Moderate deviation principles for kernel estimator of invariant density in bifurcating Markov chains models

TL;DR

This paper establishes a moderate deviation principle for the kernel estimator of invariant density in bifurcating Markov chains, revealing new insights into the estimator's behavior at large deviations and the influence of ergodic rate on bandwidth selection.

Contribution

It extends previous work by proving a moderate deviation principle, showing the impact of ergodic rate on bandwidth choice, and unifying regimes for large deviations.

Findings

Moderate deviation principle proven for the estimator.

Ergodic rate influences bandwidth selection for moderate deviations.

No regime distinction for large deviation behavior.

Abstract

Bitseki and Delmas (2021) have studied recently the central limit theorem for kernel estimator of invariant density in bifurcating Markov chains models. We complete their work by proving a moderate deviation principle for this estimator. Unlike the work of Bitseki and Gorgui (2021), it is interesting to see that the distinction of the two regimes disappears and that we are able to get moderate deviation principle for large values of the ergodic rate. It is also interesting and surprising to see that for moderate deviation principle, the ergodic rate begins to have an impact on the choice of the bandwidth for values smaller than in the context of central limit theorem studied by Bitseki and Delmas (2021).