Contiguous alternatives which preserve Cramér-type large deviations for a general class of statistics

✍ Scribed by Tiee-Jian Wu

• Publisher: Springer Japan
• Year: 1989
• Tongue: English
• Weight: 498 KB
• Volume: 41
• Category: Article
• ISSN: 0020-3157
• DOI: 10.1007/bf00057731

✦ Synopsis

Let PN and QN, N>_ 1, be two possible probability distributions of a random vector Xu = (Xm .... ,XNN), whose components are N independent. Suppose PN and QN have respective densities ps = iH=~f(xNi --N N ON) and qN ----i ~=l = f(xNi-ONi), where ON = N-1 i=1 y' ONi, such that ~i~umax [ONi-ON[ = O(N-1/2), f(x) > 0 for almost every real x, f is absolutely continuous, and sup = [f'(x -O)]2/f(x)dx < ~ for some 00 > 0. The conti-0.~ 0-< 0o guity of {qN} to {pN} is well known. In this paper it is proven that under these conditions {Qu} preserves C.-T.L.D. (Cram6r-type large deviation) from {PN} for a general class of statistics ~r which includes R-, U-and L-statistics as members. That means, for any {Su = SN(XN)} from ~, a C.-T.L.D. theorem with range C <_ x <_ o(N ~) (any C _< 0), 0 < 6 _< 4 -1, holds for {SN} under {PN}, implying that the same theorem holds for {Su} under {QN}. It also provides a quick and simple way to establish C.-T.L.D. results for statistics under {QN}.