Annals of Applied Probability

Extreme(ly) mean(ingful): Sequential formation of a quality group

Abba M. Krieger, Moshe Pollak, and Ester Samuel-Cahn

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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” (β = 1) and the “β-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 β times the average score of those already selected. Denote by k the average of the k first selected items, and by Tk the time it takes to amass them. Some of the key results obtained are: under mild conditions, for the better than average rule, k less a suitable chosen function of logk converges almost surely to a finite random variable. When 1 − F(x) = e−[xα + h(x)], α > 0 and h(x) / xαx → ∞ 0, then Tk is of approximate order k2. When β > 1, the asymptotic results for k are of a completely different order of magnitude. Interestingly, for a class of distributions, Tk, suitably normalized, asymptotically approaches 1, almost surely for relatively small β ≥ 1, in probability for moderate sized β and in distribution when β is large.

Article information

Ann. Appl. Probab., Volume 20, Number 6 (2010), 2261-2294.

First available in Project Euclid: 19 October 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G99: None of the above, but in this section
Secondary: 62F07: Ranking and selection 60F15: Strong theorems

Selection rules averages better than average sequential observations asymptotics


Krieger, Abba M.; Pollak, Moshe; Samuel-Cahn, Ester. Extreme(ly) mean(ingful): Sequential formation of a quality group. Ann. Appl. Probab. 20 (2010), no. 6, 2261--2294. doi:10.1214/10-AAP684.

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  • [1] Abramowitz, M. and Stegun, I. A. (1964). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series 55. Dover, New York.
  • [2] Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. II, 2nd ed. Wiley, New York.
  • [3] Krieger, A. M., Pollak, M. and Samuel-Cahn, E. (2007). Select sets: Rank and file. Ann. Appl. Probab. 17 360–385.
  • [4] Krieger, A. M., Pollak, M. and Samuel-Cahn, E. (2008). Beat the mean: Sequential selection by better than average rules. J. Appl. Probab. 45 244–259.
  • [5] Preater, J. (2000). Sequential selection with a better-than-average rule. Statist. Probab. Lett. 50 187–191.
  • [6] Robbins, H. and Siegmund, D. (1971). A convergence theorem for nonnegative almost supermartingales and some applications. In Optimizing Methods in Statistics (Proc. Sympos., Ohio State Univ., Columbus, Ohio, 1971) 233–257. Academic Press, New York.
  • [7] Resnick, S. I. (1987). Extreme Values, Regular Variation, and Point Processes. Applied Probability. A Series of the Applied Probability Trust 4. Springer, New York.