Article1999

Capturing the distributional behaviour of the maximum likelihood estimator of a changepoint

47 citations0/0 files storedBiometrika

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Advanced Statistical Process Monitoring Probability and Statistical Research Statistical Distribution Estimation and Applications

Abstract

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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References

1987Applied Probability and Queues 1970Inference About the Change-Point in a Sequence of Random Variables 1978The Problem of the Nile: Conditional Solution to a Changepoint Problem 1986Confidence Regions and Tests for a Change-Point in a Sequence of Exponential Family Random Variables 1987Approximating the Distribution of the Maximum Likelihood Estimate of the Change-Point in a Sequence of Independent Random Variables 1988Confidence Sets in Change-Point Problems 1994Some aspects of change-point analysis 1987Maximum likelihood estimation of a change-point in the distribution of independent random variables: General multiparameter case

Cited by

2001Maximum likelihood estimation of a change-point for exponentially distributed random variables 2010Exact asymptotic distribution of change-point mle for change in the mean of Gaussian sequences 2002Detection and estimation of abrupt changes in the variability of a process 1999Change-point methods for Weibull models with applications to detection of trends in extreme temperatures 2015A General Regression Changepoint Test for Time Series Data

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