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The weak convergence of the empirical process with random sample size

Published online by Cambridge University Press:  24 October 2008

Ronald Pyke
Affiliation:
University of Washington

Extract

In many applied probability models, one is concerned with a sequence {Xn: n > 1} of independent random variables (r.v.'s) with a common distribution function (d.f.), F say. When making statistical inferences within such a model, one frequently must do so on the basis of observations X1, X2,…, XN where the sample size N is a r.v. For example, N might be the number of observations that it was possible to take within a given period of time or within a fixed cost of experimentation. In cases such as these it is not uncommon for statisticians to use fixed-sample-size techniques, even though the random sample size, N, is not independent of the sample. It is therefore important to investigate the operating characteristics of these techniques under random sample sizes. Much work has been done since 1952 on this problem for techniques based on the sum, X1 + … + XN (see, for example, the references in (3)). Also, for techniques based on max(X1, X2, …, XN), results have been obtained independently by Barndorff-Nielsen(2) and Lamperti(9).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

REFERENCES

(1)Allen, J. L. and Beekman, J. A.A statistical test involving a random number of random variables. Ann. Math. Statist. 37 (1966), 13051311.CrossRefGoogle Scholar
(2)Barndorff-Nielsen, O.On the limit distribution of the maximum of a random number of independent random variables. Acta. Math. Acad. Sci. Hungar. 15 (1964), 399403.CrossRefGoogle Scholar
(3)Billingsley, Patrick. Limit theorems for randomly selected partial sums. Ann. Math. Statist. 33 (1962), 8592.CrossRefGoogle Scholar
(4)Darling, D. A.The Kolmogorov-Smirnov, Cramér-von Mises tests. Ann. Math. Statist. 28 (1957), 823838.CrossRefGoogle Scholar
(5)Donsker, M.Justification and extension of Doob's heuristic approach to the Kolmogorov-Smirnov theorems. Ann. Math. Statist. 23 (1952), 277281.CrossRefGoogle Scholar
(6)Doob, J. L.Stochastic processes (Wiley; New York, 1953).Google Scholar
(7)Dudley, R. M.Weak convergence of probabilities on non-separable metric spaces and empirical measures on Euclidean spaces. Illinois J. Math. 10 (1966), 109126.CrossRefGoogle Scholar
(8)Kac, M.On deviations between theoretical and empirical distributions. Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 252257.CrossRefGoogle ScholarPubMed
(9)Lamperti, John. On the maximum of a random number of random variables. (1964) (Unpublished).Google Scholar
(10)Lévy, Paul. Processus Stochastiques et Mouvement Brownien (Gauthier-Villars; Paris, 1948).Google Scholar
(11)Prokhorov, Yu. V.Convergence of random processes and limit theorems in probability theory. Theor. Probability Appl. 1 (1956), 157214.CrossRefGoogle Scholar
(12)Pyke, Ronald and Shorack, Galen. Weak convergence of the two-sample empirical process and the Chernoff-Savage theorem. (Unpublished.)Google Scholar
(13)Skorokhod, A. V.Limit theorems for stochastic processes. Theor. Probability Appl. 1 (1956), 261290.CrossRefGoogle Scholar