Sankhya: The Indian Journal of Statistics

2004, Volume 66, Pt. 3, 466--473

Ordering  Convolutions of  Gamma Random Variables


Baha-Eldin Khaledi, Statistical Research Center, Tehran and Razi University, Kermanshah, Iran
Subhash C. Kochar, Indian Statistical Institute, New Delhi, India

SUMMARY. Let  $a_{(i)}$ and $b_{(i)}$ be the $i$th smallest components of    ${\bf a}=(a_1,\ldots,a_n)$ and ${\bf b}=(b_1,\ldots,b_n)$, respectively,  where ${\bf a}, {\bf b} \in \IR $.  The vector ${\bf a}$ is said to be $p$-larger than the vector   ${\bf b}$ (denoted by ${\bf a} \stackrel {p} \succeq {\bf b}$  ) if  $\prod_{i=1}^{k}a_{(i)} \le \prod_{i=1}^{k} b_{(i)},\mbox{ for } k=1,\ldots,n$. Let $X_{\lambda_1},\ldots,X_{\lambda_n}$ be independent random variables such that  $X_{\lambda_i}$ has gamma distribution with shape parameter $a \ge 1$  and  scale parameter $ \lambda_i$, $i= 1, \ldots, n$.  It is shown that if $\mbox {\boldmath $ \lambda $} \stackrel {p} \succeq   \mbox {\boldmath $ \lambda ^*$}$, then $\sum_{i=1}^n X_{\lambda_i}$ is greater than  $\sum_{i=1}^n X_{\lambda_i^*}$ according to dispersive  as well as  hazard rate orderings. This strengthens the results of Kochar and Ma (1999) and Korwar (2002) from usual majorization to $p$-larger ordering and leads to better bounds on various quantities of interest.

AMS (1991) subject classification}. 60E15, 62N05, 62D05.

Key words and phrases. Schur functions, majorization,  $p$-larger ordering, log-concave density, dispersive  ordering and  hazard rate ordering.

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