ON THE LAST TIME AND THE NUMBER OF TIMES AN ESTIMATOR IS MORE THAN EPSILON FROM ITS TARGET VALUE
成果类型：
Article
署名作者：
HJORT, NL; FENSTAD, G
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
DOI：
10.1214/aos/1176348533
发表日期：
1992
页码：
469-489
关键词：
摘要：
Suppose theta(n) is a strongly consistent estimator for theta(0) in some i.i.d. situation. Let N(epsilon) and Q(epsilon) be, respectively, the last n and the total number of n for which theta(n) is at least epsilon away from theta(0). The limit distributions for epsilon(2)N(epsilon) and epsilon(2)Q(epsilon) as epsilon goes to zero are obtained under natural and weak conditions. The theory covers both parametric and nonparametric cases, multidimensional parameters and general distance functions. Our results are of probabilistic interest, and, on the statistical side, suggest ways in which competing estimators can be compared. In particular several new optimality properties for the maximum likelihood estimator sequence in parametric families are established. Another use of our results is ways of constructing sequential fixed-volume or shrinking-volume confidence sets, as well as sequential tests with power 1. The paper also includes limit distribution results for the last n and the number of n for which the supremum distance parallel-to F(n) - F parallel-to greater-than-or-equal-to epsilon, where F(n) is the empirical distribution function. Other results are reached for epsilon(5/2)N(epsilon) and epsilon(5/2)Q(epsilon) in the context of nonparametric density estimation, referring to the last time and the number of times where \f(n)(x) - f(x)\ greater-than-or-equal-to epsilon. Finally, it is shown that our results extend to several non-i.i.d. situations.