[Paper Review] On Stochastic Comparisons of Order Statistics from Heterogeneous Exponential Samples

Yaming Yu|arXiv (Cornell University)|Jul 22, 2016

Statistical Distribution Estimation and Applications11 references3 citations

TL;DR

This paper resolves two long-standing open problems in stochastic order theory by proving that the kth order statistic from a heterogeneous exponential sample stochastically dominates that from a homogeneous sample under star, hazard rate, and dispersive orderings. The key result establishes that $ Y_{k:n} \leq_* X_{k:n} $, confirming conjectures by Xu and Balakrishnan (2012) and Pältănea (2008), and extends to general spacings via convolution and mixture closure properties of the star order.

ABSTRACT

We show that the $k$th order statistic from a heterogeneous sample of $n\geq k$ exponential random variables is larger than that from a homogeneous exponential sample in the sense of star ordering, as conjectured by Xu and Balakrishnan (2012). As a consequence, we establish hazard rate ordering for order statistics between heterogeneous and homogeneous exponential samples, resolving an open problem of Pǎltǎnea (2008). Extensions to general spacings are also presented.

Motivation & Objective

• To resolve the conjecture by Xu and Balakrishnan (2012) that the kth order statistic from a heterogeneous exponential sample dominates the homogeneous case under star ordering.
• To confirm Pältănea's (2008) conjecture that $ Y_{k:n} \leq_{\rm hr} X_{k:n} $ under the same condition.
• To extend stochastic comparisons from order statistics to general spacings, such as $ X_{k:n} - X_{m:n} $, using distributional representations and mixture closure.
• To establish sufficient and necessary conditions for stochastic dominance in terms of elementary symmetric functions and rate parameters.

Proposed method

• Prove $ Y_{k:n} \leq_* X_{k:n} $ using the star order and a representation of order statistics via convolution with exponential distributions.
• Apply Lemma 1 (closure of star order under mixtures) to handle weighted sums of distribution functions from subsamples.
• Use the variation diminishing property of TP2 kernels to establish unique crossing points in distribution functions, ensuring strict star ordering.
• Derive distributional representations for spacings $ X_{k:n} - X_{m:n} $ as mixtures of order statistics from reduced samples.
• Characterize stochastic dominance for spacings using conditions on $ \gamma $, $ \lambda_i $, and elementary symmetric functions $ s_{k-m}^{[\mathbf{r}]} $.
• Leverage Maclaurin's inequality and log-concavity arguments to compare convolution structures of homogeneous and heterogeneous samples.

Experimental results

Research questions

• RQ1 Does the kth order statistic from a heterogeneous exponential sample stochastically dominate that from a homogeneous sample under the star order?
• RQ2 Is the condition $ \gamma \geq \left( \binom{n}{k}^{-1} s_k(\lambda_1,\ldots,\lambda_n) \right)^{1/k} $ equivalent to hazard rate ordering between $ Y_{k:n} $ and $ X_{k:n} $?
• RQ3 Can the stochastic comparison results for order statistics be extended to general spacings $ X_{k:n} - X_{m:n} $?
• RQ4 What are the necessary and sufficient conditions for $ Y_{k:n} - Y_{m:n} \leq_{\rm order} X_{k:n} - X_{m:n} $ under stochastic, hazard rate, or dispersive order?
• RQ5 Can the star order be used to derive likelihood ratio ordering results, or is a different approach required?

Key findings

• The paper proves $ Y_{k:n} \leq_* X_{k:n} $, confirming the conjecture of Xu and Balakrishnan (2012) for all $ 1 \leq k \leq n $.
• It establishes that $ Y_{k:n} \leq_{\rm hr} X_{k:n} $ if and only if $ \gamma \geq \left( \binom{n}{k}^{-1} s_k(\lambda_1,\ldots,\lambda_n) \right)^{1/k} $, resolving Pältănea's (2008) open problem.
• The star order implies both hazard rate and dispersive ordering, so $ Y_{k:n} \leq_{\rm hr} X_{k:n} $ and $ Y_{k:n} \leq_{\rm disp} X_{k:n} $ follow directly from $ Y_{k:n} \leq_* X_{k:n} $.
• For general spacings, $ Y_{k:n} - Y_{m:n} \leq_* X_{k:n} - X_{m:n} $ holds for $ 1 \leq m < k \leq n $, extending the main result.
• The condition for $ Y_{k:n} - Y_{m:n} \leq_{\rm st} X_{k:n} - X_{m:n} $ is given by a complex inequality involving permutations and symmetric functions of the $ \lambda_i $.
• In the special case $ m=1, k=n $, the condition reduces to $ \gamma \geq \left( \frac{\prod_{i=1}^n \lambda_i}{\Lambda/n} \right)^{1/(n-1)} $, recovering Theorem 4.1 of Xu and Balakrishnan (2012).