Bootstrap of the offspring mean in the critical process with a non-stationary immigration

Ibrahim Rahimov, Zayed University

Document Type

Article

Source of Publication

Stochastic Processes and their Applications

Publication Date

11-1-2009

Abstract

In applications of branching processes, usually it is hard to obtain samples of a large size. Therefore, a bootstrap procedure allowing inference based on a small sample size is very useful. Unfortunately, in the critical branching process with stationary immigration the standard parametric bootstrap is invalid. In this paper, we consider a process with non-stationary immigration, whose mean and variance vary regularly with nonnegative exponents α and β, respectively. We prove that 1 + 2 α is the threshold for the validity of the bootstrap in this model. If β < 1 + 2 α, the standard bootstrap is valid and if β > 1 + 2 α it is invalid. In the case β = 1 + 2 α, the validity of the bootstrap depends on the slowly varying parts of the immigration mean and variance. These results allow us to develop statistical inferences about the parameters of the process in its early stages. © 2009 Elsevier B.V. All rights reserved.

ISSN

0304-4149

Publisher

Elsevier BV

Volume

119

Issue

11

First Page

3939

Last Page

3954

Disciplines

Physical Sciences and Mathematics

Keywords

Branching process, Martingale theorem, Non-stationary immigration, Parametric bootstrap, Skorokhod space, Threshold

Scopus ID

70349865178

Indexed in Scopus

yes

Open Access

yes

Open Access Type

Bronze: This publication is openly available on the publisher’s website but without an open license

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