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Some methodological aspects of validation of models in nonparametric regression. (English) Zbl 1090.62531

Summary: We describe some general methods for constructing goodness of fit tests in nonparametric regression models. Our main concern is the development of statisticial methodology for the assessment (validation) of specific parametric models M as they arise in various fields of applications. The fundamental idea which underlies all these methods is the investigation of certain goodness of fit statistics (which may depend on the particular problem and may be driven by different criteria) under the assumption that a specified model (which has to be validated) holds true as well as under a broad range of scenaria, where this assumption is violated. This is motivated by the fact that outcomes of tests for the classical hypothesis:”The mode M holds true”(and their associated p-values) bear various methodological flaws. Hence, our suggestion is always to accompany such a test by an analysis of the type II error, which is in goodness of fit problems often the more serious one. We give a careful description of the methodological aspects, the required asymptotic theory, and illustrate the main principles in the problem of testing model assumptions such as a specific parametric form or homoscedasticity in nonparametric regression models.

MSC:

62G08 Nonparametric regression and quantile regression
62G10 Nonparametric hypothesis testing

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[1] Achieser N. J., Theory of approximation (1956) · Zbl 0072.28403
[2] Alcala J. T., Statistics & Probability Letters 42 pp 39– (1999)
[3] Altman D. G., British Medical Journal 311 pp 485– (1995) · doi:10.1136/bmj.311.7003.485
[4] Azzalini A., Journal of the Royal Statistical Society 55 pp 549– (1993)
[5] Berkson J., Journal of the American Statistical Association 33 pp 526– (1938)
[6] Berkson J., Journal of the American Statistical Association 37 pp 325– (1942)
[7] Berkson J., Journal of the American Statistical Association 38 pp 242– (1943)
[8] Biedermann S., Test 9 pp 417– (2000)
[9] Bierens H. J., Econometrica 58 pp 1443– (1990)
[10] Bissantz N., Comparison of parametric models with the same or a different number of parameters in noisy inhomogenoeus inverse problems - with applications to recovering the luminosity density from the milky way. Astron. & Astroph. 376 pp 735– (2001)
[11] Breusch T. S., Econometrica 47 pp 1287– (1979)
[12] Brockwell P. J., Time series: theory and methods (1991) · Zbl 0709.62080 · doi:10.1007/978-1-4419-0320-4
[13] Brodeau F., Statistics 24 pp 95– (1993)
[14] Brown L. D., Annals of Statistics 25 pp 2345– (1997)
[15] Bunke H., Series Statistics 12 pp 7– (1981) · Zbl 0468.62055 · doi:10.1080/02331888108801565
[16] Bunke H., Series Statistics 11 pp 3– (1980) · Zbl 0453.62049 · doi:10.1080/02331888008801521
[17] Carroll R.J., Transformation and weighting in regression (1988) · Zbl 0666.62062 · doi:10.1007/978-1-4899-2873-3
[18] Chen J.C., Annals of the Institute of Statistical Mathematics 46 pp 251– (1994)
[19] Chow S.C., Design and analysis of bioavailability and bioequivalence studies (1992) · Zbl 0823.62086
[20] Cook R.D., Biometrika 70 pp 1– (1983)
[21] Cox D., Annals of Statistics 18 pp 113– (1988)
[22] Dette H., Annals of Statistics 27 pp 1012– (1999)
[23] Dette H., Journal of Statistical Planning and Inference (2001)
[24] DOI: 10.1214/aos/1028144860 · Zbl 0930.62041 · doi:10.1214/aos/1028144860
[25] DOI: 10.1023/A:1003439114929 · Zbl 0903.62039 · doi:10.1023/A:1003439114929
[26] H. Dette, and A. Munk (1998 ), Testing heteroscedasticity in nonparametric regression .Journal of the Royal Statistical Society,Series B60 , 693 -708 . · Zbl 0909.62035
[27] H. Dette, A. Munk, and T. Wagner (1998 ), Estimating the variance in nonparametric regression - what is a reasonable choice?Journal of the Royal Statistical Society,Series B60 , 751 -764 . · Zbl 0944.62041
[28] Dette H., Journal of Nonparametric Statistics 12 pp 309– (2000)
[29] Diblasi A., Statistics & Probability Letters 33 pp 95– (1997)
[30] DOI: 10.1016/0378-3758(94)00045-W · Zbl 0812.62051 · doi:10.1016/0378-3758(94)00045-W
[31] Djojosugito R.A., Computational Statistics 9 pp 213– (1994)
[32] Djojosugito R.A., Communications in Statistics - Theory and Methods 24 pp 2183– (1995)
[33] Eubank R.L., Journal of the American Statistical Association 85 pp 387– (1990)
[34] Eubank R.L., Annals of Statistics 20 pp 1412– (1992)
[35] Eubank R.L., Communications in Statistics - Theory and Methods 22 pp 3327– (1993)
[36] Eubank J., Journal of the Royal Statistical Society 55 pp 145– (1993)
[37] Fan J., Journal of the American Statistical Society 87 pp 998– (1992) · doi:10.1080/01621459.1992.10476255
[38] Fan J., Journal of the American Statistical Association 91 pp 674– (1996)
[39] Fan J., Local polynomial modelling and its applications (1996) · Zbl 0873.62037
[40] Fan J., Journal of the American Statistical Association 96 pp 640– (2001)
[41] Fan J., Annals of Statistics 29 pp 153– (2001)
[42] Firth D., Biometrika 78 pp 245– (1991)
[43] Gallant A.R., Nonlinear statistical models (1987) · Zbl 0611.62071 · doi:10.1002/9780470316719
[44] Gasser L. T., Biometrika 73 pp 626– (1986)
[45] Gasser T., Journal of the Royal Statistical Society 47 pp 238– (1985)
[46] Gasser T., Kernel estimation of regression functions, in: Lecture Notes in Mathematics 757, Smoothing techniques for curve estimation (1979) · Zbl 0418.62033 · doi:10.1007/BFb0098486
[47] Gonzalez, Test 2 pp 161– (1993)
[48] Gonzalez-Manteiga W., Computational Statistics & Data Analysis 20 pp 521– (1995)
[49] Gozalo P.L., Econometric Theory 9 pp 451– (1993)
[50] Goutis C., Biometrika 85 pp 29– (1998)
[51] Hall P., Biometrika 77 pp 521– (1990) · Zbl 1033.62031
[52] Hall P., Advances in Applied Probability 23 pp 476– (1991)
[53] Hall P., Biometrika 77 pp 415– (1990) · Zbl 1033.62031
[54] Hardle W., Annals of Statistics 21 pp 1926– (1993)
[55] Hart J.D., Nonparametric smoothing and lack of fit tests (1997) · Zbl 0886.62043 · doi:10.1007/978-1-4757-2722-7
[56] Hart J.D., Journal of the American Statistical Association 87 pp 1018– (1992)
[57] Harrison M.J., Journal of the American Statistical Association 74 pp 494– (1979)
[58] J.L. Hodges, and E.L. Lehmann (1954 ), Testing the approximative validity of statistical hypotheses .Series B16 , 261 -268 . · Zbl 0057.35403
[59] Horrowitz J., An adaptive, rate-optimal test of a parametric mean-regression model against a nonparametric alternative (2001)
[60] Koenker R., Econometrika 50 pp 43– (1981)
[61] Jayashuriya B.R., Journal of the American Statistical Association 91 pp 1626– (1996)
[62] Kozek A.S., Journal of Multivariate Analysis 37 pp 66– (1991)
[63] Kuchibhatla M., Nonparametric Statistics 7 pp 1– (1996)
[64] McBride G., Australian and New Zealand Journal of Statistics 41 pp 19– (1998)
[65] McKinnon J.G., Journal of Economic Literature 30 pp 102– (1992)
[66] Muller H.G., Scandinavian Journal of Statistics 19 pp 157– (1992)
[67] Muller H.G., Annals of Statistics 15 pp 610– (1987)
[68] Muller H.G., Annals of Statistics 23 pp 946– (1995)
[69] Munk A., Journal of the Royal Statistical Society 60 pp 223– (1998)
[70] Munk A., Nonlinear Analysis 47 pp 1513– (2001) · Zbl 1042.62621
[71] Munk A., Scandinavian Journal of Statistics 29 pp 501– (2001)
[72] Munk A., A class of simple tests for heteroscedasticity in regression models based on difference estimators (2001)
[73] Neil J.W., Annals of Statistics 13 pp 1482– (1985)
[74] Neyman J., Skandinavisk Aktuarietidskrift 20 pp 149– (1937)
[75] Rice J., Annals of Statistics 12 pp 1215– (1984)
[76] Roy T., Journal of Mathematical Chemistry 21 pp 103– (1997)
[77] Sacks J., Annals of Mathematical Statistics 41 pp 2057– (1970)
[78] Seber G.A.F., Nonlinear regression (1989) · Zbl 0721.62062 · doi:10.1002/0471725315
[79] Staniswalis J.G., Journal of the American Statistical Association 86 pp 684– (1991)
[80] DOI: 10.1214/aos/1031833666 · Zbl 0926.62035 · doi:10.1214/aos/1031833666
[81] Stute W., Journal of Statistical Planning and Inference 53 pp 75– (1996)
[82] Stute W., Journal of the American Statistical Association 93 pp 141– (1998)
[83] Neumann J., Annals of Mathematical Statistics 12 pp 367– (1941)
[84] Neumann J., Annals of Mathematical Statistics 13 pp 86– (1942)
[85] Weirather G., Metrika 40 pp 367– (1993)
[86] Wooldridge J.M., Econometric Theory 8 pp 452– (1992) · Zbl 04510756 · doi:10.1017/S0266466600013165
[87] Yatchew A.J., Econometric Theory 8 pp 435– (1992) · Zbl 04510755 · doi:10.1017/S0266466600013153
[88] DOI: 10.1016/0304-4076(95)01760-7 · Zbl 0865.62030 · doi:10.1016/0304-4076(95)01760-7
[89] Zheng J.X., A consistent nonparametric test of heteroscedasticity (2000)
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