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Boundary-crossing probabilities of some random fields related to likelihood ratio tests for epidemic alternatives

Part of: Stochastic processes Limit theorems Parametric inference

Published online by Cambridge University Press:  14 July 2016

Qiwei Yao
Qiwei Yao*
Affiliation:
Southeast University, Nanjing
*
∗ Present address: Institute of Mathematics and Statistics, University of Kent, Canterbury, Kent CT2 7NF, UK.
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Abstract

We consider the likelihood ratio tests to detect an epidemic alternative in the following two cases of normal observations: (1) the alternative specifies a square wave drift in the mean value of an i.i.d. sequence; (2) the alternative permits a square wave drift in the intercept of a simple linear regression model. To develop the approximations for the significance levels leads us to consider boundary-crossing problems of some two-dimensional discrete-time Gaussian fields. By the method which was proposed originally by Woodroofe (1976) and adapted to study maxima of some random fields by Siegmund (1988), some large deviations for the conditional non-linear boundary-crossing probabilities are developed. Some results of Monte Carlo experiments confirm the accuracy of these approximations.

Keywords

LARGE DEVIATIONGAUSSIAN FIELDS

MSC classification

Primary: 60F10: Large deviations
Secondary: 60G15: Gaussian processes 62F99: None of the above, but in this section
Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 1 , March 1993 , pp. 52 - 65
Copyright
Copyright © Applied Probability Trust 1993 

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Footnotes

Research supported partially by the Alexander von Humboldt-Stiftung, the National Science Foundation of China, and the Deutsche Forschungsgemeinschaft.

References

Hogan, M. and Siegmund, D. (1986) Large deviations for the maxima of some random fields. Adv. Appl. Math. 7, 2–22.Google Scholar
Inchi, Hu (1985) Repeated significance tests for exponential families. PhD dissertation, Stanford University.Google Scholar
James, B., James, K. L. and Siegmund, D. (1987) Tests for a change-point. Biometrika 74, 71–83.Google Scholar
James, B., James, K. L. and Siegmund, D. (1988) Conditional boundary crossing probabilities with applications to change-point problems. Ann. Prob. 16, 825–839.Google Scholar
Kim, H. J. and Siegmund, D. (1989) The likelihood ratio test for a change-point in simple linear regression. Biometrika 76, 409–423.CrossRefGoogle Scholar
Lehmann, E. L. (1959) Testing Statistical Hypotheses. Wiley, New York.Google Scholar
Levin, B. and Kline, J. (1985) The cusum test of homogeneity with an application to spontaneous abortion epidemiology. Statist. Med. 4, 469–488.CrossRefGoogle ScholarPubMed
Siegmund, D. (1985) Sequential Analysis. Springer-Verlag, New York.CrossRefGoogle Scholar
Siegmund, D. (1986) Boundary crossing probabilities and statistical applications. Ann. Statist. 14, 361–404.Google Scholar
Siegmund, D. (1988) Approximate tail probabilities for the maxima of some random fields. Ann. Prob. 16, 487–501.Google Scholar
Woodroofe, M. (1976) A renewal theorem for curved boundaries and moments of first passage times. Ann. Prob. 4, 67–80.Google Scholar
Yao, Q. (1989) Large deviations for boundary crossing probabilities of some random fields. J. Math. Res. Exposition 9, 181–192.Google Scholar