Let X
1,…,X
n
be independent exponential random variables with X
i
having hazard rate . Let Y
1,…,Y
n
be a random sample of size n from an exponential distribution with common hazard rate ̃λ = (∏
i=1
n
λ
i
)1/n
, the geometric mean of the λis. Let X
n:n
= max{X
1,…,X
n
}. It is shown that X
n:n
is greater than Y
n:n
according to dispersive as well as hazard rate orderings. These results lead to a lower bound for the variance of X
n:n
and an upper bound on the hazard rate function of X
n:n
in terms of . These bounds are sharper than those obtained by Dykstra et al. ((1997), J. Statist. Plann. Inference
65, 203–211), which are in terms of the arithmetic mean of the λ
i
s. Furthermore, let X
1
*,…,X
n
∗ be another set of independent exponential random variables with X
i
∗ having hazard rate λ
i
∗, i = 1,…,n. It is proved that if (logλ1,…,logλ
n
) weakly majorizes (logλ1
∗,…,logλ
n
∗, then X
n:n
is stochastically greater than X
n:n
∗.
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