Theoretical Statistics and Mathematics Unit, ISI Delhi

Ordering convolutions of gamma random variables

by Baha-Eldin Khaledi and Subhash C. Kochar

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 ${\bf a}, {\bf b}$ are in $R $. 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 [Statistics \& Probability Letters 43 (1999), 321-324] and
Korwar [J. Multivariate Analysis 80 (2002), 344-357] from usual
majorization to $p$-larger ordering and leads to better bounds on
various quantities of interest.

isid/ms/2004/02 [fulltext]

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