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Theory & Methods: Approximations of distributions for some standardized partial sums in sequential analysis
Author(s) -
Liu Wei,
Wang Nan,
Wang Suojin
Publication year - 2002
Publication title -
australian and new zealand journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 1369-1473
DOI - 10.1111/1467-842x.00212
Subject(s) - mathematics , random variable , central limit theorem , independent and identically distributed random variables , zero (linguistics) , sequence (biology) , combinatorics , limit (mathematics) , distribution (mathematics) , approximations of π , simple (philosophy) , stopping time , discrete mathematics , statistics , mathematical analysis , philosophy , linguistics , epistemology , biology , genetics
In sequential analysis it is often necessary to determine the distributions of √t Y t and/or √a Y t , where t is a stopping time of the form t = inf{ n ≥ 1 : n+S n +ξ n > a }, Y n is the sample mean of n independent and identically distributed random variables (iidrvs) Y i with mean zero and variance one, S n is the partial sum of iidrvs X i with mean zero and a positive finite variance, and { ξ n } is a sequence of random variables that converges in distribution to a random variable ξ as n →∞ and ξ n is independent of ( X n+1 , Y n+1 ), (X n+2 , Y n+2 ), . . . for all n ≥ 1. Anscombe’s (1952) central limit theorem asserts that both √t Y t and √a Y t are asymptotically normal for large a , but a normal approximation is not accurate enough for many applications. Refined approximations are available only for a few special cases of the general setting above and are often very complex. This paper provides some simple Edgeworth approximations that are numerically satisfactory for the problems it considers.