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Dimension-reduction type test for linearity of a stochastic regression model. (English) Zbl 0927.62044
Summary: This article investigates a test for linearity of a multivariate stochastic regression model. The use of nonparametric regression procedures for developing regression diagnostics has been the subject of several recent research efforts. However, when the dimension of the regressor is large, some traditional nonparametric methods, such as kernel estimation, may be inefficient.
We suggest two test statistics based on projection pursuit technique and kernel method. The tests proposed are consistent against all fixed smooth alternatives to linearity and are asymptotically distribution-free for the distribution of the error. Furthermore, the tests are applied to an example of real-life data and some simulated data sets to demonstrate the availability of the tests proposed.

62G10 Nonparametric hypothesis testing
62J05 Linear regression; mixed models
62H99 Multivariate analysis
Full Text: DOI
[1] H.Z. An and B. Chen. A Kolmogorov-Smirnov Type Statistic with Application to Test for Nonlinearity in Time Series.International Statistical Review, 1991, 59: 287–307. · Zbl 0748.62049 · doi:10.2307/1403689
[2] R.J. Carroll and K.C. Li. Measurement Error Regression with Unknown Link: Dimension Reduction and Data Visualization.J. Amer. Statist. Assoc., 1992, 87: 1040–1050. · Zbl 0765.62002 · doi:10.2307/2290641
[3] P.D. Chen. On the Approximation ofL 2 by Ridge Functions.Bulletin of Chinese Sciences, 1991, 35: 880–884.
[4] X.R. Chen and P.R. Krishnaiah. A Test of Linearity of Regression Models.Chinese J. Appl. Probab. Statist., 1990, 6: 363–377.
[5] J.C. Davis. Statistics and Data Analysis in Geology. John Wiley & Sons Inc., New York, 1973.
[6] R.L. Eubank and J.D. Hart. Commonalty of Cusum, von Neumann and Smoothing-based Goodness-of-fit Tests.Biometrika, 1993, 80: 89–98. · Zbl 0792.62042 · doi:10.1093/biomet/80.1.89
[7] R.L. Eubank and C.H. Speiegelman. Testing the Goodness of Fit of a Linear Model via Nonparametric Regression Techniques.J. Amer. Statist. Assoc., 1990, 85: 387–392. · Zbl 0702.62037 · doi:10.2307/2289774
[8] K.T. Fang and Y. Wang. Number-Theoretic Methods and Applications in Statistics. Chapman and Hall, London, 1993.
[9] W. Härdle and E. Mammen. Comparing Nonparametric Versus Parametric Regression Fits.Annals of Statistics, 1993, 21: 1926–1947. · Zbl 0795.62036 · doi:10.1214/aos/1176349403
[10] W. Härdle and T.M. Stoker. Investigating Smooth Multiple Regression by the Method of Average Deruvatives.J. Amer. Statist. Assoc., 1989, 84: 986–995. · Zbl 0703.62052 · doi:10.2307/2290074
[11] T. Hsing and R.J. Carroll. Asymptotic Properties of Sliced Inverse Regression.Ann. Statist., 1992, 20:, 1040–1061. · Zbl 0821.62019 · doi:10.1214/aos/1176348669
[12] P. Huber. Projection Pursuit (with Discussions).Ann. Statist., 1985, 13: 435–525. · Zbl 0595.62059 · doi:10.1214/aos/1176349519
[13] W.C. Kumbein and R.L. Shreve. Some Statistical Properties of Density Channel Networks. Tech. Rept., 1970, 13: Office of Naval Research, ONR Task No. 389-150, 117. (Available from the Documents Clearing House, Arlington, Va., as document AD 705 625).
[14] K.C. Li. Sliced Inverse Regression for Dimension Reduction.J. Amer. Statist. Assoc., 1991, 84: 316–345. · Zbl 0742.62044 · doi:10.2307/2290563
[15] K.C. Li. On Principal Hessian Directions for Data Visualization and Dimension Reduction: Another Application of Stein’s Lemma.J. Amer. Statist. Assoc., 1992, 87: 1025–1039. · Zbl 0765.62003 · doi:10.2307/2290640
[16] D. Pollard. Convergence of Stochastic Processes. Springer-Verlag, New York, 1984. · Zbl 0544.60045
[17] J.L. Powell, J.H. Stock and T.M. Stoker. Semiparametric Estimation of Index Coefficients.Econometrics, 1989, 57: 1403–1430. · Zbl 0683.62070 · doi:10.2307/1913713
[18] B.L.S.P. Rao. Nonparametric Functional Estimation. Academic Press, Orlando, 1983. · Zbl 0542.62025
[19] L.X. Zhu. Convergence Rates of the Empirical Processes Indexed by the Classes of Functions and Applications.Chinese J. Sys. Sci. & Math. Scis., 1993, 13: 33–41. · Zbl 0776.60032
[20] L.X. Zhu and K.T. Fang. Asymptotic for the Kernel Estimate of Sliced Inverse Regression.Annals of Statistics, 1996, 24: 1053–1068. · Zbl 0864.62027
[21] L.X. Zhu, F.J. Hickernell and H.Z. An. A Goodness-of-fit for Linearity of a Stochastic Regression Model. Technical Report, No. 36, Dept. of Math., Hong Kong Baptist College, 1993. · Zbl 0898.62057
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