Approximations of distributions of some standardized partial sums in sequential analysis

Liu, Wei, Wang, Nan and Wang, Suojin (2002) Approximations of distributions of some standardized partial sums in sequential analysis. Australian & New Zealand Journal of Statistics, 44 (1), 109-119. (doi:10.1111/1467-842X.00212).

Record type: Article

Abstract

In sequential analysis it is often necessary to determine the distributions of ?tYt and/or ?a Yt, where t is a stopping time of the form t = inf{n ? 1 : n+Sn+ ?n> a}, Yn is the sample mean of n independent and identically distributed random variables (iidrvs) Yi with mean zero and variance one, Sn is the partial sum of iidrvs Xi with mean zero and a positive finite variance, and {?n} is a sequence of random variables that converges in distribution to a random variable ? as n?? and ?n is independent of (Xn+1, Yn+1), (Xn+2, Yn+2), . . . for all n ? 1. Anscombe's (1952) central limit theorem asserts that both ?t Yt and ?a Yt 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.

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