Abstract
The classical problem of change point is considered when the data are assumed to be correlated. The nuisance parameters in the model are the initial level μ and the common variance σ2. The four cases, based on none, one, and both of the parameters are known are considered. Likelihood ratio tests are obtained for testing hypotheses regarding the change in level, δ, in each case. Following Henderson (1986), a Bayesian test is obtained for the two sided alternative. Under the Bayesian set up, a locally most powerful unbiased test is derived for the case μ=0 and σ2=1. The exact null distribution function of the Bayesian test statistic is given an integral representation. Methods to obtain exact and approximate critical values are indicated.
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References
Anderson, T. W. (1984) An Introduction to Multivariate Statistical Analysis. 2nd ed. Wiley, New York.
Bagshaw, M., and Johnson, R.A. (1975), “The Effect of Serial Correlation on the performance of CUSUM Tests II,” Technometrics, 17, 73–80.
Box, G.E.P., and Tiao, G.C. (1965), “A change in level of a Nonstationary Time Series,” Biometrika, 52, 181–192.
Chernoff, H., and Zacks, S. (1964), “Estimating the Current Mean of a Normal Distribution which is subjected to Changes in Time,” Annals of Mathematical Statistics, 35, 999–1018.
Ferguson, T.S. (1967) Mathematical Statistics. A Decision Theoretic Approach. Academic Press, New York.
Gardner, JR., L.A. (1969), “On Detecting Changes in the Mean of Normal Variates,” Annals of Mathematical Statistics, 40, 116–126.
Gupta, S.S. (1963), “Probability Integrals of Multivariate Normal and Multivariatet,” Annals of Mathematical Statistics, 34, 792–828.
Henderson, R. (1986), “Change-Point Problem with Correlated Observations, With an Application in Material Accountancy,” Technometrics, 28, 381–389.
Johnston, J. (1984) Econometric Methods. McGraw Hill, New York.
Johnson, R.A., and Bagshaw, M. (1975), “The Effect of Serial Correlation on the Performance of CUSUM Tests,” Technometrics, 16, 103–112.
Krishnaiah, P.R., and Miao, B.Q. (1987), “Review About Estimation of Change Point,” Technical Report 87-48, Center for Multivariate Analysis, University of Pittsburgh.
Page, E.S. (1954), “Continuous Inspection Schemes,” Biometrika, 41, 100–115.
Sen, A., and Srivastava, M.S. (1975a), “Some One-Sided Tests for Change in Level,” Technometrics, 17, 61–64.
Sen, A., and Srivastava, M.S. (1975b), “On Tests for Detecting Change in Mean,” Annals of Statistics, 3, 98–108.
Shaban, S.A. (1980), “Change Point Problem and Two-Phase Regression: Annotated Bibliography,” International Statistical Review, 48, 83–93.
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Nagaraj, N.K. Two-sided tests for change in level for correlated data. Statistical Papers 31, 181–194 (1990). https://doi.org/10.1007/BF02924689
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DOI: https://doi.org/10.1007/BF02924689