CHINESE JOURNAL OF APPLIED PROBABILITY AND STATIST 2009, 25(4) 381-388 DOI:      ISSN: 1001-4268 CN: 31-1256

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On Stochastic Orders for Order Statistics\\from Normal Distributions

Huang Yongjun,Zhang Xinsheng

Department of Statistics, School of Management, Fudan University,

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
$X_{(n)}\geq_{\text{st}}X^*_{(n)}$.

Keywords�� Normal distribution   stochastic orders   majorization   order statistics.
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