ASYMPTOTIC SUPREMA OF AVERAGED RANDOM FUNCTIONS
成果类型：
Article
署名作者：
ZAMAN, A
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
DOI：
10.1214/aos/1176348390
发表日期：
1991
页码：
2145-2159
关键词：
摘要：
Suppose X(i) are i.i.d. random variables taking values in X, THETA is a parameter space and y: X x THETA --> R is a map. Define the averages S(n)(y, theta) = (1/n)SIGMA(i = 1)(n)y(X(i), theta) and the truncated expectations T(m)(y, theta) = E max(y(X1, theta), - m). Under the hypothesis of global dominance [i.e., E sup(THETA) y(X1, theta) < infinity] and some regularity conditions, the main result of the paper characterizes the asymptotic suprema of S(n) as follows. For any subset G of THETA, with probability 1, [GRAPHICS] This has immediate application to consistency of M-estimators. In particular, under global dominance, maxima of S(n) must converge to the same limit as the maxima of T(m)(y, theta) almost surely. We also obtain necessary and sufficient conditions for consistency resembling Huber's in the case of local dominance [i.e., each theta is-an-element-of THETA has a neighborhood N(theta) such that E sup(psi) is-an-element-of N(theta)y(X1, psi) < infinity]. In this case there must exist a function b(theta) greater-than-or-equal-to 1 such that y/b is globally dominated and maxima of T(m)(y/b, theta) converge.