Asymptotic expansions for array branching processes with applications to bootstrapping

Asymptotic expansions are obtained for the distribution function of a studentized estimator of the offspring mean sequence in an array branching process with immigration. The expansion result is shown to hold in a test function topology. As an application of this result, it is shown that the bootstrapping distribution of the estimator of the offspring mean in a sub-critical branching process with immigration also admits the same expansion (ia probability). From these considerations, it is concluded that the bootstrapping distribution provides a better approximation asymptotically than the normal distribution.