© 1999 by Biometrika Trust
Capturing the distributional behaviour of the maximum likelihood estimator of a changepoint
A1 Department of Pure and Applied Mathematics and Program in Statistics, Washington State University, Pullman, Washington 99164, USA jandhyala@wsu.edu A2 Department of Management and Systems and Program Statistics, Washington State University, Pullman, Washington 99164, USA fotopo@wsu.edu
We consider the problem of estimating the unknown changepoint in a sequence of time-ordered observations. Upper and lower bounds are derived for the asymptotic distribution of the maximum likelihood estimator and methods of approximation are suggested. A computationally efficient algorithm is presented for deriving the bounds and approximations for the asymptotic probabilities of the maximum likelihood estimator when the parameters before and after the changepoint are unknown. We also show an essentially exponential rate of convergence of the probability distribution of the maximum likelihood estimator from finite samples to the case of infinite samples. We apply the algorithm to the cases of normal and exponential distributions. For the exponential distribution the lower and upper bounds for the right tail probabilities of the maximum likelihood estimator, and the two approximations, are identical. This is not the case for the normal distribution. Finally, we apply our changepoint analysis for the case of the exponential distribution to data on explosions in British coal mines.
Keywords:Maximum likelihood estimator; Maximum of a random walk; Negative drift; Parameter change.
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