Asymptotic distribution of the CLSE in a critical process with immigration

I. Rahimov

doi:10.1016/j.spa.2007.11.004

Asymptotic distribution of the CLSE in a critical process with immigration

I. Rahimov^*

^*Corresponding author for this work

King Fahd University of Petroleum & Minerals

Research output: Contribution to journal › Article › peer-review

8 Scopus citations

Abstract

It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a discrete time branching process with immigration is not asymptotically normal. If the offspring variance tends to zero, it is normal with normalization factor n^{2 / 3}. We study a situation of its asymptotic normality in the case of non-degenerate offspring distribution for the process with time-dependent immigration, whose mean and variance vary regularly with non-negative exponents α and β, respectively. We prove that if β < 1 + 2 α, the CLSE is asymptotically normal with two different normalization factors and if β > 1 + 2 α, its limit distribution is not normal but can be expressed in terms of the distribution of certain functionals of the time-changed Wiener process. When β = 1 + 2 α the limit distribution depends on the behavior of the slowly varying parts of the mean and variance.

Original language	English
Pages (from-to)	1892-1908
Number of pages	17
Journal	Stochastic Processes and their Applications
Volume	118
Issue number	10
DOIs	https://doi.org/10.1016/j.spa.2007.11.004
State	Published - Oct 2008

Bibliographical note

Funding Information:
This paper is based on a part of the results obtained under research project MS/THEOREM/335 funded by King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. I am grateful to the referee and the associate editor for careful readings of the first version of the paper and for valuable comments.

Keywords

Branching process
Functional
Least squares estimator
Skorokhod space
Time-dependent immigration

ASJC Scopus subject areas

Statistics and Probability
Modeling and Simulation
Applied Mathematics

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10.1016/j.spa.2007.11.004

Cite this

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title = "Asymptotic distribution of the CLSE in a critical process with immigration",

abstract = "It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a discrete time branching process with immigration is not asymptotically normal. If the offspring variance tends to zero, it is normal with normalization factor n2 / 3. We study a situation of its asymptotic normality in the case of non-degenerate offspring distribution for the process with time-dependent immigration, whose mean and variance vary regularly with non-negative exponents α and β, respectively. We prove that if β < 1 + 2 α, the CLSE is asymptotically normal with two different normalization factors and if β > 1 + 2 α, its limit distribution is not normal but can be expressed in terms of the distribution of certain functionals of the time-changed Wiener process. When β = 1 + 2 α the limit distribution depends on the behavior of the slowly varying parts of the mean and variance.",

keywords = "Branching process, Functional, Least squares estimator, Skorokhod space, Time-dependent immigration",

author = "I. Rahimov",

note = "Funding Information: This paper is based on a part of the results obtained under research project MS/THEOREM/335 funded by King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. I am grateful to the referee and the associate editor for careful readings of the first version of the paper and for valuable comments.",

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AU - Rahimov, I.

N1 - Funding Information: This paper is based on a part of the results obtained under research project MS/THEOREM/335 funded by King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia. I am grateful to the referee and the associate editor for careful readings of the first version of the paper and for valuable comments.

PY - 2008/10

Y1 - 2008/10

N2 - It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a discrete time branching process with immigration is not asymptotically normal. If the offspring variance tends to zero, it is normal with normalization factor n2 / 3. We study a situation of its asymptotic normality in the case of non-degenerate offspring distribution for the process with time-dependent immigration, whose mean and variance vary regularly with non-negative exponents α and β, respectively. We prove that if β < 1 + 2 α, the CLSE is asymptotically normal with two different normalization factors and if β > 1 + 2 α, its limit distribution is not normal but can be expressed in terms of the distribution of certain functionals of the time-changed Wiener process. When β = 1 + 2 α the limit distribution depends on the behavior of the slowly varying parts of the mean and variance.

AB - It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a discrete time branching process with immigration is not asymptotically normal. If the offspring variance tends to zero, it is normal with normalization factor n2 / 3. We study a situation of its asymptotic normality in the case of non-degenerate offspring distribution for the process with time-dependent immigration, whose mean and variance vary regularly with non-negative exponents α and β, respectively. We prove that if β < 1 + 2 α, the CLSE is asymptotically normal with two different normalization factors and if β > 1 + 2 α, its limit distribution is not normal but can be expressed in terms of the distribution of certain functionals of the time-changed Wiener process. When β = 1 + 2 α the limit distribution depends on the behavior of the slowly varying parts of the mean and variance.

KW - Branching process

KW - Functional

KW - Least squares estimator

KW - Skorokhod space

KW - Time-dependent immigration

UR - https://www.scopus.com/pages/publications/50649084546

U2 - 10.1016/j.spa.2007.11.004

DO - 10.1016/j.spa.2007.11.004

M3 - Article

AN - SCOPUS:50649084546

SN - 0304-4149

VL - 118

SP - 1892

EP - 1908

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 10

ER -

Asymptotic distribution of the CLSE in a critical process with immigration

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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