Sankhya:
The Indian Journal of Statistics
1986, Volume 48, Series A, Pt. 3, pp. 339--353
THE ASYMPTOTIC BEHAVIOR OF THE LIKELIHOOD RATIO STATISTIC FOR TESTING A SHIFT IN MEAN IN A SEQUENCE OF INDEPENDENT NORMAL VARIATES
By
YI-CHING YAO
And
RICHARD A. DAVIS, Colorado State University
SUMMARY. Let X1, …Xn be an independent sequence of random variables such that X1,… Xr ~ i.i.d N (m, s2) and Xr+1, …, Xn ~ i.i.d N(m + q, s2) where m, q ¹ 0, s2 and r are unknown parameters. The asymptotic properties of the likelihood ratio in testing H0 : r = n (no change point) vs. H1 : r < n are derived. It is shown, using a result of Darling and Erdos, that the likelihood ratio, suitably normalized and under H0 converges in distribution to the double exponential extreme value distribution. The asymptotic operating characteristics of the likelihood ratio test are studied and comparisons are made between the likelihood ratio test and a Bayes test. In particular, the Bayes test outperforms the likelihood ratio test as long as the change point does not occur very early or late in the sequence. Also, an approximation procedure to the null distribution is proposed which is not only correct in the limit and easy to compute, but provides a good approximation for 20 £ n £ 50.
AMS (1980) subject classification. 62F05, 62E20
Key words and phrases. Change point , Normalized Brownian bridge, Ornstein-Uhlenbeck process, Extreme value, Asymptotics