Limit Distribution of the &amp;amp;phi;-Divergence Based Change Point Estimator

Mwelu Susan,Anthony G Waititu,Peter N Mwita,Charity Wamwea

doi:10.4236/ojs.2021.113020

Abstract

The assumption of stationarity is too restrictive especially for long time series. This paper studies the change point problem through a change point estimator based on the φ-divergence which provides a rich set of distance like measures between pairs of distributions. The change point problem is considered in the following sub-fields: the problem of divergence estimation, testing for the homogeneity between two samples as well as estimating the time of change. The asymptotic distribution of the change point estimator is estimated by the limiting distribution of a stochastic process within given bounds through asymptotic theory surrounding the likelihood theory. The distribution is found to converge to that of a standardized Brownian bridge process.

Highlights

Many real world data are made up of consecutive regimes that are separated by abrupt changes [1]
This paper studies the change point problem through a change point estimator based on the φ -divergence which provides a rich set of distance like measures between pairs of distributions
A divergence based estimator has been used for estimating change in the parameters of any given parametric distribution

Summary

Introduction

Many real world data are made up of consecutive regimes that are separated by abrupt changes [1]. They are limited in different ways and their suitability depend on the underlying assumptions. [10] introduced the change point problem within the off-line setting Since this pioneering work, methodologies used for change point detection have been widely researched on with methods extending to techniques for higher order moments within time series data. With the assumption that change time is unknown, [4] gives eight limiting conditions that yields the null distribution of the likelihood ratio test statistic as the supremum of a standardized Brownian bridge.

Single Change Point Detection and Estimation

Main Result

Simulation Study

Conclusion