theorem 5.5 the result can be extended from joint likelihood ratio order to likelihood ratio order. The answer is negative as illustrated by a counterexample in Misra and van der Meulen [105].

Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having failure rate 𝜆_𝑖 for 𝑖=1,…,𝑛. Hu and Xie [77] proved the following result.

Theorem 5.6. If 𝜆_𝑛+1≤[≥]𝜆_𝑘 for 𝑘=1,…,𝑛, then 𝐷_𝑛∶𝑛≤_𝑙𝑟𝐷_{𝑛+1∶𝑛+1},𝐷_1∶𝑛≤_𝑙𝑟𝐷_2∶𝑛+1𝐷_2∶𝑛+1≤_𝑙𝑟𝐷_2∶𝑛. (5.6)
If 𝜆_𝑖+𝜆_𝑗≥𝜆_𝑘 for all distinct 𝑖,𝑗 and 𝑘 then 𝐷_{𝑛−1∶𝑛}≤_𝑙𝑟𝐷_𝑛∶𝑛,𝐷_{𝑛∶𝑛+1}≤_𝑙𝑟𝐷_𝑛∶𝑛. (5.7)

They also showed that 𝐷_2∶3≤_lr𝐷_3∶3 for all 𝜆_𝑖’s, and 𝐷_1∶𝑛≤_lr𝐷_2∶𝑛,𝑛≥2. (5.8)

Kochar and Korwar [20] conjectured that (5.3) can be extended to the case when 𝑋₁,…,𝑋_𝑛 are independent exponential random variables with possibly different scale parameters. They proved their conjecture for 𝑛=3 and Torrado et al. [106] provided a proof of this conjecture for 𝑛=4, but the general case still remains an open problem. Torrado et al. [106] also proved that for any 𝑛, 𝐷^∗_2∶𝑛≤_hr𝐷^∗_3∶𝑛. This topic has been extensively studied by Khaledi and Kochar [30], Wen et al. [107], Xu et al. [108], and Chen and Hu [109] in single-outlier or multiple-outlier exponential models.

5.2. Comparing Spacings of Two Random Samples

How does a variability ordering between two probability distributions affect the relative positioning of the observations in random samples from the two distributions? To answer this question, let 𝑋_1∶𝑛≤𝑋_2∶𝑛≤⋯≤𝑋_𝑛∶𝑛 denote the order statistics of a random sample 𝑋₁,𝑋₂,…,𝑋_𝑛 from a distribution with distribution function 𝐹. Similarly, let 𝑌_1∶𝑛≤𝑌_2∶𝑛≤⋯≤𝑌_𝑛∶𝑛 denote the order statistics of an independent random sample 𝑌₁,𝑌₂,…,𝑌_𝑛 from a distribution with distribution function 𝐺. The corresponding spacings are defined by 𝐷_𝑖∶𝑛≡𝑋_𝑖∶𝑛−𝑋_{𝑖−1∶𝑛} and 𝐻_𝑖∶𝑛≡𝑌_𝑖∶𝑛−𝑌_{𝑖−1∶𝑛}, for i=1,2,…,𝑛, where 𝑋_0∶𝑛=𝑌_0∶𝑛≡0.Bartoszewic [42] proved the following result.

Theorem 5.7. One has 𝑌≤_{𝑑𝑖𝑠𝑝}𝑋⇒𝐻_1𝑛,𝐻_2∶𝑛,…,𝐻_𝑛∶𝑛≤_𝑠𝑡𝐷_1∶𝑛,𝐷_2∶𝑛,…,𝐷_𝑛∶𝑛. (5.9)

Some consequences of the this result as reported in Kochar [25] are as follows.(a)For 1≤𝑖<𝑗≤𝑛, 𝑋_𝑗∶𝑛−𝑋_𝑖∶𝑛≥_st𝑌_𝑗∶𝑛−𝑌_𝑖∶𝑛. (5.10) In particular, 𝑋_𝑛∶𝑛−𝑋_1∶𝑛≥_st𝑌_𝑛∶𝑛−𝑌_1∶𝑛. (5.11)(b)One has 𝑠²_𝑋≥_st𝑠²_𝑌, (5.12) where 𝑠²_𝑋 and 𝑠²_𝑌 are the sample variances of the two samples.(c)One has 𝜂_𝑋≥_st𝜂_𝑌, (5.13) where 𝜂_𝑋=𝑛2^{−1𝑖<𝑗}𝑋_𝑗∶𝑛−𝑋_𝑖∶𝑛 (5.14) is the Gini’s mean difference for the 𝑋-sample. Similarly we define 𝜂_𝑌.

Kochar et al. [110] extended the implication (5.10) when dispersive ordering between 𝐹 and 𝐺 is replaced by either excess wealth order. They proved the following result.

Theorem 5.8. If 𝑌≤_𝑒𝑤𝑋, then, for all 1≤𝑘≤𝑛−1, 𝑋_𝑛∶𝑛−𝑋_𝑘∶𝑛≥_𝑖𝑐𝑥𝑌_𝑛∶𝑛−𝑌_𝑘∶𝑛⇒E𝑋_𝑛∶𝑛−𝑋_𝑘∶𝑛≤E𝑌_𝑛∶𝑛−𝑌_𝑘∶𝑛. (5.15)

Taking 𝑘=1 in this theorem gives us a result on the sample spacings, 𝑌≤_ew𝑋⇒𝑋_𝑛∶𝑛−𝑋_1∶𝑛≥_icx𝑌_𝑛∶𝑛−𝑌_1∶𝑛. (5.16)

It is well known that a random variable 𝑋 is NBUE (new better than used in expectation) if and only if 𝑋≤_ew𝑌, where 𝑌 is exponential with the same mean as that of 𝑋 (see [111]). It follows directly from Theorem 5.8 that, if 𝑋₁,…,𝑋_𝑛 is a random sample from a distribution which is NBUE, then, 𝐸𝑋_𝑛∶𝑛−𝑋_𝑘∶𝑛≤11+⋯+𝐸[𝑋]𝑛−𝑘,forany1≤𝑘≤𝑛−1. (5.17)

5.3. Comparing Spacings of Heterogeneous Exponential Samples

Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛, and let 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆. Kochar and Korwar [20] proved that the distribution function of 𝐷^∗_𝑘∶𝑛 for 𝑖=2,⋯,𝑛 is a mixture of independent exponential random variables with the density function 𝑓_{𝐷∗𝑘∶𝑛}(𝑥)=_𝐫∏^𝑛_𝑖=1𝜆_𝑖∏^𝑛_𝑖=1∑^𝑛_𝑗=𝑖𝜆_𝑟𝑗∑^𝑛_𝑗=𝑘𝜆_𝑟𝑗𝑛−𝑖+1exp−𝑥∑^𝑛_𝑗=𝑘𝜆_𝑟𝑗𝑛−𝑖+1, (5.18) where 𝐫 extends over all of the permutations of {1,2,…,𝑛}. Hence, the distribution of 𝐷^∗_𝑘∶𝑛 could be represented as 𝐹_{𝐷𝑘∶𝑛}(𝑥)=_𝑗∈𝐫𝑝_𝑗𝐹_𝜆^∗_𝑘𝑗(𝑥), (5.19) where 𝑗 denotes a permutation of (𝜆₁,…,𝜆_𝑛) belonging to 𝐫 and 𝑝_𝑗=∏^𝑛_𝑖=1𝜆_𝑖∏^𝑛_𝑖=1∑^𝑛_𝑗=𝑖𝜆_𝑟𝑗,𝜆^∗_𝑘𝑗=∑^𝑛_𝑗=𝑘𝜆_𝑟𝑗(𝑛−𝑘+1)², (5.20) and 𝐹_𝜆^∗_𝑘𝑗 means an exponential distribution with hazard rate 𝜆^∗_𝑘𝑗.

Pledger and Proschan [15] considered the problem of stochastically comparing the spacings of nonidentical independent exponential random variables with those corresponding to stochastically comparable independent and identically distributed exponential random variables.

Theorem 5.9 (see [15]). If 𝑋₁,…,𝑋_𝑛 are independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛, and 𝑌₁,…,𝑌_𝑛 is a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆=∑^𝑛_𝑖=1𝜆_𝑖/𝑛, then 𝐻^∗_𝑘∶𝑛≤_𝑠𝑡𝐷^∗_𝑘∶𝑛, (5.21) where 𝐻^∗_𝑘∶𝑛=(𝑛−𝑘+1)(𝑌_𝑘∶𝑛−𝑌_{𝑘−1∶𝑛}) for 𝑘=1,…,𝑛 are the normalized spacings of 𝑌_𝑖’s, and 𝑌_0∶𝑛≡0.

Kochar and Korwar [20] strengthened and extended this result from stochastic ordering to likelihood ratio ordering and dispersive ordering.

Theorem 5.10. Under the conditions of Theorem 5.9, 𝐻^∗_𝑘∶𝑛≤_𝑙𝑟𝐷^∗_𝑘∶𝑛,𝐻^∗_𝑘∶𝑛≤_{𝑑𝑖𝑠𝑝}𝐷^∗_𝑘∶𝑛,𝑘=1,…,𝑛. (5.22)

Kochar and Rojo [21] further strengthened Theorem 5.10 to multivariate likelihood ratio order.

Theorem 5.11. Under the conditions of Theorem 5.9, 𝐻^∗_1∶𝑛,…,𝐻^∗_𝑛∶𝑛≤_𝑙𝑟𝐷^∗_1∶𝑛,…,𝐷^∗_𝑛∶𝑛. (5.23)

As a consequence, it follows that 𝑌_𝑗∶𝑛−𝑌_𝑖∶𝑛≤_st𝑋_𝑗∶𝑛−𝑋_𝑖∶𝑛∀1≤𝑖<𝑗≤𝑛. (5.24)

Kochar and Xu [112] established the following results providing sufficient and necessary conditions for stochastically comparing 𝐷_𝑘∶𝑛 and 𝐻_𝑘∶𝑛 according to likelihood ratio and reverse hazard rate orderings.

Theorem 5.12. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛, and 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆. Then, for 𝑘≥2, (a)𝐻_𝑘∶𝑛≤_𝑙𝑟𝐷_𝑘∶𝑛 or (b)𝐻_𝑘∶𝑛≤_𝑟ℎ𝐷_𝑘∶𝑛; if and only if 1𝜆≥∑𝑛−𝑘+1^𝐫∏^𝑛_𝑖=1𝜆_𝑖/∏^𝑛_𝑖=1∑^𝑛_𝑗=𝑖𝜆_𝑟𝑗∑^𝑛_𝑗=𝑘𝜆_𝑟𝑗²∑_𝐫∏^𝑛_𝑖=1𝜆_𝑖/∏^𝑛_𝑖=1∑^𝑛_𝑗=𝑖𝜆_𝑟𝑗∑^𝑛_𝑗=𝑘𝜆_𝑟𝑗. (5.25)

Analogously, we have an equivalent necessary and sufficient condition for hazard rate order, dispersive order, and stochastic order between 𝐷_𝑘∶𝑛 and 𝐻_𝑘∶𝑛 as proved in Kochar and Xu [112].

Theorem 5.13. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛, and 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆. Then, for 𝑘≥2, (a)𝐻_𝑘∶𝑛≤_ℎ𝑟D_𝑘∶𝑛, (b)𝐻_𝑘∶𝑛≤_{𝑑𝑖𝑠𝑝}𝐷_𝑘∶𝑛 or (c)𝐻_𝑘∶𝑛≤_𝑠𝑡𝐷_𝑘∶𝑛 if and only if 1𝜆≥𝑛−𝑘+1_𝐫∏^𝑛_𝑖=1𝜆_𝑖∏^𝑛_𝑖=1∑^𝑛_𝑗=𝑖𝜆_𝑟𝑗^𝑛_𝑗=𝑘𝜆_𝑟𝑗. (5.26)

The next result is also from Kochar and Xu [112].

Theorem 5.14. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛, and 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆. Then, for 𝑘≥2, (a)𝐻_𝑘∶𝑛≤_𝑚𝑟𝑙𝐷_𝑘∶𝑛, (b)𝐻_𝑘∶𝑛≤_𝑅𝑆𝐷_𝑘∶𝑛 or (c)E𝐻_𝑘∶𝑛≤E𝐷_𝑘∶𝑛if and only if 1𝜆≥1𝑛−𝑘+1∑_𝐫∏^𝑛_𝑖=1𝜆_𝑖/∑^𝑛_𝑗=𝑘𝜆_𝑟𝑗∏^𝑛_𝑖=1∑^𝑛_𝑗=𝑖𝜆_𝑟𝑗. (5.27)

Example 5.15. Suppose 𝑋₁,𝑋₂,𝑋₃ are independent exponential random variables with parameters 𝜆₁,𝜆₂,𝜆₃, respectively; and 𝑌₁,𝑌₂,𝑌₃ are i.i.d. exponential random variables with parameter 𝜆. Then, from Theorem 5.13, 𝑌_3∶3−𝑌_2∶3≤_hr𝑋_3∶3−𝑋_2∶3 (5.28) if and only if 𝜆≥𝜆_hr=2𝜆₁𝜆₂𝜆₃𝜆₁+𝜆₂+𝜆₃1𝜆₂+𝜆₃+1𝜆₁+𝜆₃+1𝜆₁+𝜆₂. (5.29) From Theorem 5.12, 𝑌_3∶3−𝑌_2∶3≤_lr𝑋_3∶3−𝑋_2∶3 (5.30) if and only if 𝜆≥𝜆_lr=321𝜆₂+𝜆₃+1𝜆₁+𝜆₃+1𝜆₁+𝜆₂⁻¹. (5.31) From Theorem 5.14𝑌_3∶3−𝑌_2∶3≤_RS𝑋_3∶3−𝑋_2∶3 (5.32) if and only if 𝜆≥𝜆_RS=𝜆₁𝜆₂𝜆₃𝜆₁+𝜆₂+𝜆₃𝜆²₂+𝜆²₃(𝜆₂+𝜆₃)𝜆²₂𝜆²₃+𝜆²₁+𝜆²₃(𝜆₁+𝜆₃)𝜆²₁𝜆²₃+𝜆²₂+𝜆²₁(𝜆₂+𝜆₁)𝜆²₂𝜆²₁⁻¹. (5.33) It is worth noting that from (2.7), it follows that 𝜆_𝑙𝑟≥𝜆_hr≥𝜆_RS. (5.34) For example, let 𝜆₁=1,𝜆₂=2,𝜆₃=3. (5.35) Then, 𝜆_lr=1.91489>𝜆_hr=1.56667>𝜆_RS=1.30435. (5.36)

Recently Balakrishnan and Xu [113] have established the following interesting result which compares the degree of skewness between sample spacings according to star ordering.

Theorem 5.16. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛, and 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆. Then, for 𝑘≥2, 𝐻_𝑘∶𝑛≤_∗𝐷_𝑘∶𝑛. (5.37)

Torrado and Lillo [114] considered the problem of stochastically comparing the spacings of two heterogenous samples with independent exponential random variables. They obtained sufficient conditions on the parameters for such comparisons to hold.

5.4. Stochastic Comparisons of Sample Ranges of Two Heterogeneous Samples

Sample range is one of the simple criteria for comparing variabilities among distributions and hence it is important to study its stochastic properties. First we study the stochastic properties of the range of a random sample from a continuous distribution. Let 𝑋₁,…,𝑋_𝑛 be a random sample from 𝐹 and let 𝑌₁,…,𝑌_𝑛 be an independent random sample from another distribution 𝐺. It follows from Lemma 3(c) of Bartoszewic [42] that 𝑋≥_disp𝑌⇒𝑋_𝑛∶𝑛−𝑋_1∶𝑛≥_st𝑌_𝑛∶𝑛−𝑌_1∶𝑛. This observation along with Theorem 2.12 (a) leads to the following theorem.

Theorem 5.17. Let 𝑋≥_ℎ𝑟𝑌 and let either 𝐹 or 𝐺 be DFR. Then 𝑌_𝑛∶𝑛−𝑌_1∶𝑛≤_𝑠𝑡𝑋_𝑛∶𝑛−𝑋_1∶𝑛. (5.38)

Kochar and Rojo [21] proved the following result on stochastic properties of the sample range when the observations are independent exponentials with unequal parameters.

Theorem 5.18. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛, and let 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆=∑^𝑛_𝑖=1𝜆_𝑖/𝑛, then, 𝑌_𝑛∶𝑛−𝑌_1∶𝑛≤_𝑠𝑡𝑋_𝑛∶𝑛−𝑋_1∶𝑛. (5.39)

The next theorem due to Khaledi and Kochar [28] improves upon this result.

Theorem 5.19. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛, and let 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆=(∏^𝑛_𝑖=1𝜆_𝑖)^1/𝑛, the geometric mean of the 𝜆’s. Then, 𝑌_𝑛∶𝑛−𝑌_1∶𝑛≤_𝑠𝑡𝑋_𝑛∶𝑛−𝑋_1∶𝑛. (5.40)

As a consequence of this result we get the following upper bound on the distribution function of the sample range in terms 𝜆, the geometric mean of the parameters.

Corollary 5.20. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛. Then for 𝑥>0, 𝑃𝑋_𝑛∶𝑛−𝑋_1∶𝑛≤𝑥≤1−𝑒⁻𝜆_𝑥^𝑛−1. (5.41)

This bound is better than the one obtained in Kochar and Rojo [21] in terms of 𝜆, since the expression on the R.H.S. of (5.41) is increasing in 𝜆 and 𝜆≤𝜆.

Recently, Zhao and Li [90] obtained a necessary and sufficient condition for stochastically comparing sample ranges from heterogeneous and homogeneous exponential samples.

Theorem 5.21. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛, and let 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆, then, 𝑌_𝑛∶𝑛−𝑌_1∶𝑛≤_𝑠𝑡𝑋_𝑛∶𝑛−𝑋_1∶𝑛∏⟺𝜆≥𝜆=^𝑛𝑖=1𝜆_𝑖𝜆^1/(𝑛−1), (5.42) where 𝜆=∑^𝑛_𝑖=1𝜆_𝑖/𝑛.

Note that 𝜆≤𝜆, which improves upon the results of Theorem 5.19.

Kochar and Xu [85] partially strengthened Theorem 5.18 from stochastic order to reverse hazard rate order.

Theorem 5.22. 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=∑^𝑛_𝑖=1𝜆_𝑖/𝑛. Then 𝑌_𝑛∶𝑛−𝑌_1∶𝑛≤_𝑟ℎ𝑋_𝑛∶𝑛−𝑋_1∶𝑛. (5.43)

They also proved the following theorem for the PHR model.

Theorem 5.23. Let 𝑋₁,…,𝑋_𝑛 be independent random variables with 𝑋_𝑖 having survival function 𝐹^𝜆𝑖, 𝑖=1,…,𝑛. Let 𝑌₁,…,𝑌_𝑛 be a random sample with common population survival distribution 𝐹^𝜆, where 𝜆=∑^𝑛_𝑖=1𝜆_𝑖/𝑛, then 𝑌_𝑛∶𝑛−𝑌_1∶𝑛≤_𝑠𝑡𝑋_𝑛∶𝑛−𝑋_1∶𝑛. (5.44)

The following result of Genest et al. [60] establishes likelihood ration ordering between the sample ranges and improves upon all the previously known results on this problem.

Theorem 5.24. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛, and 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆=∑^𝑛_𝑖=1𝜆_𝑖/𝑛. Let 𝑟_𝑋̃(𝑥) and 𝑟_𝑌̃(𝑥) be reversed hazard rates of 𝑋_𝑛∶𝑛−𝑋_1∶𝑛 and 𝑌_𝑛∶𝑛−𝑌_1∶𝑛, respectively. Then 𝑟_𝑋̃(𝑥)𝑟_𝑌̃(𝑥)𝑖𝑠𝑖𝑛𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔. (5.45)

The following results follow as consequences as shown in that paper.

Corollary 5.25. 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, then,
(a) 𝑌_𝑛∶𝑛−𝑌_1∶𝑛≤_𝑙𝑟𝑋_𝑛∶𝑛−𝑋_1∶𝑛, (5.46)
(b) 𝑌_𝑛∶𝑛−𝑌_1∶𝑛≤_{𝑑𝑖𝑠𝑝}𝑋_𝑛∶𝑛−𝑋_1∶𝑛. (5.47)

Balakrishnan and Xu [113] established a star ordering result between sample ranges as reported in the next theorem.

Theorem 5.26. Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛 and let 𝑌₁,…,𝑌_𝑛 be a random sample of size 𝑛 from an exponential distribution with common hazard rate 𝜆. Then 𝑌_𝑛∶𝑛−𝑌_1∶𝑛≤_∗𝑋_𝑛∶𝑛−𝑋_1∶𝑛. (5.48)

Let 𝑋₁,…,𝑋_𝑛 be independent exponential random variables with 𝑋_𝑖 having hazard rate 𝜆_𝑖, 𝑖=1,…,𝑛 and let 𝑌₁,…,𝑌_𝑛 be an another set of independent exponential random variables with hazard rates 𝜇₁,…,𝜇_𝑛, respectively. Kochar and Rojo [20] proved that 𝜆₁,…,𝜆≽_𝑛^𝑚𝜇₁,…,𝜇_𝑛⇒𝑋_𝑛∶𝑛−𝑋_1∶𝑛≥_st𝑌_𝑛∶𝑛−𝑌_1∶𝑛. (5.49)

Recently Ding et al. [115] strengthened (5.49) from stochastic ordering to reverse hazard rate ordering. Then also proved that under the above set up, log𝜆₁,…,log𝜆≽_𝑛^𝑚𝜇₁,…,𝜇_𝑛⇒𝑋_𝑛∶𝑛−𝑋_1∶𝑛≥_st𝑌_𝑛∶𝑛−𝑌_1∶𝑛. (5.50)

They also extended (5.49) to the PHR model.

5.5. Dependence Orderings among Spacings

It is known that spacings of a random sample from a DFR (decreasing failure rate) distribution are conditionally increasing in sequence (cf., [49, page 151]). Karlin and Rinott [54] have pointed out that if the DFR assumption is strengthened to assume that the parent distribution has a log-convex density, then the spacings have the corresponding stronger property of being MTP₂ dependent. Khaledi and Kochar [116] extended this result to the case when the random variables 𝑋₁,…,𝑋_𝑛 are dependent. It is proved that if the joint pdf of 𝑋_𝑖’s is permutation symmetric, TP₂ in pairs and log-convex in each argument, then their spacings are MTP₂ dependent. They also studied the dependence properties of spacings of independent but non-identically distributed exponential random variables and showed with the help of a counterexample that in this case the spacings may not be MTP₂ dependent. Khaledi and Kochar [116] proved that in the case of a single-outlier when all except one of the parameters are equal, the spacings are MTP₂ dependent. A consequence of this result is that in this case var(𝑋_1∶𝑛)≤var(𝑋_2∶𝑛)≤⋯≤var(𝑋_𝑛∶𝑛). They also proved that in the case of multiple-outliers model any pair of consecutive spacings, 𝐷_𝑖∶𝑛 and 𝐷_{𝑖+1∶𝑛} are TP₂ dependent for 𝑖=1,…,𝑛−1.

6. Applications

6.1. Type-II Censoring

If 𝑛 items are put on life test and the test terminated at the time of 𝑟th failure, then 𝑇_𝑟𝑛=^𝑟_𝑖=1(𝑛−𝑖+1)𝑋_𝑖∶𝑛−𝑋_{𝑖−1∶𝑛}=^𝑟_𝑖=1𝐷^∗_𝑖∶𝑛 (6.1) represents the total time on test (TTT), which is a well-known concept in statistical reliability. As pointed out in Epstein [117], if the component lifetimes are i.i.d. exponential with mean 𝜃 or equivalently the failure rate 𝜆=1/𝜃, the best estimator (minimum variance unbiased estimator) of 𝜃 is 𝑇𝜃=^𝑟𝑛𝑟. (6.2) It is well-known that (cf., [117]), 2𝑇_𝑟𝑛𝑟^𝐿→−𝜒²_2𝑟, (6.3) where 𝜒²_2𝑟 denotes a chi-square distribution with 2𝑟 degrees of freedom. Based on this result, many estimation and testing statistical procedures have been established. However, if the components are actually from different exponential distributions with failure rates 𝜆^∗₁,…,𝜆^∗_𝑛, then it follows from Theorem 5.11 and Theorem 6.E.8 of Shaked and Shanthikumar [2] that 𝜃^∗≥_st𝜃, (6.4) where 𝜃^∗=∑^𝑟_𝑖=1𝐷^∗_𝑖∶𝑛𝑟 (6.5) and 𝜆=∑^𝑛_𝑖=1𝜆^∗_𝑖/𝑛. This means that based on TTT statistic, one would overestimate the average lifetime of components using heterogeneous exponential components.

6.2. Reliability

An 𝑛-component system that fails if and only if at least 𝑘 of the 𝑛 components fail is called a 𝑘-out-of-𝑛: F system. The lifetime of such a system could be represented as 𝑋_𝑘∶𝑛. The 𝑘-out-of-𝑛 system structure is a very popular type of redundancy in fault tolerant systems with wide applications in industrial and military systems. For two different systems say a 𝑘-out-of-𝑛: F system and a 𝑘+1-out-of-𝑛: F system, the engineer may be interested in the additional lifetime 𝑋_{𝑘+1∶𝑛}−𝑋_𝑘∶𝑛 for the system design and the cost purpose. Due to the complicated distribution form, one may provide a sharp bound on the survival function of this based on Theorem 5.13.

For example, a plane has four engines, and a minimum of three engines are required for the plane to work. Hence, this plane is a 2-out-of-4: F system. If the engineer wants to improve the system to the 3-out-of-4: F system, he/she has to consider the cost and the reliability of improvement simultaneously. Hence, it is important to estimate the survival probability and the mean additional lifetime 𝑋_3∶4−X_2∶4. Theorem 5.13 provides a lower bound for the survival function of the additional lifetime. Let (𝜆₁,𝜆₂,𝜆₃,𝜆₄)=(1,2,2.5,3), then, 𝑃𝑋_3∶4−𝑋_2∶4>𝑥≥𝑒^−0.955𝑥. (6.6) Using Theorem 5.14, the following sharp lower bound could be established, 𝐸𝑋_3∶4−𝑋_2∶4≥1.0879. (6.7)

In Table 1, we list the lower bounds on 𝜆^′s for various partial orders to hold in Theorems 5.12–5.14.

Table 1 

The values above are comparable to the arithmetic mean 2.125, geometric mean 1.96799, and harmonic mean 1.791045 of 𝜆_𝑖’s. It is interesting to note that all of the values are less than the arithmetic mean, which coincides with the condition in Theorem 5.11.