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Invalidity of bootstrap for critical branching processes with immigration. (English) Zbl 0807.62063
In this article it is assumed that the Galton-Watson process has the initial size 1 and the offspring and immigration probability mass functions are of the type \[ p_ \theta (n)= a(n) \theta^ n/ A(\theta), \qquad q_ \varphi (k)= b(k)\varphi^ k/B (\varphi), \] respectively, where \(\{a(n)\}\) and \(\{b(k)\}\) are known nonnegative sequences, \(A(\theta)= \sum_{n=0}^ \infty a(n) \theta^ n\) and \(B(\varphi)= \sum_{k=0}^ \infty b(k) \varphi^ k\). Let \(m\) be the offspring mean. This article considers the case when the observations consist of generation sizes of the branching process with immigration together with the immigration component of each generation.
It is shown that the bootstrap version of the standardized maximum likelihood estimator (s.m.l.e.) does not have the same limiting distribution as the s.m.l.e. under the assumption that \(m=1\) (critical case). In fact, given the sample, the value of the conditional distribution function of the bootstrap version of the s.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 result [in \(D^ + [0, \infty)\)] is established using operator semigroup convergence theorems.
Reviewer: V.Topchij (Omsk)

62M05 Markov processes: estimation; hidden Markov models
62F12 Asymptotic properties of parametric estimators
62G09 Nonparametric statistical resampling methods
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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