NECESSARY AND SUFFICIENT CONDITIONS FOR ASYMPTOTIC NORMALITY OF L-STATISTICS

• 成果类型: Article
• 署名作者: MASON, DM; SHORACK, GR
• 署名单位: University of Washington; University of Washington Seattle
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/aop/1176989529
• 发表日期: 1992
• 页码: 1779-1804
• 关键词: order statistics; sums

摘要

It is now classical that the sample mean YBAR is known to be asymptotically normal with square-root n norming if and only if 0 < Var[Y] < infinity and with arbitrary norming if and only if the df of Y is in the domain of attraction of the normal df. Now let T(n) = n-1SIGMAc(ni)h(X(n:i)) for order statistics X(n:i) from a df F denote a general L-statistic subject to a bit of regularity; the key condition introduced into this problem in this paper is the regular variation of the score function J defining the c(ni)'s. We now define a rv Y by Y = K(xi), where xi is uniform (0, 1) and where dK = J dh(F-1). Then T(n) is shown to be asymptotically normal with square-root n norming if and only if 0 < Var[Y] < infinity and with arbitrary norming if and only if the df of Y is in the domain of attraction of the normal df. As it completely parallels the classical theorem, this theorem gives the right conclusion for L-statistics. In order to establish the necessity above, we also obtain a nice necessary and sufficient condition for the stochastic compactness of T(n) and give a representation formula for all possible subsequential limit laws.