Abstract
Random coefficient regressions have been applied in a wide range of fields, from biology to economics, and constitute a common frame for several important statistical models. A nonparametric approach to inference in random coefficient models was initiated by Beran and Hall. In this paper we introduce and study goodness of fit tests for the coefficient distributions; their asymptotic behavior under the null hypothesis is obtained. We also propose bootstrap resampling strategies to approach these distributions and prove their asymptotic validity using results by Giné and Zinn on bootstrap empirical processes. A simulation study illustrates the properties of these tests.
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REFERENCES
Beran, R. (1995). Prediction in random coefficient regression, J. Statist. Plann. Inference, 43, 205-213.
Beran, R. and Hall, P. (1992). Estimating coefficient distributions in random coefficient regressions, Ann. Statist., 20, 1970-1984.
Beran, R. and Millar, P. W. (1994). Minimum distance estimation in random coefficient regression, Ann. Statist., 22, 1976-1992.
Beran, R., Feuerverger, A. and Hall, P. (1996). On nonparametric estimation of intercept and slope distributions in random coefficient regression, Ann. Statist., 24, 2569-2592.
Chow, G. C. (1983). Random and changing coefficient models, Handbook of Econometrics, Vol. II (eds. M. D. Intriligator and Z. Griliches), 1213-1245, North-Holland, Amsterdam.
Dudley, R. M. (1984). A course on empirical processes, école d'été de Probabilités de Saint-Flour XII—1982, Lecture Notes in Math., 1097, 2-142, Springer, New York.
Fan, J. (1991). On the optimal rates of convergence for nonparametric deconvolution problems, Ann. Statist., 19, 1275-1272.
Fisher, N. J., Mammen, E. and Marron, J. S. (1994). Testing for multimodality. Comp. Statist. Data Anal., 18, 499-512.
Giné, E. and Zinn, J. (1986). Lectures on the central limit theorem for empirical processes, Probabily and Banach Spaces, Lecture Notes in Math., 1221, 50-113, Springer, Berlin.
Giné, E. and Zinn, J. (1990). Bootstrapping general empirical measures, Annals of Probability, 18, 851-869.
Giné, E. and Zinn, J. (1991). Gaussian characterization of uniform Donsker classes of functions, Annals of Probability, 19, 758-782.
Longford, N. T. (1993). Random Coefficient Models, Clarendon Press, Oxford.
Mardia, K. V., Kent, J. T. and Bibby, J. M. (1979). Multivariate Statistics, Academic Press, London.
Nicholls, D. F. and Pagan, A. R. (1985). Varying coefficient regression, Handbook of Statistics 5 (eds. E. J. Hannan, P. R. Krishnaiah and M. M. Rao), 413-449, North-Holland, Amsterdam.
Nicholls, D. F. and Quinn, B. N. (1982). Random Coefficient Autoregressive Models: An Introduction, Springer, New York.
Pollard, D. (1982). A central limit theorem for empirical processes, J. Austral. Math. Soc. Ser. A, 33, 235-248.
Pollard, D. (1984). Convergence of Stochastic Processes, Springer, New York.
Pollard, D. (1989). Asymptotics via empirical processes, Statist. Sci., 4, 341-366.
Raj, B. and Ullah, A. (1981). Econometrics, a Varying Coefficients Approach, Croom-Helm, London.
Scheffé, H. (1959). The Analysis of Variance, Wiley, New York.
van Es, A. J. (1991). Uniform deconvolution: Nonparametric maximum likelihood and inverse estimation, Nonparametric Functional Estimation and Related Topics, Kluwer, Dordrecht.
Yang, R. and Chen, M. H. (1995). Bayesian analysis for random coefficient regression models using noninformative priors, J. Multivariate Anal., 55, 283-311.
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Delicado, P., Romo, J. Goodness of Fit Tests in Random Coefficient Regression Models. Annals of the Institute of Statistical Mathematics 51, 125–148 (1999). https://doi.org/10.1023/A:1003887303233
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DOI: https://doi.org/10.1023/A:1003887303233