Abstract
A sequential procedure is proposed for constructing a fixed-size confidence region for the parameters of a linear regression model. The procedure is based on certain regression analogues of trimmed means, as formulated by Welsh (1987,Ann. Statist.,15, 20–36), rather than least squares estimates. For error distributions with continuous, symmetric density and some moment higher than fourth finite, if the design points of the model are bounded, then the procedure is both asymptotically consistent and asymptotically efficient as the size of the region approaches zero.
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References
Albert, A. (1966). Fixed size confidence ellipsoids for linear regression parameters,Ann. Math. Statist.,37, 1602–1630.
Anseombe, F. (1952). Large sample theory of sequential estimation,Proc. Cambr. Philos. Soc.,48, 600–607.
Brown, B. M. (1971). A note on convergence of moments,Ann. Math. Statist.,42, 777–779.
Chow, Y. S. and Robbins, H. (1965). On the asymptotic theory of fixed-width confidence intervals for the mean,Ann. Math. Statist.,36, 457–462.
Chow, Y. S. and Studden, W. J. (1969). Monotonieity of the variance under truncation and variations of Jensen's inequality,Ann. Math. Statist.,40, 1106–1108.
Chow, Y. S. and Teicher, H. (1978).Probability Theory, Springer, New York.
Finster, M. (1985). Estimation in the general linear model when the accuracy is specified before data collection,Ann. Statist.,13, 663–675.
Gleser, L. (1965). On the asymptotic theory of fixed-size sequential confidence bounds for linear regression parameters,Ann. Math. Statist.,36, 463–467.
Koenker, R. W. and Bassett, G. W. (1978). Regression quantiles,Econometrica,46, 33–50.
Lai, T. L. and Wei, C. Z. (1982). Least squares estimates in stechastic regression models with applications to identification and control of dynamic systems,Ann. Statist.,10, 154–166.
Martinsek, A. T. (1984). Sequential determination of estimator as well as sample size.Ann. Statist.,12, 533–550.
Martinsek, A. T. (1987). Sequential point estimation in regression models: The nonparametric case, Tech, Report, University of Illinois.
Ruppert, D. and Carroll, R. J. (1980). Trimmed least squares estimation in the linear model.J. Amer. Statist. Assoc.,75, 828–838.
Shu, W. Y. (1987). Limit theorems for processes and stopping rules in adaptive sequential estimation, Ph.D. Dissertation, University of Illinois.
Srivastava, M. S. (1967). On fixed-width confidence bounds for regression parameters and the mean vector,J. Roy. Statist. Soc. Ser. B,29, 132–140.
Srivastava, M. S. (1971). On fixed-width confidence bounds for regression parameters,Ann. Math. Statist.,42, 1403–1411.
Welsh, A. H. (1987). The trimmed mean in the linear model,Ann. Statist.,15, 20–36.
Woodroofe, M. (1982).Nonlinear Renewal Theory in Sequential Analysis, Society for Industrial and Applied Mathematics, Philadelphia.
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Research supported in part by the National Science Foundation under Grants DMS 85-03321 and 88-02556 and by the Air Force under Grant AFOSR-87-0041.
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Martinsek, A.T. Sequential estimation in regression models using analogues of trimmed means. Ann Inst Stat Math 41, 521–540 (1989). https://doi.org/10.1007/BF00050666
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DOI: https://doi.org/10.1007/BF00050666