Mathematics – Probability
Scientific paper
2010-11-15
Annals of Applied Probability 2010, Vol. 20, No. 6, 2261-2294
Mathematics
Probability
Published in at http://dx.doi.org/10.1214/10-AAP684 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Inst
Scientific paper
10.1214/10-AAP684
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.
Krieger Abba M.
Pollak Moshe
Samuel-Cahn Ester
No associations
LandOfFree
Extreme(ly) mean(ingful): Sequential formation of a quality group does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Extreme(ly) mean(ingful): Sequential formation of a quality group, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Extreme(ly) mean(ingful): Sequential formation of a quality group will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-298456