Bootstrap for Critical Branching Process with Non-Stationary Immigration

2000 Mathematics Subject Classification: Primary 60J80, Secondary 62F12, 60G99.

In the critical branching process with a stationary immigration the standard parametric bootstrap for an estimator of the offspring mean is invalid. We consider the process with non-stationary immigration, whose mean and variance α(n) and β(n) are finite for each n ≥ 1 and are regularly varying sequences with nonnegative exponents α and β, respectively. It turns out that if α(n) → ∞ and β(n) = o(nα2(n)) as n → ∞, then the standard parametric bootstrap procedure leads to a valid approximation for the distribution of the conditional least squares estimator. We state a theorem which justifies the validity of the bootstrap. By Monte-Carlo and bootstrap simulations for the process we confirm the theoretical findings. The simulation study highlights the validity and utility of the bootstrap in this model as it mimics the Monte-Carlo pivots even when generation size is small.