# On the change-point problem

TL;DR: In this article, 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 is considered.

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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TL;DR: In this paper, a binary search procedure was proposed 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.

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.

4 citations

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TL;DR: Shoutir Kishore Chatterjee (SKC) as discussed by the authors was the National Lecturer in Statistics (1985-1986), 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.

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.”

1 citations

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TL;DR: Shoutir Kishore Chatterjee (SKC) as mentioned in this paper 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 Council of Scientific and Industrial Research, India.

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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TL;DR: In this paper, the identity of two univariate distribution functions F1(x) and F9(x), when there is a sample of fixed size from F1 (x), and the observations (Y) from F2 (x) are drawn sequentially, is investigated.

Abstract: The problem considered in this paper is that of testing the identity of two univariate distribution functions F1(x) and F9(x) when there is a sample of fixed size from F1 (x) and the observations (Y) from F2(x) are drawn sequentially. Two score functions 4> 1(u), 0 < u < 1, l= I, 2, which reflect departures from the null hypothesis in specific ways are taken. Writing F~. (x) for a certain uniformly consistent estimate of F1 (x) based on the first sample, sums of the type S,. 01 = £ 4> 1 (Ft. (Y;)) are calculated for successive observations ;-1 on Y. Observation is stopped as soon as S,. ( 11 reaches a pre-fixed upper bound. The hypothesis is rejected or accepted on the basis of the terminal value of Sn <•>. Different large sample procedures based on this approach are formulated and examined.

18 citations

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01 Jan 1983TL;DR: In this paper, a class of tests for change-points based on recursive U-statistics has been considered, along with some invariance principles for recursive U -statistics, asymptotic properties of the proposed tests are studied.

Abstract: Tests for change—points for the location as well as regression models are often based on cumulative sums of recursive residuals. These recursive residuals are also employed in the sequential detection problem. In the context of general estimable parameters (funtionals of the underlying distribution functions), such recursive residuals may be defined in terms of recursive U—statistics. A class of tests for the change—points based on recursive U—statistics has been considered. Along with some invariance principles for recursive U—statistics, asymptotic properties of the proposed tests are studied.

17 citations

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TL;DR: In this article, three parametric models of distributions are considered; the one parameter exponential family, the location parameter family, and the scale parameter family and the test statistics are shown to converge (in distribution) to the supremum of Brownian motion process, with or without a change-point according to the alternative and the null hypothesis respectively.

Abstract: Retrospective tests are constructed to detect a local shift of parameter of a distribution function occuring at unknown point of time between consecutive independent observations. Three parametric models of distributions are considered; the one parameter exponential family, the location parameter family and the scale parameter family. The class of tests considered is based on the Generalized Likelihood Ratio (GLR) tests, appropriately adapted for such a change-point problem. Asymptotic techniques are used to obtain the limiting distribution of the test statistics, under both the null hypothesis of no change and the change-point alternative. The test statistics, being maximum likelihood type statistics, are shown to converge (in distribution) to the supremum of Brownian motion process, with or without a change-point according to the alternative and the null hypothesis respectively. Analytical expressions for the asymptotic power functions of the proposed tests are provided. These results are then used to provide power comparisons of the tests with those of the Chernoff-Zacks' quasi-Bayesian test.

8 citations