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Journal ArticleDOI

Some One-Sided Tests for Change in Level

01 Feb 1975-Technometrics (Taylor & Francis Group)-Vol. 17, Iss: 1, pp 61-64
TL;DR: In this paper, a simple application using traffic accident data is presented, where Bayesian tests as well as a test based on the maximum likelihood estimate of γ are considered and their powers are compared by Monte Carlo methods.
Abstract: We consider tests based on one observation on each of N ≥ 2 random variables X l, …, XN to decide if the means μ of the xi 's are all equal against the one-sided alternative that a shift has occurred at some unknown point γ, (i.e. μ1, = μ2 = … = μ r < μ r+1 = … = μ N ). The x i 's are considered to be normally distributed with a common unknown variance. Bayesian tests as well as a test based on the maximum likelihood estimate of γ are considered and their powers are compared by Monte Carlo methods. The exact distribution of a Bayesian test statistic is derived. A simple application using traffic accident data is presented.
Citations
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Journal ArticleDOI
TL;DR: In this paper, the authors considered a mean shift with an unknown shift point in a linear process and estimated the unknown point (change point) by the method of least squares, and established the consistency and the rate of convergence for the estimated change point.
Abstract: This paper considers a mean shift with an unknown shift point in a linear process and estimates the unknown shift point (change point) by the method of least squares. Pre-shift and post-shift means are estimated concurrently with the change point. The consistency and the rate of convergence for the estimated change point are established. The asymptotic distribution for the change point estimator is obtained when the magnitude of shift is small. It is shown that serial correlation affects the variance of the change point estimator via the sum of the coefficients (impulses) of the linear process. When the underlying process is an ARMA, a mean shift causes overestimation of its order. A simple procedure is suggested to mitigate the bias in order estimation.

483 citations

Journal ArticleDOI
TL;DR: In this paper, a possible alternative to the hypothesis that the sequence X 1, X 2, X 3, X 4, X 5, X 6, X 7, X 8, X 9, X 10, X 11, X 12, X 13, X 14, X 15, X 16, X 17, X 18, X 19, X 20, X 21, X 22, X 23, X 24, X 25, X 26, X 27, X 28, X 29, X 30, X
Abstract: A possible alternative to the hypothesis that the sequence X 1, X 2, …, Xn are iid N(ξ, σ2) random variables is that at some unknown instant the expectation ξ shifts. The likelihood ratio test for the alternative of a location shift is studied and its distribution under the null hypothesis found. Tables of standard fractiles are given, along with asymptotic results.

328 citations

Journal ArticleDOI
TL;DR: In this article, a study is made about a shift in the mean of a set of independent normal random variables with unknown common variance, and the marginal and joint posterior distributions of the unknown time point and the amount of shift are derived.
Abstract: In this article, a study is made about a shift in the mean of a set of independent normal random variables with unknown common variance. The marginal and joint posterior distributions of the unknown time point and the amount of shift are derived. Small and large sample results are presented.

204 citations

Journal ArticleDOI
TL;DR: In this paper, the authors compute the asymptotic distribution of the maximum likelihood ratio test when they want to check whether the parameters of normal observations have changed at an unknown point and prove that the limit distribution is based on the largest deviation between a $d$-dimensional Ornstein-Uhlenbeck process and the origin.
Abstract: We compute the asymptotic distribution of the maximum likelihood ratio test when we want to check whether the parameters of normal observations have changed at an unknown point. The proof is based on the limit distribution of the largest deviation between a $d$-dimensional Ornstein-Uhlenbeck process and the origin.

164 citations

Book ChapterDOI
S. Zacks1
01 Jan 1983
TL;DR: In this article, the authors discuss the sequential procedures of testing and estimation and describe classical and Bayesian approaches to the change-point problem and present Bayesian and maximum likelihood estimation of the location of the shift points.
Abstract: Publisher Summary This chapter discusses fixed sample and the sequential procedures of testing and estimation and describes classical and Bayesian approaches to the change-point problem. It presents Bayesian and maximum likelihood estimation of the location of the shift points. The Bayesian approach is based on modeling the prior distribution of the unknown parameters, adopting a loss function and deriving the estimator, which minimizes the posterior risk. The chapter discusses this approach with an example of a shift in the mean of a normal sequence. The estimators obtained are generally nonlinear complicated functions of the random variables. From the Bayesian point of view, these estimators are optimal. The maximum likelihood estimation of the location parameter of the change point is an attractive alternative to the Bayes estimators. Regression relationship can change at unknown epochs, resulting in different regression regimes that should be detected and identified.

163 citations

References
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Journal ArticleDOI
TL;DR: In this paper, the problem of making inference about the point in a sequence of zero-one variables at which the binomial parameter changes is discussed, and the asymptotic distribution of the maximum likelihood estimate of the change-point is derived in computable form using random walk results.
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)

766 citations


"Some One-Sided Tests for Change in ..." refers background in this paper

  • ...The allied problem of estimating the value of r has been considered by, among others, Hinkley [6, 7]....

    [...]

Journal ArticleDOI
E. S. Page1

674 citations

Journal ArticleDOI
TL;DR: In this paper, a Bayesian approach is used to estimate the current mean of an object in a given trajectory from a series of observations, and a sequence of tests are designed to locate the last time point of change.
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)

554 citations

Journal ArticleDOI
TL;DR: In this article, 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 $r$.
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.

365 citations

Journal ArticleDOI
TL;DR: In this paper, the authors consider the problem of making inferences about a possible shift in level of the series associated with the occurrance of an event at some particular time, for example, the observations might be of some economic indicator and one might suspect a change in level to occur in a particular interval because of change in fiscal policy.
Abstract: : Suppose that observations zsubt of a time series are available at equally spaced time intervals The authors consider the problem of making inferences about a possible shift in level of the series associated with the occurrance of an event E at some particular time For example, the observations might be of some economic indicator and one might suspect a change in level to occur in a particular interval because of a change in fiscal policy Alternatively, zsubt could be the daily output of a chemical process and the event E might be a change in the supplier of raw material

255 citations