# On the change-point problem

Abstract: We consider the problem of testing a shift in the level of a process occurring at an unknown time point given an intial sample of fixed size from the original unchanged process. Various large sample results for the proposed test are formulated and examined.

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Abstract: This article studies the problem of testing and locating changepoints in likelihood ratios of two multinomial probability vectors. We propose a binary search procedure to detect the changepoints in the sequence of the ratios of probabilities and obtain the maximum likelihood estimators of two multinomial probability vectors under the assumption that the probability ratio sequence has a changepoint. We also give a strongly consistent estimator for the changepoint location. An information theoretic approach is used to test the equality of two discrete probability distributions against the alternative that their ratios have a changepoint. Approximate critical values of the test statistics are provided by simulation for several choices of model parameters. Finally, we examine a real life data set pertaining to average daily insulin dose from the Boston Collaborative Drug Surveillance Program and locate the changepoints in the probability ratios.

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Abstract: Shoutir Kishore Chatterjee was born in Ranchi, a small hill station in India, on November 6, 1934. He received his B.Sc. in statistics from the Presidency College, Calcutta, in 1954, and M.Sc. and Ph.D. degrees in statistics from the University of Calcutta in 1956 and 1962, respectively. He was appointed a lecturer in the Department of Statistics, University of Calcutta, in 1960 and was a member of its faculty until his retirement as a professor in 1997. Indeed, from the 1970s he steered the teaching and research activities of the department for the next three decades. Professor Chatterjee was the National Lecturer in Statistics (1985–1986) of the University Grants Commission, India, the President of the Section of Statistics of the Indian Science Congress (1989) and an Emeritus Scientist (1997–2000) of the Council of Scientific and Industrial Research, India. Professor Chatterjee, affectionately known as SKC to his students and admirers, is a truly exceptional person who embodies the spirit of eternal India. He firmly believes that “fulfillment in man’s life does not come from amassing a lot of money, after the threshold of what is required for achieving a decent living is crossed. It does not come even from peer recognition for intellectual achievements. Of course, one has to work and toil a lot before one realizes these facts.”

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Abstract: Shoutir Kishore Chatterjee was born in Ranchi, a small hill station in India, on November 6, 1934. He received his B.Sc. in statistics from the Presidency College, Calcutta, in 1954, and M.Sc. and Ph.D. degrees in statistics from the University of Calcutta in 1956 and 1962, respectively. He was appointed a lecturer in the Department of Statistics, University of Calcutta, in 1960 and was a member of its faculty until his retirement as a professor in 1997. Indeed, from the 1970s he steered the teaching and research activities of the department for the next three decades. Professor Chatterjee was the National Lecturer in Statistics (1985--1986) of the University Grants Commission, India, the President of the Section of Statistics of the Indian Science Congress (1989) and an Emeritus Scientist (1997--2000) of the Council of Scientific and Industrial Research, India. Professor Chatterjee, affectionately known as SKC to his students and admirers, is a truly exceptional person who embodies the spirit of eternal India. He firmly believes that ``fulfillment in man's life does not come from amassing a lot of money, after the threshold of what is required for achieving a decent living is crossed. It does not come even from peer recognition for intellectual achievements. Of course, one has to work and toil a lot before one realizes these facts.''

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Abstract: : The report discusses the problem of making inference about the point in a sequence of zero-one variables at which the binomial parameter changes. The asymptotic distribution of the maximum likelihood estimate of the change-point is derived in computable form using random walk results. The asymptotic distributions of likelihood ratio statistics are obtained for testing hypotheses about the change-point. Some exact numerical results for these asymptotic distributions are given and their accuracy as finite sample approximations is discussed. (Author)

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Abstract: : A tracking problem is considered. Observations are taken on the successive positions of an object traveling on a path, and it is desired to estimate its current position. The objective is to arrive at a simple formula which implicitly accounts for possible changes in direction and discounts observations taken before the latest change. To develop a reasonable procedure, a simpler problem is studied. Successive observations are taken on n independently and normally distributed random variables X sub 1, X sub 2, ..., X sub n with means mu sub 1, mu sub 2, ..., mu sub n and variance 1. Each mean mu sub i is equal to the preceding mean mu sub (i-1) except when an occasional change takes place. The object is to estimate the current mean mu sub n. This problem is studied from a Bayesian point of view. An 'ad hoc' estimator is described, which applies a combination of the A.M.O.C. Bayes estimator and a sequence of tests designed to locate the last time point of change. The various estimators are then compared by a Monte Carlo study of samples of size 9. This Bayesian approach seems to be more appropriate for the related problem of testing whether a change in mean has occurred. This test procedure is simpler than that used by Page. The power functions of the two procedures are compared. (Author)

539 citations

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01 Feb 1955

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Abstract: Procedures are considered for testing whether the means of each variable in a sequence of independent random variables can be taken to be the same, against alternatives that a shift might have occurred after some point $r$. Bayesian test statistics as well as some statistics depending on estimates of $r$ are presented and their powers compared. Exact and asymptotic distribution functions are derived for some of the Bayesian statistics.

347 citations