A Unified Limit Theory via Bootstrap for Branching Processes with Immigration

In this paper we consider bootstrap approximation to the sampling distribution of the maximum likelihood estimator (m.l.e.) of the offspring mean m in a branching process with immigration. A clever 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 (1992). Furthermore, the modified bootstrap is shown to provide valid approximations for other values of m (=f:. 1) as well. Thus, in this example, the modified bootstrap provides a unified solution while the form of the limit distribution of the m.l.e. via classical asymptotic theory depends on m. It is argued that similar modifications will be useful more generally. AMS(1990) Subject Classifications: Primary 60J80,62G09.