关于正态分布的次序统计量的随机序

设$X_1,X_2,cdots,X_n$和$X^*_1,X^*_2,cdots,X^*_n$分别服从正态分布$N(mu_i,sigma^2)$和$N(mu^*_i,sigma^2)$,以$X_{(1)}$,$X^*_{(1)}$分别表示$X_1,cdots,X_n$和$X^*_1,cdots,X^*_n$的极小次序统计量,以$X_{(n)}$, $X^*_{(n)}$分别表示$X_1,cdots,X_n$和$X^*_1,cdots$,$X^*_n$的极大次序统计量. 我们得到了如下结果:(i),如果存在严格单调函数$f$使得$(f(mu_{1}),cdots,f(mu_{n}))succeq_{text{m}}$ $(f(mu^{*}_{1}),cdots,f(mu^{*}_{n}))$,且$f'(x)f'(x)!geq!0$, 则$X_{(1)}!leq_{text{st}}!X^*_{(1)}$;(ii),如果存在严格单调函数$f$使得$(f(mu_{1})$,$cdots,f(mu_{n}))succeq_{text{m}}(f(mu^{*}_{1}),cdots,f(mu^{*}_{n}))$,且$f'(x)f'(x)leq 0$, 则$X_{(n)}geq_{text{st}}X^*_{(n)}$.(iii),设$X_{1},X_{2},cdots,X_{n}$和, $X^*_{1},X^*_{2},cdots,X^*_{n}$分别服从正态分布$N(mu,sigma_i^2)$和$N(mu,sigma_i^{*2})$,若$({1}/{sigma_{1}},cdots,{1}/{sigma_{n}})succeq_{text{m}}({1}/{sigma^{*}_{1}},cdots,{1}/{sigma^{*}_{n}})$,则有$X_{(1)}leq_{text{st}}X^*_{(1)}$和$X_{(n)}geq_{text{st}}X^*_{(n)}$同时成立.

On Stochastic Orders for Order Statisticsfrom Normal Distributions

In this paper we obtain some new results on stochasticorders for order statistics from normal distributions. Let$X_1,cdots,X_n,X^*_1,cdots,X^*_n$ be independent normal randomvariables with $X_{i}sim N(mu_i,sigma^2)$ and $X^*_{i}simN(mu^*_i,sigma^2)$, $i=1,cdots,n$. Suppose that there exists astrictly monotone function $f$ such that$(f(mu_{1}),cdots,f(mu_{n}))succeq_{text{m}}(f(mu^{*}_{1}),cdots,f(mu^{*}_{n}))$,we prove that: (i) if $f'(x)f'(x)geq 0$, then$X_{(1)}leq_{text{st}}X^*_{(1)}$; (ii) if $f'(x)f'(x)leq 0$,then $X_{(n)}geq_{text{st}}X^*_{(n)}$. Moreover, let $X_{i}simN(mu,sigma_i^2)$ and $X^*_{i}sim N(mu,sigma_i^{*2})$,$i=1,cdots,n$. We obtain that$({1}/{sigma_{1}},cdots,{1}/{sigma_{n}})succeq_{text{m}}({1}/{sigma^{*}_{1}},cdots,{1}/{sigma^{*}_{n}})$ implies that$X_{(1)}leq_{text{st}}X^*_{(1)}$ and$X_{(n)}geq_{text{st}}X^*_{(n)}$.