摘要:
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.
Abstract:
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 $(f(\mu_{1}),\cdots,f(\mu_{n}))\succeq_{\text{m}}(f(\mu^{*}_{1}),\cdots,f(\mu^{*}_{n}))$, 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 $X_{i}\sim N(\mu,\sigma_i^2)$ and $X^*_{i}\sim N(\mu,\sigma_i^{*2})$, $i=1,\cdots,n$. We obtain that $({1}/{\sigma_{1}},\cdots,{1}/{\sigma_{n}})\succeq_{\text{m}} ({1}/{\sigma^{*}_{1}},\cdots,{1}/{\sigma^{*}_{n}})$ implies that $X_{(1)}\leq_{\text{st}}X^*_{(1)}$ and $X_{(n)}\geq_{\text{st}}X^*_{(n)}$.
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