×

Optimal pseudo-Gaussian and rank-based random coefficient detection in multiple regression. (English) Zbl 1455.62135

Summary: Random coefficient regression (RCR) models are the regression versions of random effects models in analysis of variance and panel data analysis. Optimal detection of the presence of random coefficients (equivalently, optimal testing of the hypothesis of constant regression coefficients) has been an open problem for many years. The simple regression case has been solved recently and the multiple regression case is considered here. The latter poses several theoretical challenges: (a) a nonstandard ULAN structure, with log-likelihood gradients vanishing at the null; (b) cone-shaped alternatives under which traditional optimality concepts are no longer adequate; (c) nuisance parameters that are not identified under the null but have a significant impact on local powers. We propose a new (local and asymptotic) concept of optimality for this problem and, for specified error densities, derive parametrically optimal procedures. A suitable modification of the Gaussian version of the latter is shown to qualify as a pseudo-Gaussian test. The asymptotic performances of those pseudo-Gaussian tests, however, are quite poor under skewed and heavy-tailed densities. We therefore also construct rank-based tests, possibly based on data-driven scores, the asymptotic relative efficiencies of which are remarkably high with respect to their pseudo-Gaussian counterparts.

MSC:

62J05 Linear regression; mixed models
62G10 Nonparametric hypothesis testing
62F03 Parametric hypothesis testing
PDF BibTeX XML Cite
Full Text: DOI Euclid

References:

[1] R. P. Abelson and J. W. Tukey. Efficient utilization of non-numerical information in quantitative analysis general theory and the case of simple order., The Annals of Mathematical Statistics, 34 :1347-1369, 1963. · Zbl 0121.13907
[2] A. Akharif and M. Hallin. Efficient detection of random coefficients in autoregressive models., Annals of Statistics, 31:675-704, 2003. · Zbl 1039.62081
[3] A. Azzalini and A. Capitanio. Distributions generated by perturbation of symmetry with emphasis on a multivariate skew t-distribution., Journal of the Royal Statistical Society: Series B, 65:367-389, 2003. · Zbl 1065.62094
[4] R. Beran and P. G. Hall. Estimating coefficient distributions in random coefficient regressions., Annals of Statistics, 20 :1970-1984, 1992. · Zbl 0774.62068
[5] R. Beran and P. W. Millar. Minimum distance estimation in random coefficient regression., Annals of Statistics, 22 :1976-1992, 1994. · Zbl 0828.62031
[6] R. Beran, A. Feuerverger, and P. G. Hall. On nonparametric estimation of intercept and slope distributions in random coefficient regression., Annals of Statistics, 24 :2569-2592, 1996. · Zbl 0867.62021
[7] P. J. Bickel, C. A. J. Klaassen, Y. Ritov, and J. A. Wellner., Efficient and Adaptive Estimation for Semiparametric Models. Johns Hopkins Univ. Press, Baltimore, 1993. · Zbl 0786.62001
[8] T. S. Breusch and A. R. Pagan. A simple test for heteroscedasticity and random coefficient variation., Econometrica, 47 :1287-1294, 1979. · Zbl 0416.62021
[9] H. Chernoff and I. R. Savage. Asymptotic normality and efficiency of certain nonparametric test statistics., The Annals of Mathematical Statistics, 29:972-994, 1958. · Zbl 0092.36501
[10] P. Delicado and J. Romo. Goodness-of-fit tests in random coefficient regression models., Annals of the Institute of Statistisitical Mathematics, 51:125-148, 1999. · Zbl 1049.62506
[11] P. Delicado and J. Romo. Random coefficient regressions: parametric goodness of fit., Journal of Statistical Planning and Inference, 119:377-400, 2004. · Zbl 1157.62333
[12] Y. Dodge and J. Jurecková., Adaptive Regression. New York: Springer, 2000. · Zbl 0943.62063
[13] M. Fihri, A. Akharif, A. Mellouk, and M. Hallin. Efficient pseudo-Gaussian and rank-based detection of random regression coefficients., Journal of Nonparametric Statistics, 32:367-402, 2020. · Zbl 1451.62043
[14] B. Garel and M. Hallin. Local asymptotic normality of multivariate arma processes with a linear trend., Annals of the Institute of Statistical Mathematics, 47:551-579, 1995. · Zbl 0841.62076
[15] J. Gu. Neyman’s \(C(\alpha)\) test for unobserved heterogeneity., Econometric Theory, 32 :1483-1522, 2016. · Zbl 1385.62007
[16] J. Gu, R. Koenker, and S. Volgushev. Testing for homogeneity in mixture models., Econometric Theory, 34:850-895, 2018. · Zbl 1393.62018
[17] J. Hájek. Local asymptotic minimax and admissibility in estimation., In Proceedings of the sixth Berkeley Symposium on Mathematical Statistics and Probability, 1:175-194, 1972. · Zbl 0281.62010
[18] J. Hájek and Z. Šidák., Theory of Rank Tests. Academic Press, New York-London, 1967. · Zbl 0161.38102
[19] M. Hallin. On the Pitman-nonadmissibility of correlogram-based methods., Journal of Time Series Analysis, 15:607-612, 1994. · Zbl 0807.62068
[20] M. Hallin and D. La Vecchia. R-estimation in semiparametric dynamic location-scale models., Journal of Econometrics, 196:233-247, 2017. · Zbl 1403.91274
[21] M. Hallin and C. Mehta. R-estimation for asymmetric independent component analysis., Journal of the American Statistical Association, 110:218-232, 2015. · Zbl 1381.62145
[22] M. Hallin and D. Paindaveine. Optimal tests for multivariate location based on interdirections and pseudo-Mahalanobis ranks., Annals of Statistics, 30 :1103-1133, 2002a. · Zbl 1101.62348
[23] M. Hallin and D. Paindaveine. Optimal procedures based oninterdirections and pseudo-Mahalanobis ranks for testing multivariate elliptic white noise against VARMA dependence., Bernoulli, 8:787-816, 2002b. · Zbl 1018.62046
[24] M. Hallin and D. Paindaveine. Chernoff-Savage and Hodges-Lehmann results for Wilks’ test of independence. In, Beyond Parametrics in Interdisciplinary Research: Festschrift in Honor of Professor Pranab K. Sen, pages 184-196. I.M.S. Lecture Notes-Monograph Series, 2008.
[25] M. Hallin and O. Tribel. The efficiency of some nonparametric rank-based competitors to correlogram methods. In F. T. Bruss and L. LeCam, editors, Game Theory, Optimal Stopping, Probability, and Statistics: Papers in Honor of T.S. Ferguson on the Occasion of his 70th Birthday, pages 249-262. I.M.S. Lecture Notes-Monograph Series, 2000. · Zbl 0980.62036
[26] M. Hallin and B. J. M. Werker. Semiparametric efficiency, distribution-freeness and invariance., Bernoulli, 9:137-165, 2003. · Zbl 1020.62042
[27] C. Hildreth and J. P. Houck. Some estimators for a linear model with random coefficients., Journal of the American Statistical Association, 63:584-595, 1968. · Zbl 0162.49804
[28] J. Jurecková. Asymptotic linearity of a rank statistic in regression parameter., The Annals of Mathematical Statistics, 40 :1889-1900, 1969. · Zbl 0188.51003
[29] J. P. Kreiss. On adaptive estimation in stationary ARMA processes., Annals of Statistics, 15:112-133, 1987. · Zbl 0616.62042
[30] A. Kudo. A multivariate analogue of the one-sided test., Biometrika, 50:403-418, 1963. · Zbl 0121.13906
[31] L. Le Cam., Asymptotic Methods in Statistical Decision Theory. Springer series in Statistics. Springer-Verlag, New York, 1986. · Zbl 0605.62002
[32] L. Le Cam and G. L. Yang., Asymptotics in Statistics: Some Basic Concepts. Springer-Verlag, New York, 2nd edition, 2000. · Zbl 0952.62002
[33] B. Lind and G. Roussas. A remark on quadratic mean differentiability., The Annals of Mathematical Statistics, 43 :1030-1034, 1972. · Zbl 0265.60058
[34] F. Lombard. An elementary proof of asymptotic normality for linear rank statistics., South African Statistical Journal, 20:29-35, 1986. · Zbl 0611.62015
[35] A. Mallet. A maximum likelihood estimation method for random coefficient regression models., Biometrika, 73:645-656, 1986. · Zbl 0615.62083
[36] P. Newbold and T. Bos., Stochastic Parameter Regression Models. Beverly Hills California Sage Publications, 1985.
[37] D. F. Nicholls and A. R. Pagan. Varying coefficient regression., E. J. Hannan, P. R. Krishnaiah, M. M. Rao, eds., Handbook of Statistics, 5:413-449, 1985.
[38] P. A. Novikov. A locally directionally maximin test for a multidimensional parameter with order-restricted alternatives., Russian Mathematics, 55:33-41, 2011. · Zbl 1227.62010
[39] D. Paindaveine. A Chernoff-Savage result for shape. on the non-admissibility of pseudo-Gaussian methods., Journal of Multivariate Analysis, 97 :2206-2220, 2006. · Zbl 1101.62045
[40] B. Raj. Linear regression with random coefficients: The finite sample and convergence properties., Journal of the American Statistical Association, 70:127-137, 1975. · Zbl 0313.62049
[41] B. Raj and A. Ullah., Econometrics, A Varying Coefficient Approach. Croom-Helm, London, 1981. · Zbl 0506.62097
[42] T. Ramanathan and M. Rajarshi. Rank tests for testing randomness of a regression coefficient in a linear regression model., Metrika, 39:113-124, 1992. · Zbl 0743.62035
[43] W. Schaafsma and L. Smid. Most stringent somewhere most powerful tests against alternatives restricted by a number of linear inequalities., The Annals of Mathematical Statistics, 37 :1161-1172, 1966. · Zbl 0146.40103
[44] P. A. V. B. Swamy., Statistical Inference in Random Coefficient Regression Models. Springer-Verlag, New York, 1971. · Zbl 0231.62113
[45] A. R. Swensen. The asymptotic distribution of the likelihood ratio for autoregressive time series with a regression trend., Journal of Multivariate Analysis, 16:54-70, 1985. · Zbl 0563.62065
[46] A. Wald. A note on regression analysis., The Annals of Mathematical Statistics, 18:586-589, 1947. · Zbl 0029.30703
[47] J.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.