The Annals of Statistics

Invalidity of Bootstrap for Critical Branching Processes with Immigration

T. N. Sriram

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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 $D^+\lbrack 0, \infty)$] result is established using operator semigroup convergence theorems.

Article information

Ann. Statist. Volume 22, Number 2 (1994), 1013-1023.

First available in Project Euclid: 11 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Secondary: 62G09: Resampling methods

Critical branching processes asymptotic invalidity operator semigroup convergence theorems diffusion process generator parametric bootstrap


Sriram, T. N. Invalidity of Bootstrap for Critical Branching Processes with Immigration. Ann. Statist. 22 (1994), no. 2, 1013--1023. doi:10.1214/aos/1176325509.

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