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Some asymptotic results for robust procedures for testing the constancy of regression models over time. (English) Zbl 0722.62024
For \(X_ 1,...,X_ n\) independent random variables, where \(X_ i\) is distributed according to \(F(x-c'_ i\theta_ i)\), with \(c_ i\) known regression p-vectors and \(\theta_ i\) unknown vector parameters, the problem of robust testing \(H_ 0:\theta_ 1=...=\theta_ n\) against \(H_ 1:\theta_ 1=...=\theta_ m\neq \theta_{m+1}=...=\theta_ n\) is considered. A recursive M-procedure used for the testing problem is based on a normalization of \[ \max_{p<k\leq n}\{k^{-1/2} | \sum^{k}_{j=p+1}\Psi [x_ i-c'_ i\theta_{i-1}(\Psi)]| \}, \] where \(\Psi\) is a score function and \(\theta_{i-1}(.)\) is the M- estimator based on \(X_ 1,...,X_{i-1}\). The author studies the asymptotics of the above statistics under the null hypotheses.

MSC:
62F35 Robustness and adaptive procedures (parametric inference)
62F05 Asymptotic properties of parametric tests
62E20 Asymptotic distribution theory in statistics
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References:
[1] J. Antoch, M. Hušková: Some M-tests for detection of a change in linear models. Proceedings of the Fourth Prague Symposium on Asymptotic Statistics (P. Mandl, M. Hušková, Charles University, Prague 1989, pp. 123-136. · Zbl 0705.62030
[2] R. L. Brown J. Durbin, J. M. Evans: Techniques for testing the constancy of regression relationships over time (with discussion). J. Roy. Statist. Soc. Ser. B 37 (1975), 149-182. · Zbl 0321.62063
[3] M. Csörgö, L. Horváth: Nonparametric methods for changepoint problems. Handbook of Statistics, vol. 7 (P. R. Krishnaiah and C. R. Rao. North Holland, Amsterdam 1988, pp. 403-425.
[4] D. A. Darling, P. Erdös: A limit theorem for the maximum of normalized sums of independent random variables. Duke Math. J. 23 (1956), 143-155. · Zbl 0070.13806 · doi:10.1215/S0012-7094-56-02313-4
[5] P. Hackl: Testing the Constancy of Regression Models over Time. Vandenhoeck and Ruprecht, Gottingen 1980. · Zbl 0435.62089
[6] M. Hušková: Stochastic approximation type estimators in linear models. Submitted, 1989.
[7] M. Hušková: Recursive M-tests for change point problem. Structural Change: Analysis and Forecasting (A. H. Westlund, School of Economics, Stockholm 1989.
[8] M. Hušková, P. K. Sen: Nonparametric tests for shift and change in regression at an unknown time point. The Future of the World Economy: Economic Growth and Structural Changes (P. Hackl, Springer-Verlag, Berlin-Heidelberg-New York 1989, pp. 73-87.
[9] P. R. Krishnaiah, B. Q. Miao: Review estimates about change point. Handbook of Statistics, vol. 7 (P. R. Krishnaiah and C R. Rao. North Holland, Amsterdam 1988, pp. 390-402.
[10] V. V.Petrov: Sums of Independent Random Variables. Springer-Verlag, Berlin-Heidelberg - New York 1975. · Zbl 0336.60050
[11] P. K. Sen: Recursive M-tests for the constancy of multivariate regression relationships overtime. Sequential Anal. 5(1984), 191 - 211. · Zbl 0588.62139
[12] S. Zacks: Survey of classical and Bayesian approaches to the change point problem: fixed sample and sequential procedures of testing and estimation. Recent Advances in Statistics. Papers in Honour of Herman Chernoff’s Sixtieth Birthday, Acad. Press, New York, 1983, pp. 245-269. · Zbl 0563.62062
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