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X^*_n$�ֱ������̬�ֲ�$N(\mu_i,\sigma^2)$��$N(\mu^*_i,\sigma^2)$,
��$X_{(1)}$,
$X^*_{(1)}$�ֱ��ʾ$X_1,\cdots,X_n$��$X^*_1,\cdots,X^*_n$�ļ�С����ͳ����,
��$X_{(n)}$, $X^*_{(n)}$�ֱ��ʾ$X_1,\cdots,X_n$��$X^*_1,\cdots$,
$X^*_n$�ļ������ͳ����. ���ǵõ������½��:
(i)\,��������ϸ�����$f$ʹ��$(f(\mu_{1}),\cdots,f(\mu_{n}))
\succeq_{\text{m}}$ $(f(\mu^{*}_{1}),\cdots,f(\mu^{*}_{n}))$,
��$f'(x)f''(x)\!\geq\!0$, ��$X_{(1)}\!\leq_{\text{st}}\!X^*_{(1)}$;
(ii)\,��������ϸ�����$f$ʹ��$(f(\mu_{1})$,
$\cdots,f(\mu_{n}))\succeq_{\text{m}}(f(\mu^{*}_{1}),\cdots,f(\mu^{*}_{n}))$,
��$f'(x)f''(x)\leq 0$, ��$X_{(n)}\geq_{\text{st}}X^*_{(n)}$.
(iii)\,��$X_{1},X_{2},\cdots,X_{n}$��\, $X^*_{1},X^*_{2},\cdots,
X^*_{n}$�ֱ������̬�ֲ�$N(\mu,\sigma_i^2)$��$N(\mu,\sigma_i^{*2})$,
��$({1}/{\sigma_{1}},\cdots,{1}/{\sigma_{n}})\succeq_{\text{m}}
({1}/{\sigma^{*}_{1}},\cdots,{1}/{\sigma^{*}_{n}})$,
����$X_{(1)}\leq_{\text{st}}X^*_{(1)}$��$X_{(n)}\geq_{\text{st}}X^*_{(n)}$ͬʱ����.
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
$X_{(n)}\geq_{\text{st}}X^*_{(n)}$.