On Stochastic Orders for Order Statistics\\from Normal Distributions

In this paper we obtain some new results on stochastic orders for order statistics from normal distributions. Let X_1,\cdots,X_n,X^*_1,\cdots,X^*_n be independent normal random variables with X_i\sim N(\mu_i,\sigma^2) and X^*_i\sim N(\mu^*_i,\sigma^2), i=1,\cdots,n. Suppose that there exists a strictly monotone function f such that (f(\mu_1),\cdots,f(\mu_n))\succeq_\textm(f(\mu^*_1),\cdots,f(\mu^*_n)), we prove that: (i) if f'(x)f''(x)\geq 0, then X_(1)\leq_\textstX^*_(1); (ii) if f'(x)f''(x)\leq 0, then X_(n)\geq_\textstX^*_(n). Moreover, let X_i\sim N(\mu,\sigma_i^2) and X^*_i\sim N(\mu,\sigma_i^*2), i=1,\cdots,n. We obtain that (1/\sigma_1,\cdots,1/\sigma_n)\succeq_\textm (1/\sigma^*_1,\cdots,1/\sigma^*_n) implies that X_(1)\leq_\textstX^*_(1) and X_(n)\geq_\textstX^*_(n).