On the Rate of Approximations for Maximum Likelihood Tests in Change-Point Models

✍ Scribed by Edit Gombay; Lajos Horváth

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 886 KB
• Volume: 56
• Category: Article
• ISSN: 0047-259X
• DOI: 10.1006/jmva.1996.0007

✦ Synopsis

We study the asymptotics of maximum-likelihood ratio-type statistics for testing a sequence of observations for no change in parameters against a possible change while some nuisance parameters remain constant over time. We obtain extreme value as well as Gaussian-type approximations for the likelihood ratio. We get necessary and sufficient conditions for the weak convergence of supremum and L p -functionals of the likelihood ration process. We also approximate the maximum likelihood ratio with Ornstein Uhlenbeck processes and obtain bounds for the rate of approximation. We show that the Ornstein Uhlenbeck approach is superior to the extreme value limit in case of moderate sample sizes.

1996 Academic Press, Inc.

1. Approximations for the Likelihood Ratio Process

Let X 1 , X 2 , ..., X n be independent random vectors in R m with distribution functions F(x; % 1 , ' 1 ), ..., F(x; % n , ' n ), where % i # 3 (1) R d and ' i # 3 (2) R p , 1 i n. We want to test

against the change-point alternative

article no.