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Asymptotic Representations of the Multivariate Empirical Distribution Function and Applications
Author(s) -
Ralescu Stefan
Publication year - 1986
Publication title -
australian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.434
H-Index - 41
eISSN - 1467-842X
pISSN - 0004-9581
DOI - 10.1111/j.1467-842x.1986.tb00705.x
Subject(s) - asymptotic distribution , mathematics , weak convergence , estimator , iterated logarithm , asymptotic analysis , point process , convergence of random variables , context (archaeology) , brownian motion , remainder , central limit theorem , law of the iterated logarithm , iterated function , statistics , logarithm , random variable , mathematical analysis , computer science , paleontology , computer security , arithmetic , asset (computer security) , biology
Summary Often, many complicated statistics used as estimators or test statistics take the form of the (multivariate) empirical distribution function evaluated at a random vector (V n ). Denote such statistics by S n . This paper describes methods for the study of various asymptotic properties of S n . First, under minimal assumptions, a weak asymptotic representation for S n is derived. This result may be used to show the asymptotic normality of S n . Second, under slightly more stringent regularity conditions, an almost sure representation of S n , with suitable order (as.) of the remainder term is studied and then a law of the iterated logarithm for S n , is derived. In this context, strong convergence results from a sequential point of view are also studied. Finally, weak convergence to a Brownian motion process is established. As an application, we show the limiting normality of S n , for a random number of summands.