  The present paper studies the limiting behavior of the average score of a
sequentially selected group of items or individuals, the underlying
distribution of which, $F$, belongs to the Gumbel domain of attraction of
extreme value distributions. This class contains the Normal, Lognormal, Gamma,
Weibull and many other distributions. The selection rules are the "better than
average" ($\beta=1$) and the "$\beta$-better than average" rule, defined as
follows. After the first item is selected, another item is admitted into the
group if and only if its score is greater than $\beta$ times the average score
of those already selected. Denote by $\bar{Y}_k$ the average of the $k$ first
selected items, and by $T_k$ the time it takes to amass them. Some of the key
results obtained are: under mild conditions, for the better than average rule,
$\bar{Y}_k$ less a suitable chosen function of $\log k$ converges almost surely
to a finite random variable. When $1-F(x)=e^{-[x^{\alpha}+h(x)]}$, $\alpha>0$
and $h(x)/x^{\alpha}\stackrel{x\rightarrow \infty}{\longrightarrow}0$, then
$T_k$ is of approximate order $k^2$. When $\beta>1$, the asymptotic results for
$\bar{Y}_k$ are of a completely different order of magnitude. Interestingly,
for a class of distributions, $T_k$, suitably normalized, asymptotically
approaches 1, almost surely for relatively small $\beta\ge1$, in probability
for moderate sized $\beta$ and in distribution when $\beta$ is large.
