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Testing goodness of fit for the distribution of errors in multivariate linear models. (English) Zbl 1070.62029

Summary: To test goodness of fit to any fixed distribution of errors in multivariate linear models, we consider a weighted integral of the squared modulus of the difference between the empirical characteristic function of the residuals and the characteristic function under the null hypothesis. We study the limiting behaviour of this test statistic under the null hypothesis and under alternatives. In the asymptotics, the rank of the design matrix is allowed to grow with the sample size.

MSC:

62G10 Nonparametric hypothesis testing
62J05 Linear regression; mixed models
62J20 Diagnostics, and linear inference and regression
62H15 Hypothesis testing in multivariate analysis
62E20 Asymptotic distribution theory in statistics
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[1] Arnold, S. F., The Theory of Linear Models and Multivariate Analysis (1981), Wiley: Wiley New York
[2] Bai, J., Testing parametric conditional distributions of dynamic models, Rev. Econ. Statist., 85, 531-549 (2003)
[3] Baltagi, B. H.; Li, Q., A consistent test for the parametric distribution of regression disturbances, Adv. Econometrics, 14, 3-24 (2000) · Zbl 0982.62010
[4] Baringhaus, L.; Henze, N., A consistent test for multivariate normality based on the empirical characteristic function, Metrika, 35, 339-348 (1988) · Zbl 0654.62046
[5] Bickel, P. J.; Rosenblatt, M., On some global measures of the deviations of density function estimates, Ann. Statist., 1, 1071-1095 (1973) · Zbl 0275.62033
[6] Csörgő, S., Testing for normality in arbitrary dimension, Ann. Statist., 14, 708-723 (1986) · Zbl 0615.62060
[7] Eaton, M. L.; Perlman, M. D., The non-singularity of generalized sample covariance matrices, Ann. Statist., 4, 710-717 (1973) · Zbl 0261.62037
[8] Epps, T. W., Limiting behavior of the ICF test for normality under Gram-Charlier alternatives, Statist. Probab. Lett., 42, 175-184 (1999) · Zbl 1057.62512
[9] Epps, T. W.; Pulley, L. B., A test for normality based on the empirical characteristic function, Biometrika, 70, 723-726 (1983) · Zbl 0523.62045
[10] Fan, Y., Testing the goodness-of-fit of a parametric density function by kernel method, Econometric Theory, 10, 316-356 (1994)
[11] Fan, Y., Bootstrapping a consistent nonparametric goodness-of-fit test, Econometric Rev., 14, 367-382 (1995) · Zbl 0832.62038
[12] Y. Fan, Goodness-of-fit tests for a multivariate distribution by the empirical characteristic function, J. Multivariate Anal., doi:10.1006/jmva.1997.1672.; Y. Fan, Goodness-of-fit tests for a multivariate distribution by the empirical characteristic function, J. Multivariate Anal., doi:10.1006/jmva.1997.1672. · Zbl 0949.62044
[13] Fan, Y., Goodness-of-fit tests based on kernel density estimators with fixed smoothing parameters, Econometric Theory, 14, 604-621 (1998)
[14] W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, Wiley, New York, 1971.; W. Feller, An Introduction to Probability Theory and its Applications, vol. 2, Wiley, New York, 1971. · Zbl 0219.60003
[15] Gregory, G. G., Large sample theory for \(U\)-statistics and test of fit, Ann. Statist., 5, 110-123 (1977) · Zbl 0371.62033
[16] Hall, P., Central limit theorem for integrated square error of multivariate nonparametric density estimators, J. Multivariate Anal., 14, 1-16 (1984) · Zbl 0528.62028
[17] Hall, P.; Welsh, A. H., A test for normality based on the empirical characteristic function, Biometrika, 70, 485-489 (1983) · Zbl 0519.62037
[18] N. Henze, T. Wagner, A new approach to the BHEP tests for multivariate normality, J. Multivariate Anal., doi:10.1006/jmva.1997.1684.; N. Henze, T. Wagner, A new approach to the BHEP tests for multivariate normality, J. Multivariate Anal., doi:10.1006/jmva.1997.1684. · Zbl 0874.62043
[19] Huber, P. J., Robust regressionasymptotics, conjectures and Monte Carlo, Ann. Statist., 1, 799-821 (1973) · Zbl 0289.62033
[20] Jiang, J., Goodness-of-fit tests for mixed model diagnostics, Ann. Statist., 29, 1137-1164 (2001) · Zbl 1041.62062
[21] M.D. Jiménez-Gamero, J. Muñoz-García, M.V. Alba-Fernández, Goodness-of-fit tests based on the empirical characteristic function, submitted for publication.; M.D. Jiménez-Gamero, J. Muñoz-García, M.V. Alba-Fernández, Goodness-of-fit tests based on the empirical characteristic function, submitted for publication.
[22] Koul, H., Weighed Empiricals and Linear Models (1992), IMS: IMS Hayward, CA · Zbl 0998.62501
[23] Koutrouvelis, I. A., A goodness-of-fit test of simple hypothesis based on the empirical characteristic function, Biometrika, 67, 238-240 (1980) · Zbl 0425.62024
[24] Koutrouvelis, I. A.; Kellermeier, J., A goodness-of-fit test based on the empirical characteristic function when parameters must be estimated, J. R. Statist. Soc. B, 43, 173-176 (1981) · Zbl 0473.62037
[25] Mammen, E., Asymptotics with increasing dimension for robust regression with applications to the bootstrap, Ann. Statist., 17, 382-400 (1989) · Zbl 0674.62017
[26] Mammen, E., Empirical process of residuals for high-dimensional linear models, Ann. Statist., 24, 307-335 (1996) · Zbl 0853.62042
[27] Murota, K.; Takeuchi, K., The studentized empirical characteristic function and its application to test the shape of distribution, Biometrika, 68, 55-65 (1981) · Zbl 0463.62041
[28] Pierce, D. A.; Kopecky, K. J., Testing goodness of fit for the distribution of errors in regression models, Biometrika, 66, 1-5 (1979) · Zbl 0395.62049
[29] Portnoy, S., Asymptotic behaviour of the empiric distribution of \(M\)-estimated residuals from a regression model with many parameters, Ann. Statist., 14, 1152-1170 (1986) · Zbl 0612.62072
[30] Serfling, R. J., Approximation Theorems of Mathematical Statistics (1980), Wiley: Wiley New York · Zbl 0456.60027
[31] Stephens, M. A., Asymptotic results for goodness-of-fit statistics with unknown parameters, Ann. Statist., 4, 357-369 (1976) · Zbl 0325.62014
[32] Welsh, A. H., A note on scale estimates based on the empirical characteristic function and their application to test for normality, Statist. Probab. Lett., 2, 345-348 (1984) · Zbl 0557.62045
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