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A consistent test for the functional form of a regression based on a difference of variance estimators. (English) Zbl 0957.62036
Summary: We study the problem of testing the functional form of a given regression model. A consistent test is proposed which is based on the difference of the least squares variance estimator in the assumed regression model and a nonparametric variance estimator. The corresponding test statistic can be shown to be asymptotically normal under the null hypothesis and under fixed alternatives with different rates of convergence corresponding to both cases. This provides a simple asymptotic test, where the asymptotic results can also be used for the calculation of the type II error of the procedure at any particular point of the alternative and for the construction of tests for precise hypotheses. Finally, the finite sample performance of the new test is investigated in a detailed simulation study, which also contains a comparison with the commonly used tests.

MSC:
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
62J02 General nonlinear regression
62G08 Nonparametric regression and quantile regression
Software:
KernSmooth; nlmdl
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[1] Azzalini, A. and Bowman, A. (1993). On the use of nonparametric regression for checking linear relationships. J. Roy. Statist. Soc. Ser. B 55 549-559. JSTOR: · Zbl 0800.62222 · links.jstor.org
[2] Berger, J. O. and Delampade, M. (1987). Testing precise hypotheses. Stat. Sci. 2 317-352. · Zbl 0955.62545 · doi:10.1214/ss/1177013238
[3] Brodeau, F. (1993). Tests for the choice of approximative models in nonlinear regression when the variance is unknown. Statistics 24 95-106. · Zbl 0808.62059 · doi:10.1080/02331888308802396
[4] De Jong, P. (1987). A central limit theorem for generalized quadratic forms. Probab. Theory Related Fields 75 261-277. · Zbl 0596.60022 · doi:10.1007/BF00354037
[5] Dette, H. and Munk, A. (1998). Validation of linear regression models. Ann. Statist. 26 778-800. · Zbl 0930.62041 · doi:10.1214/aos/1028144860
[6] Dette, H., Munk, A. and Wagner, T. (1998). A review of variance estimators with applications to multivariate nonparametric regression models. In Multivariate Analysis, Design of Experiments and Survey Sampling (S. Ghosh, ed.) 469-498. Dekker, New York. · Zbl 0946.62056
[7] Elfving, G. (1952). Optimum allocation in linear regression theory. Ann. Math. Statist. 23 255- 262. · Zbl 0047.13403 · doi:10.1214/aoms/1177729442
[8] Eubank, R. L. and Hart, J. D. (1992). Testing goodness of fit in regression via order selection criteria. Ann. Statist. 20 1412-1425. · Zbl 0776.62045 · doi:10.1214/aos/1176348775
[9] Fan, J. (1992). Design-adaptive nonparametric regression. J. Amer. Statist. Soc. 87 998-1004. JSTOR: · Zbl 0850.62354 · doi:10.2307/2290637 · links.jstor.org
[10] Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and Its Applications. Chapman and Hall, London. · Zbl 0873.62037
[11] Gallant, A. R. (1987). Nonlinear Statistical Models. Wiley, New York. · Zbl 0611.62071
[12] Gasser, T. and M üller, H.-G. (1979). Kernel estimation of regression functions. Smoothing Techniques for Curve Estimation. Lecture Notes in Math. 757 23-68. Springer, New York. · Zbl 0418.62033 · doi:10.1007/BFb0098489
[13] Gasser, T., M üller, H.-G. and Mamitzsch, V. (1985). Kernels for nonparametric curve estimation. J. Roy. Statist. Soc. Ser. B 47 238-252. JSTOR: · Zbl 0574.62042 · links.jstor.org
[14] Gasser, T., Skroka, L. and Jennen-Steinmetz, G. (1986). Residual variance and residual pattern in nonlinear regression. Biometrika 73 626-633. JSTOR: · Zbl 0649.62035 · doi:10.1093/biomet/73.3.625 · links.jstor.org
[15] Hall, P. and Marron, J. S. (1990). On variance estimation in nonparametric regression. Biometrika 77 415-419. JSTOR: · Zbl 0711.62035 · doi:10.1093/biomet/77.2.415 · links.jstor.org
[16] Härdle, W. and Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. Ann. Statist. 21 1926-1947. · Zbl 0795.62036 · doi:10.1214/aos/1176349403
[17] Kutchibhatla, M. and Hart, J. D. (1996). Smoothing-based lack-of-fit tests: variations on a theme. Nonparametr. Statist. 7 1-22. · Zbl 0877.62041 · doi:10.1080/10485259608832685
[18] Nadaraya, E. A. (1964). On estimating regression. Theory Probab. Appl. 10 186-190. · Zbl 0136.40902
[19] Neil, J. W. and Johnson, D. E. (1985). Testing linear regression function adequacy without replication. Ann. Statist. 13 1482-1489. · Zbl 0582.62056 · doi:10.1214/aos/1176349749
[20] Rice, J. (1984). Bandwidth choice for nonparametric regression. Ann. Statist. 12 1215-1230. · Zbl 0554.62035 · doi:10.1214/aos/1176346788
[21] Sacks, J. and Ylvisaker, D. (1970). Designs for regression problems for correlated errors. Ann. Math. Statist. 41 2057-2074. · Zbl 0234.62025 · doi:10.1214/aoms/1177696705
[22] Seber, G. A. F. and Wild, G. J. (1989). Nonlinear Regression. Wiley, New York. · Zbl 0721.62062
[23] Shillington, E. R. (1979). Testing lack of fit in regression without replication. Canad. J. Statist. 7 137-146. · Zbl 0445.62075 · doi:10.2307/3315113
[24] Stute, W. (1997). Nonparametric model checks for regression. Ann. Statist. 25 613-641. · Zbl 0926.62035 · doi:10.1214/aos/1031833666
[25] Stute, W., Gonzáles Manteiga, W. and Presedo Quindimil, M. (1998). Bootstrap approximation in model checks for regression. J. Amer. Statist. Assoc. 93 141-149. JSTOR: · Zbl 0902.62027 · doi:10.2307/2669611 · links.jstor.org
[26] Wand, M. P. and Jones, M. C. (1995). Kernel Smoothing. Chapman and Hall, London. · Zbl 0854.62043
[27] Watson, G. S. (1964). Smooth regression analysis. Sankhy\?a Ser. A 26 359-372. · Zbl 0137.13002
[28] Weirather, G. (1993). Testing a linear regression model against nonparametric alternatives. Metrika 40 367-379. · Zbl 0785.62049 · doi:10.1007/BF02613703 · eudml:176486
[29] Whittle, P. (1964). On the convergence to normality of quadratic forms in independent variables. Theory. Probab. Appl. 9 103-108. · Zbl 0146.40905
[30] Wooldridge, J. M. (1992). A test for a functional form against nonparametric alternatives. Econometric Theory 8 452-475. JSTOR: · links.jstor.org
[31] Yatchew, A. J. (1992). Nonparametric regression tests based on least squares. Econometric Theory 8 435-451. JSTOR: · links.jstor.org
[32] Zheng, J. X. (1996). A consistent test of a functional form via nonparametric estimation techniques. J. Econometrics 75 263-289. · Zbl 0865.62030 · doi:10.1016/0304-4076(95)01760-7
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