Cram\'{e}r moderate deviations for a supercritical Galton-Watson process

TL;DR

This paper derives Cramér moderate deviation results for the Lotka-Nagaev estimator in supercritical Galton-Watson processes, providing theoretical insights and applications to confidence interval construction.

Contribution

It introduces new Cramér moderate deviation results for the estimator using martingale methods, enhancing understanding of its probabilistic behavior.

Findings

Cramér moderate deviation results established for the estimator

Applications demonstrated in confidence interval construction

Martingale techniques used for proofs

Abstract

Let $(Z_n)_{n\geq0}$ be a supercritical Galton-Watson process. The Lotka-Nagaev estimator $Z_{n+1}/Z_n$ is a common estimator for the offspring mean.In this paper, we establish some Cram\'{e}r moderate deviation results for the Lotka-Nagaev estimator via a martingale method. Applications to construction of confidence intervals are also given.