EXTREME(LY) MEAN(INGFUL): SEQUENTIAL FORMATION OF A QUALITY GROUP

• 成果类型: Article
• 署名作者: Krieger, Abba M.; Pollak, Moshe; Samuel-Cahn, Ester
• 署名单位: University of Pennsylvania; Hebrew University of Jerusalem
• 刊物名称: ANNALS OF APPLIED PROBABILITY
• ISSN/ISSBN: 1050-5164
• DOI: 10.1214/10-AAP684
• 发表日期: 2010
• 页码: 2261-2294
• 关键词: selection

摘要

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 (Y) over bar (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, (Y) over bar (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) ->(x ->infinity) 0, then T(k) is of approximate order k(2). When beta > 1, the asymptotic results for (Y) over bar (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 >= 1, in probability for moderate sized beta and in distribution when beta is large.