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Testing the equality of linear single-index models. (English) Zbl 1185.62076
Summary: Comparison of nonparametric regression models has been extensively discussed in the literature for the one-dimensional covariate case. The comparison problem largely remains open for completely nonparametric models with multi-dimensional covariates. We address this issue under the assumption that the models are single-index models (SIMs). We propose a test for checking the equality of the mean functions of two (or more) SIMs. The asymptotic normality of the test statistic is established and an empirical study is conducted to evaluate the finite-sample performance of the proposed procedure.

MSC:
 62G08 Nonparametric regression and quantile regression 62H15 Hypothesis testing in multivariate analysis 62G20 Asymptotic properties of nonparametric inference
gss
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References:
 [1] Hall, P.; Hart, J.D., Bootstrap test for difference between means in nonparametric regression, Journal of the American statistical association, 85, 1039-1049, (1990) · Zbl 0717.62037 [2] King, E.C.; Hart, J.D.; Wehrly, T.E., Testing the equality of two regression curves using linear smoothers, Statistics & probability letters, 12, 239-247, (1991) [3] Kulasekera, K.B., Testing the equality of regression curves using quasi residuals, Journal of the American statistical association, 90, 1085-1093, (1995) · Zbl 0841.62039 [4] Kulasekera, K.B.; Wang, J., Smoothing parameter selection for power optimality in testing of regression curves, Journal of the American statistical association, 92, 500-511, (1997) · Zbl 0894.62047 [5] Dette, H.; Neumeyer, N., Nonparametric analysis of covariance, The annals of statistics, 29, 1361-1400, (2001) · Zbl 1043.62033 [6] Neumeyer, N.; Dette, H., Nonparametric comparison of regression curves: an empirical process approach, The annals of statistics, 31, 880-920, (2003) · Zbl 1032.62037 [7] Eubank, R.L., Spline smoothing and nonparametric regression, (1998), Marcel Dekker NY [8] Fan, J.; Gijbels, I., Local polynomial modelling and its applications, (1996), Chapman and Hall London · Zbl 0873.62037 [9] Hastie, T.; Tibshirani, R., Generalized additive models, (1990), Chapman and Hall London · Zbl 0747.62061 [10] Horowitz, J.; Mammen, E., Nonparametric estimation of an additive model with a link function, The annals of statistics, 32, 2412-2443, (2004) · Zbl 1069.62035 [11] Mammen, E.; Park, B.U., Bandwidth selection in smooth backfitting in additive models, The annals of statistics, 33, 1260-1294, (2005) · Zbl 1072.62025 [12] Mammen, E.; Park, B.U., A simple smooth backfitting method for additive models, The annals of statistics, 34, 2252-2271, (2006) · Zbl 1106.62042 [13] Yu, K.; Park, B.U.; Mammen, E., Smooth backfitting in generalized additive models, The annals of statistics, 36, 228-260, (2008) · Zbl 1132.62028 [14] Gu, C., Smoothing spline ANOVA models, (2002), Springer New York · Zbl 1051.62034 [15] Zhang, H.H., Variable selection for support vector machines via smoothing spline ANOVA, Statistica sinica, 16, 659-674, (2006) · Zbl 1096.62072 [16] Lin, Y.; Zhang, H.H., Component selection and smoothing in smoothing spline analysis of variance models, The annals of statistics, 34, 2272-2297, (2006) · Zbl 1106.62041 [17] Lin, Y.; Zhang, H.H., Component selection and smoothing for nonparametric regression in exponential families, Statistica sinica, 16, 1021-1041, (2006) · Zbl 1107.62036 [18] Hart, J.D., Nonparametric smoothing and lack-of-fit tests, (1997), Springer New York · Zbl 0886.62043 [19] Stoker, T.M., Consistent estimation of scaled coefficients, Econometrica, 54, 1461-1481, (1986) · Zbl 0628.62105 [20] Ichimura, H., Semiparametric least squares (SLS) and weighted SLS estimation of single-index models, Journal of econometrics, 58, 71-120, (1993) · Zbl 0816.62079 [21] Hristache, M.; Juditsky, A.; Spokoiny, V., Direct estimation of the index coefficient in a single-index model, The annals of statistics, 29, 595-623, (2001) · Zbl 1012.62043 [22] Yin, X.; Cook, R.D., Direction estimation in single-index regression, Biometrika, 92, 371-384, (2005) · Zbl 1094.62054 [23] Powell, J.L.; Stock, J.H.; Stoker, T.M., Semiparametric estimation of index coefficients, Econometrica, 57, 1403-1430, (1989) · Zbl 0683.62070 [24] Härdle, W.; Hall, P.; Ichimura, H., Optimal smoothing in single-index models, The annals of statistics, 21, 157-178, (1993) · Zbl 0770.62049 [25] Yu, Y.; Ruppert, D., Penalized spline estimation for partially linear single-index models, Journal of the American statistical association, 97, 1042-1054, (2002) · Zbl 1045.62035 [26] Xia, Y.C.; Härdle, W., Semi-parametric estimation of partially linear single-index models, Journal of multivariate analysis, 97, 1162-1184, (2006) · Zbl 1089.62050 [27] Xia, Y.; Li, W.K.; Tong, H.; Zhang, D., A goodness-of-fit test for single-index models (with discussion), Statistica sinica, 14, 1-39, (2004) · Zbl 1040.62034 [28] Stute, W.; Zhu, L.X., Nonparametric checks for single-index models, The annals of statistics, 33, 1048-1083, (2005) · Zbl 1080.62023 [29] Lin, W.; Kulasekera, K.B., Identifiability of single-index models and additive-index models, Biometrika, 94, 496-501, (2007) · Zbl 1132.62050 [30] Yatchew, A., An elementary nonparametric differencing test of equality of regression functions, Economics letters, 62, 271-278, (1999) · Zbl 0917.90065 [31] Neumeyer, N.; Sperlich, S., Comparison of separable components in different samples, Scandinavian journal of statistics, 33, 477-501, (2006) · Zbl 1114.62054 [32] Aneiros-Perez, G., Semi-parametric analysis of covariance under dependence conditions within each group, Australian & New Zealand journal of statistics, 50, 97-123, (2008) · Zbl 1145.62352 [33] Young, S.G.; Bowman, A.W., Nonparametric analysis of covariance, Biometrics, 51, 920-931, (1995) · Zbl 0875.62312 [34] Pardo-Fernandez, J.C.; Van Keilegom, I.; Gonzalez-Manteiga, W., Testing for the equality of $$k$$ regression curves, Statistica sinica, 17, 1115-1137, (2007) · Zbl 1133.62031 [35] Bowman, A.; Young, S., Graphical comparison of nonparametric curves, Applied statistics, 45, 83-98, (1996) · Zbl 0858.62003 [36] P. Cubas, Testing for the comparison of non-parametric regression curves, Preprint 99-29, IRMAR, Univ. Rennes, France, 2000 [37] W. Lin, K.B. Kulasekera, On variance estimation for the single-index models, Technical Report, Department of Mathematical Sciences, Clemson University. http://www.math.clemson.edu/reports/TR2006_06_LK.pdf, Australian and New Zealand Journal of Statistics (2008) (in press) · Zbl 1337.62058 [38] Müller, H.G., Nonparametric regression analysis of longitudinal data, (1984), Springer NY [39] Härdle, W.; Mammen, E., Comparing nonparametric versus parametric regression fits, The annals of statistics, 21, 1926-1947, (1993) · Zbl 0795.62036 [40] Xia, Y.C.; Tong, H.; Li, W.K., On extended partially linear single-index models, Biometrika, 86, 831-842, (1999) · Zbl 0942.62109 [41] Fan, J.; Yao, Q., Efficient esitmation of conditional variance functions in stochastic regression, Biometrika, 85, 645-660, (1998) · Zbl 0918.62065 [42] W. Lin, On Completely data-driven bandwidth selection for single-index models (2009) (manuscript submitted for publication) [43] D. Thibodeaux, H. Senter, J. Knowlton, D. McAlister, X. Cui, Measuring the short fiber content of cotton, in: Cotton: Nature’s High-tech Fiber, Proc. World Cotton Res. Conf. -4, Lubbock, TX, 7-11 Sept. 2007, 2007 [44] Heyde, C.C.; Brown, B.M., On the departure from normality of a certain class of martingales, The annals of mathematical statistics, 41, 2161-2165, (1970) · Zbl 0225.60026
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