Glivenko--Cantelli-type theorems for weighted empirical distribution functions based on uniform spacings
Let U1, U2,... be a sequence of independent r.v.'s having the uniform distribution on (0, 1). Let Fn be the empirical distribution based on the transformed uniform spacings Di,n:=G(nDi,n), i = 1, 2,..., n, where G is the exp(1) d.f. and Di,n is the ith spacing based on U1, U2,...,Un-1. The main purpose of this paper is the study of the almost sure behaviour of lim supn --> [infinity] [Delta]n,[alpha](q, ) and lim supn-->[infinity] [Lambda]n,r(q, ), where [Delta]n,[alpha](q, ) = sup0<t<1[n[alpha]Vb;Fn(t) - tVb;/(q(t)(1 - t))] and [Lambda]n,r(q, ) = [integral operator]10(vb;Fn(t) - tvb;/(q(t)(1 - t)))r dt for [alpha] [epsilon] [0, ), r > 0 and certain weight functions q and . Moreover, the weak behaviour of the statistics will be examined briefly. It turns out that compared with the uniform empirical process (i.i.d. case) the considered weighted Kolmogorov--Smirnov- and Cramer--von Mises-type statistics behave differently in the right tail only as far as almost sure convergence is concerned. There is no difference in the weak sense. The results can be applied to the study of linear combinations of functions of ordered spacings.
Year of publication: |
1992
|
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Authors: | Einmahl, John H. J. ; Zuijlen, Martien C. A. van |
Published in: |
Statistics & Probability Letters. - Elsevier, ISSN 0167-7152. - Vol. 13.1992, 5, p. 411-419
|
Publisher: |
Elsevier |
Keywords: | Glivenko-Cantelli theorems order statistics strong convergence uniform spacings weak convergence weighted empirical process |
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