INVALIDITY OF BOOTSTRAP FOR CRITICAL BRANCHING-PROCESSES WITH IMMIGRATION
成果类型：
Article
署名作者：
SRIRAM, TN
署名单位：
University of North Carolina; University of North Carolina Chapel Hill
刊物名称：
ANNALS OF STATISTICS
ISSN/ISSBN：
0090-5364
DOI：
10.1214/aos/1176325509
发表日期：
1994
页码：
1013-1023
关键词：
摘要：
This article considers a case of parametric bootstrap when the observations consist of generation sizes of the branching process with immigration together with the immigration component of each generation. Suppose we estimate the offspring mean m by the maximum likelihood estimator (m.l.e.). It is then shown that the bootstrap version of the standardized m.l.e. does not have the same limiting distribution as the standardized m.l.e., under the assumption that m = 1 (critical case). In other words, the asymptotic validity does not hold for the parametric bootstrap in the critical case, In fact, given the sample, the value of the conditional distribution function of the bootstrap version of standardized m.l.e. defines a sequence of random variables whose limit (in distribution) is also shown to be a random variable, when m = 1. The approach used here is via a sequence of branching processes for which a general weak convergence [in D+[0, infinity)] result is established using operator semigroup convergence theorems.