On a representation of fractional Brownian motion and the limit distributions of statistics arising in cusp statistical models

TL;DR

This paper introduces a new representation of fractional Brownian motion to analyze the limit distributions of estimators in cusp-shaped signal models, extending previous results to the entire range of Hurst parameters.

Contribution

It provides a novel fractional Brownian motion representation that enables deriving limit distributions for estimators in cusp models across all Hurst parameters.

Findings

Limit distributions expressed in terms of fBm for all H in (0,1)

Simulation results for densities and variances of estimators

Extension of previous results to full Hurst parameter range

Abstract

We discuss some extensions of results from the recent paper by Chernoyarov et al. (Ann. Inst. Stat. Math., October 2016) concerning limit distributions of Bayesian and maximum likelihood estimators in the model "signal plus white noise" with irregular cusp-type signals. Using a new representation of fractional Brownian motion (fBm) in terms of cusp functions we show that as the noise intensity tends to zero, the limit distributions are expressed in terms of fBm for the full range of asymmetric cusp-type signals correspondingly with the Hurst parameter H, 0<H<1. Simulation results for the densities and variances of the limit distributions of Bayesian and maximum likelihood estimators are also provided.