Abstract
It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a branching process with stationary immigration is not asymptotically normal. If the offspring variance tends to zero, it is normal with norming factor n3/2. We study the process with a non degenerate offspring distribution and time-dependent immigration, whose mean and variance vary regularly with non negative exponents, and β, respectively. We propose new weighted CLSE using more flexible weights and prove that if β < 1 + 2, it is asymptotically normal with two different norming factors and if β > 1 + 2, its limiting distribution is not normal but can be expressed in terms of certain functionals of the time-changed Wiener process. When β = 1 + 2, the limiting distribution depends on the behavior of the slowly varying parts of the mean and variance. Conditions guaranteeing the strong consistency of the proposed estimator will be derived.
| Original language | English |
|---|---|
| Pages (from-to) | 13-28 |
| Number of pages | 16 |
| Journal | Communications in Statistics - Theory and Methods |
| Volume | 38 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jan 2009 |
Bibliographical note
Funding Information:This article is based on a part of results obtained under research project SF2008/01 funded by KFUPM, Dhahran, Saudi Arabia. My sincere thanks to King Fahd University of Petroleum and Minerals for all their support and facilities. I am grateful to the referees for their careful reading of the first version of the article and for their valuable comments.
Keywords
- Branching process
- Consistency
- Offspring mean
- Skorokhod space
- Time-dependent immigration
- Weighted estimator
ASJC Scopus subject areas
- Statistics and Probability