1 Introduction

Order statistics have received a great amount of attention in the literature since they are widely used in reliability theory, data analysis, extreme value theory, goodness-of-fit tests, statistical inference and other applied probability and statistical areas. Please refer to David and Nagaraja (2003) and Balakrishnan and Rao (1998a, 1998b) for comprehensive discussion. Studies of order statistics from heterogeneous samples began in early 70s, motivated by robustness issues. After that, a lot of work has been done in single-outlier and multiple-outlier models. Balakrishnan (2007) synthesized recent developments on order statistics arising from independent but non-identically distributed random variables. One may also refer to Kochar and Xu (2007) and Xu (2010) for reviews on various recent developments.

Let X 1:n ≤X 2:n ≤⋯≤X n:n denote the order statistics of random variables X 1,X 2,…,X 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 . It is seen that a parallel system is a special case of the k-out-of-n system.

Skewness describes the departure of a distribution from symmetry, where one tail of the density is more “stretched out” than the other. Skewed data is observed in many areas such as economics, engineering, medicine, insurance and psychology. It may be easy to recognize symmetric distributions but not so easy to determine whether one non-symmetric distribution is more skewed than another. Several partial orders have been introduced in the literature to compare the relative skewness of probability distributions. Van Zwet (1964) introduced the concept of convex transform order to compare two distributions according to skewness (see Definition 2.1). For example, gamma distributions, which plays a prominent role in actuarial science due to its skewness, 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 (see Definition 2.2). This ordering is weaker than the convex transform order.

In the literature, stochastic comparisons of order statistics from heterogeneous samples have been extensively studied. However, most of them focused on the effect of heterogeneity on the magnitude or dispersion of order statistics. Recently, Kochar and Xu (2009) revealed an interesting relation between largest order statistics from heterogeneous exponential random variables and the one from homogeneous exponential random variables based on skewness order. It is shown that the largest order statistics from heterogeneous exponential samples are more skewed than the one from homogeneous exponential samples in the sense of convex transform order without any restriction on the parameters. As a consequence, it holds that

(1.1)

where X n:n is the largest order statistic from any heterogeneous exponential sample with size n, and cv means the coefficient of variation.

The research in this direction is of great interest in both theoretic and practical point of view. The significance of this research is highlighted in the following aspects:

  • Revealing the heterogeneous effect on the skewness. The skewness is an important index in statistics since it reflexes the departure of a distribution from symmetry. There are many interesting questions. For examples, how the heterogeneity of a random sample affects the skewness? Are order statistics from heterogeneous random samples more skewed than the one from corresponding homogeneous samples? If not, under what condition, this statement is true? Our research is devoted to answer those questions.

  • Detecting the heterogeneity with limited information. The research is useful in detecting the heterogeneity of a random sample with limited information. For example, suppose a “black box” parallel system is composted of independent exponential components (which is a common assumption in engineering). The available observations are only the lifetimes of the “black box”. Then an interesting question is whether the types of composing components are the same based on the available data? One “dirty” but quick way to check this answer is to look at the sample coefficient of variation. If it is far larger from the value in Eq. (1.1), we then reject the homogeneity assumption. Of course, a formal test can be developed based on this idea.

  • Unifying and simplifying the study on stochastic comparisons. Under the restriction of some skewness order, stochastic comparisons of order statistics may be equivalent for different stochastic orders. For example, under the star ordering, stochastic comparisons of order statistics based on stochastic order, hazard rate order or dispersive order may be equivalent, and right spread order may be just equivalent to expect value order (cf. Kochar and Xu 2009, 2011). Moreover, under some skewness order restriction, the proof of stochastic comparison may be greatly simplified. For example, Kochar and Xu (2012) proved that for two nonnegative random variables X and Y with distribution functions F and G, respectively, if X≤⋆ Y, then

    This result reveals that under the star order restriction, stochastic order between two random variables is determined by the magnitudes of distribution functions near origins. This fact can be used to simply the proof.

  • Improving estimates of distribution functions of order statistics. Estimation under order restrictions has been well studied in the literature. One may refer to Barlow et al. (1972) for the comprehensive discussion on this topic. Our research in this direction can be used to improve the estimates of distribution functions of order statistics under shape restrictions. For example, assume X 1,…,X n are heterogeneous exponential random variables. According to Kochar and Xu (2009), we have

    where Y n:n can be assumed to be the largest order statistic from standard exponential random variables. Mimicking the procedure in Barlow et al. (1972), one may derive the estimator \(\hat{F}_{n}\) for the distribution function of X n:n under the restriction

    where G is the distribution function of Y n:n .

Kochar and Xu (2011) compared order statistics from multiple-outlier exponential models according to star ordering. They showed that under some suitable conditions, there exists a star ordering between order statistics from two heterogeneous exponential samples. The result has a direct consequence that order statistics from multiple-outlier exponential samples are more skewed than the one from homogeneous exponential samples in the sense of star ordering.

Now, a natural question is whether we can have similar results for other distributions? In this paper, we will answer this question by studying the skewness of order statistics from two samples under the general distribution framework. The rest of this paper is organized as following. In Sect. 2, we recall some stochastic orders. Section 3 contains the main results. Section 4 gives some examples and applications. In the last section, we present some further discussion on this topic.

2 Preliminaries

In this section, we recall some stochastic orders which will be used in the sequel.

Assume random variables X and Y have distribution functions F and G, survival functions \(\bar{F}=1-F\) and \(\bar{G}=1-G\), density functions f and g, and failure rate functions \(r_{X}=f/\bar{F}\) and \(r_{Y}=g/\bar{G}\), respectively.

Definition 2.1

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

If X≤ c Y, then Y is more skewed than X as explained in Van Zwet (1964) and Marshall and Olkin (2007). 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 −1 F(x) means that

is increasing in u∈[0,1]. Thus X≤ c Y can be interpreted to mean that X ages faster than Y in some sense.

Definition 2.2

X is said to be smaller than Y in the star order, denoted by X≤⋆ Y (or F≤⋆ G) if the function G −1 F(x) is star shaped in the sense that G −1 F(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

Thus F≤⋆ G can be interpreted in terms of average failure rates as

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.

It is known in the literature (Marshall and Olkin 2007, p. 69) that,

where \(\mathit{cv}(X)=\sqrt{\mathrm{Var}(X)}/\mathrm{E}(X)\) denote the coefficient of variation of X.

All the above partial orders are scale invariant. A good discussion of those orders can be found in Barlow and Proschan (1981) and Marshall and Olkin (2007).

Definition 2.3

X is said to be smaller than Y in the usual stochastic order (denoted by X≤ st Y), if \(\bar{F}(x) \le\bar{G}(x)\) for all x.

For more discussion on various stochastic orders, please refer to Shaked and Shanthikumar (2007) and references therein.

3 Main results

We need the following lemmas to present the main results. The first lemma is due to Kochar and Xu (2009).

Lemma 3.1

Let X 1,…,X n be independent exponential random variables with X i having hazard rate λ i , i=1,…,n. Let Y 1,…,Y n be the other random sample from an exponential distribution with common hazard rate λ. Then,

(3.1)

The following lemma, which is a modified version of Lemma 2.1 in Kochar (2006), plays a key role in the proof of main results.

Lemma 3.2

Let ϕ be a differentiable star-shaped function on [0,∞) such that ϕ(x)≥x for all x≥0. Let ψ be an increasing differentiable function such that

Then the function

Proof

Note that ϕ is star-shaped if and only if

which can be represented as

(3.2)

Hence, for the required result, it is sufficient to show

(3.3)

Using (3.2), the left side of (3.3) satisfies

So, it is enough to prove

i.e.,

as ψ is increasing. Using the assumptions

the required result follows immediately. □

Now, we are ready to present the following result.

Theorem 3.3

Let X 1,…,X n be independent random variables with X i having survival function \(\bar{F}^{\lambda_{i}}\), i=1,…,n, and let Y 1,…,Y n be a random sample from a distribution with the common survival distribution \(\bar{F}^{\lambda}\) where \(\lambda\ge\tilde{\lambda}= \sqrt[n]{\prod_{i=1}^{n} \lambda_{i}}\), the geometric mean of λ i ’s. If

(3.4)

then

where \(R(x)=-\log\bar{F}(x)\) is the cumulative hazard rate function, and \(r(x)=f(x)/\bar{F}(x)\) is the hazard rate function of F.

Proof

Since R(x) is increasing and

it holds that, for x≥0, i=1,…,n,

So, making the transform

it follows that \(X'_{i}\) is exponential with hazard rate λ i for i=1,…,n. Similarly, let \(Y'_{i}=R(Y_{i})\) be exponential with hazard rate λ for i=1,…,n.

Observing that

it holds that

where \(G'_{n:n}(\cdot)\), \(F'_{n:n}(\cdot)\) are distribution functions of \(Y'_{n:n}\) and \(X'_{n:n}\). Now, we need to prove

By Lemma 3.1, \(F'^{-1}_{n:n}G'_{n:n}(x)\) is star-shaped on [0,∞). From Khaledi and Kochar (2000), it is known that \(\lambda\ge\tilde{\lambda}\) implies

By Lemma 3.2, it is enough to show

(3.5)

i.e.,

which follows from the assumption. □

One may wonder whether the similar result is true for other order statistics? The question can be partly answered by using the following two lemmas.

Lemma 3.4

(Kochar and Xu 2011)

Let X 1,…,X p be i.i.d. (independent and identically distributed) exponential random variables with hazard rate λ 1, and let X p+1,…,X n be another set of i.i.d. exponential random variables with hazard rate λ 2. Let Y 1,…,Y n be i.i.d. exponential random variables. Then, for k=1,…,n,

Lemma 3.5

(Bon and Pǎltǎnea 2006)

Let X 1,…,X p be independent exponential random variables with hazard rates λ i , i=1,…,n. Let Y 1,…,Y n be independent exponential random variables with a common hazard rate λ. Then,

where

Mimicking the proof of Theorem 3.3, and using Lemmas 3.4 and 3.5, one may prove the following result.

Theorem 3.6

Let X 1,…,X p be i.i.d. random variables with the common survival distribution \(\bar{F}^{\lambda_{1}}\), and let X p+1,…,X n be another set of i.i.d. random variables with common survival distribution \(\bar{F}^{\lambda_{2}}\), and let Y 1,…,Y n be a random sample from a distribution with common survival distribution \(\bar{F}^{\lambda}\) where \(\lambda\ge\hat{\lambda}\). If

then

4 Examples and applications

In the following subsections, we will present some distributions for which Theorem 3.3 and Theorem 3.6 are applicable.

4.1 Weibull distribution

Weibull distribution is the most commonly used distribution in reliability and life testing. A random variable X i is said to follow Weibull distribution W(a,b i ) if it has a survival function

(4.1)

It is seen that R(x)=x α and \(\lambda_{i}=b_{i}^{-\alpha}\), and

Hence, Eq. (3.4) is satisfied. Further, note that λ i depends on the scale parameter b i , and the star order is scale invariant. The following results follow from Theorems 3.3 and 3.6, respectively.

Corollary 4.1

Let X 1,…,X n be independent Weibull random variables W(α,b i ). Let Y 1,…,Y n be a random sample of size n from a Weibull distribution W(α,b). Then,

Corollary 4.2

Let X 1,…,X p be independent Weibull random variables W(α,b 1), and X p+1,…,X n be other independent Weibull random variables W(α,b 2). Let Y 1,…,Y n be a random sample of size n from a Weibull distribution W(α,b). Then,

In the following, we present an example to illustrate Corollaries 4.1 and 4.2.

Example 4.3

Assume X 1,X 2,X 3 are independent Weibull random variables W(1.5,8), W(1.5,3) and W(1.5,1), respectively. Assume Y 1,Y 2,Y 3 are i.i.d. Weibull random variables W(1.5,4). Figure 1 plots the density functions of X 3:3 and Y 3:3. It is seen that the density function of X 3:3 is more skewed than that of Y 3:3, which coincides with Corollary 4.1. Using Mathematica, it can be computed that E(X 3:3)≈7.58, and Var(X 3:3)≈21.39. Therefore, the coefficient of variation for X 3:3 is

Similarly, the coefficient of variation for Y 3:3 is

Hence, cv(X 3:3)>cv(Y 3:3).

Fig. 1
figure 1

Density function plots of X 3:3 and Y 3:3

Full size image

Now, assume that W 1,W 2,W 3 are independent Weibull random variables W(1.5,8), W(1.5,8) and W(1.5,1), respectively. Assume Y 1,Y 2,Y 3 are i.i.d. Weibull random variables W(1.5,17/3). Figure 2 plots the density functions of W 2:3 and Y 2:3. It is seen that the density function of W 2:3 is more skewed than that of Y 2:3, which coincides with Corollary 4.2. The coefficient of variations, by Mathematica, are

Therefore, cv(W 2:3)>cv(Y 2:3).

Fig. 2
figure 2

Density function plots of W 2:3 and Y 2:3

Full size image

4.2 Pareto distribution

Pareto distribution (also known as power-law distribution) is one of the most important distributions in extreme value theory. It is known that for any distribution with regular variation tails, the tail of the distribution function can be approximated by Pareto distribution (cf. Resnick 2007).

Recall that a random variable X i is said to follow Pareto distribution, if its survival function can be represented as

(4.2)

It is easy to check that

is increasing in x. Therefore, the following two results follow from Theorems 3.3 and 3.6, respectively.

Corollary 4.4

Let X 1,…,X n be independent Pareto random variables with shape parameter λ i for i=1,…,n. Let Y 1,…,Y n be a random sample from a Pareto distribution with \(\lambda\ge\tilde{\lambda}\). Then,

Corollary 4.5

Let X 1,…,X p be independent Pareto random variables with shape parameter λ 1, and X p+1,…,X n be independent Pareto random variables with shape parameter λ 2. Let Y 1,…,Y n be a random sample from a Pareto distribution with parameter \(\lambda\ge\hat{\lambda}\). Then,

One may wonder whether the restriction on λ in Corollary 4.4 can be removed? Unfortunately, the following example gives a negative answer.

Example 4.6

Assume X 1,X 2,X 3 are independent Pareto random variables with (λ 1,λ 2,λ 3)=(5,5,8), and assume Y 1,Y 2,Y 3 are i.i.d. Pareto random variables with (λ 1,λ 2,λ 3)=(3,3,3). It is seen that \(\tilde{\lambda}= (\lambda_{1}\lambda_{2}\lambda_{3} )^{1/3}=5.84\). It can be computed that

Hence, cv(X 3:3)<cv(Y 3:3), which reveals that the restriction can not be dropped.

4.3 Lomax distribution

Lomax distribution (also known as Pareto Type II distribution) is often used in the extreme value theory since it is a heavy tailed distribution. The survival function of a Lomax random variable X i can be presented as, for x≥0,

It can be verified that

Taking derivative with respect to x, it holds that

Therefore, the following two results follow from Theorems 3.3 and 3.6, respectively.

Corollary 4.7

Let X 1,…,X n be independent Lomax random variables with parameter λ i for i=1,…,n. Let Y 1,…,Y n be a random sample from a Lomax distribution with parameter \(\lambda\ge\tilde{\lambda}\). Then,

Corollary 4.8

Let X 1,…,X p be independent Lomax random variables with parameter λ 1, and X p+1,…,X n be independent Lomax random variables with parameter λ 2. Let Y 1,…,Y n be a random sample from a Lomax distribution with parameter \(\lambda\ge\hat{\lambda}\). Then,

5 Discussion

Note that in Eq. (4.2) we assume that X i ’s have different shape parameters. One natural question is whether we can have a similar result for different scale parameters. We answer the question in this section. Assume that random variable X i has the survival function

Theorem 5.1

Let X 1,…,X n be independent Pareto random variables with the same shape parameter λ, and different scale parameters b i for i=1,…,n. Let Y 1,…,Y n be a random sample from a Pareto distribution with shape parameter λ and scale parameter b. Then,

Proof

The distribution function of X n:n is, for x≥max{b 1,…,b n },

Similarly, the distribution function of Y n:n is, for x≥b,

According to definition, it is enough to prove that

is decreasing in x≥max{b 1,…,b n }, which is equivalent to proving

(5.1)

Denoting x i =(b i /x)λ, and simplifying Eq. (5.1), we have

i.e.,

Therefore, it is enough to prove that

which is obvious since x i ∈(0,1). Hence, the result follows. □

Similarly, one can prove the following result directly from the definition.

Theorem 5.2

Let X 1,…,X n be independent Pareto random variables with the same shape parameter λ, and different scale parameters b i for i=1,…,n. Let Y 1,…,Y n be a random sample from a Pareto distribution with shape parameter λ and scale parameter b. Then,

There are many interesting problems on this topic. For example, under the assumption of Lemma 3.1, whether one can prove X k:n ≥⋆ Y k:n ? Numerical evidence makes us believe that it is correct. However, we are unable to prove it so far. Another interesting problem is if we assume X 1,…,X n are independent random variables with X i having distribution function F(λ i t), i=1,…,n, and Y 1,…,Y n is a random sample from a distribution with the common distribution F(λt), under what conditions, we can compare order statistics from those two samples in the sense of star ordering?