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June, 1994 Invalidity of Bootstrap for Critical Branching Processes with Immigration
T. N. Sriram
Ann. Statist. 22(2): 1013-1023 (June, 1994). DOI: 10.1214/aos/1176325509


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.


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T. N. Sriram. "Invalidity of Bootstrap for Critical Branching Processes with Immigration." Ann. Statist. 22 (2) 1013 - 1023, June, 1994.


Published: June, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0807.62063
MathSciNet: MR1292554
Digital Object Identifier: 10.1214/aos/1176325509

Primary: 60J80
Secondary: 62G09

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.22 • No. 2 • June, 1994
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