Estimating changes in a multi-parameter exponential family

Abstract

A two-stage procedure is studied for estimating changes in the parameters of the multi-parameter exponential family, given a sample X ₁,…,X _n. The first step is a likelihood ratio test of the hypothesis H_oof no change. Upon rejection of this hypothesis, the change point index and pre- and post-change parameters are estimated by maximum likelihood. The asymptotic (n → ∞) distribution of the log-likelihood ratio statistic is obtained under both H_oand local alternatives. The m.l.e.fs o of the pre- and post-change parameters are shown to be asymptotically jointly normal. The distribution of the change point estimate is obtained under local alternatives. Performance of the procedure for moderate samples is studied by Monte Carlo methods.

Keywords:

• Likelihood ratio test
• maximum likelihood estimator
• weak convergence
• standardized tied-down bessel process