Estimation of the mean for critical branching process and its bootstrap approximation

This article focuses attention on the estimation of the mean of a sequence of branching process with immigration in the critical case. We get the limiting distribution of the pivot, and adopt a bootstrap procedure to bootstrap the least-square estimator with bootstrap sample size <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$$m$$</EquationSource> </InlineEquation> less than the size <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$$n$$</EquationSource> </InlineEquation> of the original sample. Under the assumptions that <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$$m\rightarrow \infty $$</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$$m/n\rightarrow 0$$</EquationSource> </InlineEquation>, the convergence in probability of the bootstrap distribution function is also established. Copyright Springer-Verlag Berlin Heidelberg 2013