Variance estimators in critical branching processes with non-homogeneous immigration

The asymptotic normality of conditional least squares estimators for the offspring variance in critical branching processes with non-homogeneous immigration is established, under moment assumptions on both reproduction and immigration. The proofs use martingale techniques and weak convergence results in Skorokhod spaces.Comment: Accepted for publication in Math Population Studie

Statistical inference for partially observed branching processes with immigration

Copyright © 2017 Applied Probability Trust. In the paper we consider the following modification of a discrete-time branching process with stationary immigration. In each generation a binomially distributed subset of the population will be observed. The number of observed individuals constitute a partially observed branching process. After inspection both observed and unobserved individuals may change their offspring distributions. In the subcritical case we investigate the possibility of using the known estimators for the offspring mean and for the mean of the stationary-limiting distribution of the process when the observation of the population sizes is restricted. We prove that, if both the population and the number of immigrants are partially observed, the estimators are still strongly consistent. We also prove that the \u27skipped\u27 version of the estimator for the offspring mean is asymptotically normal and the estimator of the stationary distribution\u27s mean is asymptotically normal under additional assumptions

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

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 \u3c 1 + 2 α, the standard bootstrap is valid and if β \u3e 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

Approximation of a sum of martingale differences generated by a bootstrap branching process

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{Z(k), k\geq 0\}$$\end{document} be a branching stochastic process with non-stationary immigration given by offspring distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{p_{j}(\theta),j\geq 0\}$$\end{document} depending on unknown parameter \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\theta\in \Theta$$\end{document}. We estimate θ by an estimator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\theta}_{n}$$\end{document} based on sample \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X}_{n}=\{Z(i), i=1, {\ldots}, n\}$$\end{document}. Given \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal{X}_{n}$$\end{document}, we generate bootstrap branching process \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{Z^{\mathcal{X}_{n}}(k), k\geq 0\}$$\end{document} for each \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n=1, 2, {\ldots}$$\end{document} with offspring distribution \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{p_{j}(\hat{\theta}_{n}), j\geq 0\}$$\end{document}. In the paper we address the following question: How good must be estimator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hat{\theta}_{n}$$\end{document}, the bootstrap process to have the same asymptotic properties as the original process? We obtain conditions for the estimator which are sufficient and necessary for this in critical case. To derive these conditions we investigate a weighted sum of martingale differences generated by an array of branching processes. We provide a general functional limit theorem for this sum, which includes critical or nearly critical processes with increasing or stationary immigration and with large or fixed number of initial ancestors. It also includes processes without immigration with increasing random number of initial individuals. Possible applications in estimation theory of branching processes are also be provided

On a Maximal Sequence Associated with Simple Branching Processes

[Rahimov Ibrahim; Рахимов Ибрахим]; [Yanev George; Янев Георги]AMS 1991 subject classification: primary 60J80, secondary 60G70