Abstract
In the critical branching process with a stationary immigration, the standard parametric bootstrap for an estimator of the offspring mean is invalid. We consider the process with non-stationary immigration, whose mean and variance α (n) and β(n) are finite for each n ≥ 1 and are regularly varying sequences with non-negative exponents α and β, respectively. We prove that if α(n) → ∞ and β(n) = o(nα2(n)) as n→ ∞, then the standard parametric bootstrap procedure leads to a valid approximation for the distribution of the conditional least-squares estimator in the sense of convergence in probability. Monte Carlo and bootstrap simulations for the process confirm the theoretical findings in the paper and highlight the validity and utility of the bootstrap as it mimics the Monte Carlo pivots even when generation size is small.
| Original language | English |
|---|---|
| Pages (from-to) | 1-19 |
| Number of pages | 19 |
| Journal | Journal of Nonparametric Statistics |
| Volume | 23 |
| Issue number | 1 |
| DOIs | |
| State | Published - Mar 2011 |
Bibliographical note
Funding Information:This paper is partially based on results obtained under research project No IN080396 funded by KFUPM, Dhahran, Saudi Arabia. Our sincere thanks to King Fahd University of Petroleum and Minerals for all the support and facilities we had. We would also like to thank two anonymous referees, the editor, and associate editor for their comments on a previous version of the paper.
Keywords
- Branching process
- Martingale theorem
- Non-stationary immigration
- Parametric bootstrap
- Skorokhod space
- Threshold
ASJC Scopus subject areas
- Statistics and Probability
- Statistics, Probability and Uncertainty
