branching process non-stationary immigration parametric bootstrap threshold martingale theorem Skorokhod space
Issue Date:
2009
Publisher:
Institute of Mathematics and Informatics Bulgarian Academy of Sciences
Citation:
Pliska Studia Mathematica Bulgarica, Vol. 19, No 1, (2009), 229p-244p
Abstract:
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.