A modified bootstrap for branching processes with immigration

In this paper we consider bootstrap approximation to the sampling distribution of an estimator of the offspring mean m in a branching process with immigration. A modification of the standard parametric bootstrap procedure is shown to eliminate the invalidity of the standard bootstrap for the case m = 1, as reported in Sriram (1994). Furthermore, the modified bootstrap is shown to provide valid approximations for other values of m (≠ 1) as well. Thus, in this example, the modified bootstrap provides a unified solution while the form of the limit distribution of the estimator of m via classical asymptotic theory depends on m. It is argued that similar modifications will be useful more generally.