On the conjecture of Kochar and Korwar (Q962221)

Let \(X_1,X_2,\ldots,X_n\) be independent heterogeneous exponential random variables, and let \(X_{1:n}\leq X_{2:n}\leq\cdots\leq X_{n:n}\) be the corresponding order statistics. The authors study the random variables \(D_{i:n}\doteq X_{i:n}-X_{i-1:n}\) and \(D_{i:n}^*\doteq(n-i-1)X_{i:n}-X_{i-1:n}\) for \(i=1,2,\ldots,n\), with \(X_{0:n}\equiv0\), which are called the spacings and the normalized spacings, respectively. Denoting the hazard rate stochastic order by \(\leq_{\text{hr}}\), the authors show that \(D_{2:n}\leq_{\text{hr}}D_{3:n}\) and \(D_{2:n}^*\leq_{\text{hr}}D_{3:n}^*\) for any \(n\). Furthermore, they also show that \(D_{3:4}\leq_{\text{hr}}D_{4:4}\) and \(D_{3:4}^*\leq_{\text{hr}}D_{4:4}^*\).