## The Annals of Statistics

### Invalidity of Bootstrap for Critical Branching Processes with Immigration

T. N. Sriram

#### Abstract

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

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

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aos/1176325509

Digital Object Identifier
doi:10.1214/aos/1176325509

Mathematical Reviews number (MathSciNet)
MR1292554

Zentralblatt MATH identifier
0807.62063

JSTOR