Premium
Asymptotics for multisample statistics
Author(s) -
Vaillancourt Jean
Publication year - 1995
Publication title -
canadian journal of statistics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.804
H-Index - 51
eISSN - 1708-945X
pISSN - 0319-5724
DOI - 10.2307/3315443
Subject(s) - mathematics , law of the iterated logarithm , estimator , infinity , dimension (graph theory) , central limit theorem , separable space , sequence (biology) , combinatorics , iterated function , logarithm , statistics , iterated logarithm , mathematical analysis , biology , genetics
Consider a family of square‐integrable R d ‐valued statistics S k = S k (X 1,k1 ; X 2,k2 ;…; X m,km ), where the independent samples X i,kj respectively have k i i.i.d. components valued in some separable metric space X i . We prove a strong law of large numbers, a central limit theorem and a law of the iterated logarithm for the sequence {S k }, including both the situations where the sample sizes tend to infinity while m is fixed and those where the sample sizes remain small while m tends to infinity. We also obtain two almost sure convergence results in both these contexts, under the additional assumption that S k is symmetric in the coordinates of each sample X i,kj. Some extensions to row‐exchangeable and conditionally independent observations are provided. Applications to an estimator of the dimension of a data set and to the Henze‐Schilling test statistic for equality of two densities are also presented.