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Unified approach to testing functional hypotheses in semiparametric contexts. (English) Zbl 1334.62080
Summary: We suggest a new method, with very wide applicability, for testing semiparametric hypotheses about functions such as regression means and probability densities. The technique is based on characterising hypotheses in terms of functionals which can be estimated root-nn consistently, and constructing test statistics in terms of estimators of the functionals. Since the tests are semiparametric it is appropriate to assess them on the basis of their ability to detect departures of size \(n^{-1/2}\) from the null hypothesis. We show that they do indeed have this property. Unlike tests constructed in a nonparametric setting their power does not depend critically on choice of a bandwidth, and in particular, smoothing parameter selection is not an issue that has to be addressed by users of the tests. Bootstrap methods are suggested for calibrating the tests. In a regression setting, applications include tests of specification (such as partial linear and index models) against nonparametric or semiparametric alternatives, and tests of monotonicity, concavity, separability, equality of regression functions and base-independence of equivalence scales. In a density setting, they include tests of radial symmetry and stochastic dominance.

MSC:
62G10 Nonparametric hypothesis testing
62G09 Nonparametric statistical resampling methods
62F03 Parametric hypothesis testing
Software:
KernSmooth
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[1] Abadie, A., Bootstrap tests for distributional treatment effects in instrumental variable models, Journal of the American statistical association, 97, 284-292, (2002) · Zbl 1073.62530
[2] Anderson, N.H.; Hall, P.; Titterington, D.M., Two-sample test statistics for measuring discrepancies between two multivariate probability density functions using kernel-based density estimates, Journal of multivariate analysis, 50, 41-54, (1998) · Zbl 0798.62055
[3] Azzalini, A.; Bowman, A., On the use of nonparametric regression for checking linear relationships, Journal of the royal statistical society series B, 55, 549-557, (1993) · Zbl 0800.62222
[4] Baltagi, B.H.; Hidalgo, J.; Li, Q., A nonparametric test for poolability using panel data, Journal of econometrics, 75, 345-367, (1996) · Zbl 0864.62029
[5] Barrett, G.F.; Donald, S.G., Consistent tests for stochastic dominance, Econometrica, 71, 71-104, (2003) · Zbl 1137.62332
[6] Barry, D., Testing for additivity of a regression function, Annals of statistics, 21, 235-254, (1993) · Zbl 0771.62033
[7] Bierens, H., Consistent model specification tests, Journal of econometrics, 20, 105-134, (1982) · Zbl 0549.62076
[8] Bierens, H., A consistent conditional moment test of functional form, Econometrica, 58, 1443-1458, (1990) · Zbl 0737.62058
[9] Bierens, H.; Ploeberger, W., Asymptotic theory of integrated conditional moments, Econometrica, 65, 1129-1151, (1997) · Zbl 0927.62085
[10] Bowman, A.W.; Jones, M.C.; Gijbels, I., Testing monotonicity of regression, Journal of computational and graphical statistics, 7, 489-500, (1998)
[11] Delecroix, M., Hall, P., Vial, C., 2002. Tests for single index models that are powerful against local alternatives. Manuscript.
[12] Dette, H., A consistent test for the functional form of a regression based on a difference of variance estimators, Annals of statistics, 27, 1012-1040, (1999) · Zbl 0957.62036
[13] Ellison, G.; Ellison, S., A simple framework for nonparametric specification testing, Journal of econometrics, 96, 1-23, (2000) · Zbl 0968.62046
[14] Eubank, R.; Hart, J., Testing goodness-of-fit in regression via order selection criteria, Annals of statistics, 20, 1412-1425, (1992) · Zbl 0776.62045
[15] Eubank, R.; Spiegelman, C., Testing the goodness of fit of a linear model via nonparametric regression techniques, Journal of the American statistical association, 85, 387-392, (1990) · Zbl 0702.62037
[16] Eubank, R.; Hart, J.; Simpson, D.; Stefanski, L., Testing for additivity in nonparametric regression, Annals of statistics, 23, 1896-1920, (1995) · Zbl 0858.62036
[17] Fan, Y.; Li, Q., Consistent model specification testsomitted variables and semiparametric functional forms, Econometrica, 64, 865-890, (1996) · Zbl 0854.62038
[18] Fan, J.Q.; Lin, S.K., Test of significance when data are curves, Journal of the American statistical association, 93, 1007-1021, (1998) · Zbl 1064.62525
[19] Friedman, J.H.; Tibshirani, R.J., The monotone smoothing of scatterplots, Technometrics, 26, 243-250, (1984)
[20] Gijbels, I.; Hall, P.; Jones, M.C.; Koch, I., Tests for monotonicity of a regression Mean with guaranteed level, Biometrika, 87, 663-673, (2000) · Zbl 0956.62039
[21] Gozalo, P.; Linton, O., Testing additivity in generalized nonparametric regression models, Journal of econometrics, 104, 1-48, (2001) · Zbl 0978.62032
[22] Härdle, W.; Mammen, E., Comparing nonparametric vs parametric regression fits, Annals of statistics, 21, 1926-1947, (1993) · Zbl 0795.62036
[23] Härdle, W.; Hall, P.; Ichimura, H., Optimal smoothing in single-index models, Annals of statistics, 21, 157-178, (1993) · Zbl 0770.62049
[24] 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
[25] Hall, P.; Heckman, N.E., Testing for monotonicity of a regression Mean by calibrating for linear functions, Annals of statistics, 28, 20-39, (2000) · Zbl 1106.62324
[26] Hall, P., Huang, L.-S. 2001. Nonparametric kernel regression subject to monotonicity constraints. Annals of Statistics, to appear. · Zbl 1012.62030
[27] Hall, P.; Huber, C.; Speckman, P.L., Covariate-matched one-sided tests for the difference between functional means, Journal of the American statistical association, 92, 1074-1083, (1997) · Zbl 0889.62033
[28] Hart, J., Nonparametric smoothing and lack-of-fit tests, (1997), Springer New York · Zbl 0886.62043
[29] Hastie, T.; Tibshirani, R., Generalized additive models, (1991), Chapman & Hall London · Zbl 0747.62061
[30] Hong, Y.; White, H., Consistent specification testing via nonparametric series regression, Econometrica, 63, 1133-1160, (1995) · Zbl 0941.62125
[31] Horowitz, J.; Härdle, W., Testing a parametric model against a semiparametric alternative, Econometric theory, 10, 821-848, (1994)
[32] Horowitz, J.; Spokoiny, V., An adaptive, rate-optimal test of a parametric Mean-regression model against a nonparametric alternative, Econometrica, 69, 599-631, (2001) · Zbl 1017.62012
[33] Ichimura, H., Semiparametric least squares (SLS) and weighted SLS estimation of single-index models, Journal of econometrics, 58, 71-120, (1993) · Zbl 0816.62079
[34] Inglot, T.; Kallenberg, W.C.M.; Ledwina, T., Power approximations to and power comparison of smooth goodness-of-fit tests, Scandinavian journal of statistics, 21, 131-145, (1994) · Zbl 0799.62042
[35] Inglot, T.; Kallenberg, W.C.M.; Ledwina, T., Vanishing shortcoming and asymptotic relative efficiency, Annals of statistics, 28, 215-238, (2000) · Zbl 1106.62328
[36] Ingster, Y.I., Minimax distinguishability of families of nonparametric hypotheses, Doklady akademii nauk SSSR (Russian), 267, 536-539, (1982)
[37] Ingster, Y.I., Asymptotically minimax hypothesis testing for nonparametric alternatives. I, Mathematical methods of statistics, 2, 85-114, (1993), Erratum Mathematical Methods of Statistics 2, 268 · Zbl 0798.62057
[38] Ingster, Y.I., Asymptotically minimax hypothesis testing for nonparametric alternatives. II, Mathematical methods of statistics, 2, 171-189, (1993), Erratum Mathematical Methods of Statistics 2, 268 · Zbl 0798.62058
[39] Kelly, C.; Rice, J., Monotone smoothing with application to dose – response curves and the assessment of synergism, Biometrics, 46, 1071-1085, (1990)
[40] Kendall, M., Stuart, A., 1979. The Advanced Theory of Statistics, vol. 2, fourth edn. Griffin, London. · Zbl 0416.62001
[41] King, E.; Hart, J.D.; Wehrly, T.E., Testing the equality of two regression curves using linear smoothers, Statistics and probability letters, 12, 239-247, (1991)
[42] Klein, R.; Spady, R., An efficient semiparametric estimator for binary response models, Econometrica, 61, 387-422, (1993) · Zbl 0783.62100
[43] Koul, H.L.; Schick, A., Testing for the equality of two nonparametric regression curves, Journal of statistical planning and inference, 65, 293-314, (1997) · Zbl 0908.62057
[44] 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
[45] Lavergne, P.; Vuong, Q., Nonparametric significance testing, Econometric theory, 16, 576-601, (2000) · Zbl 0968.62047
[46] Lepski, O.; Spokoiny, V., Minimax nonparametric hypothesis testingthe case of an inhomogeneous alternative, Bernoulli, 5, 333-358, (1999) · Zbl 0946.62050
[47] Li, Q.; Wang, S., A simple consistent bootstrap test for a parametric regression function, Journal of econometrics, 87, 145-165, (1998) · Zbl 0943.62031
[48] Linton, O.B., Efficient estimation of additive nonparametric regression models, Biometrika, 84, 469-474, (1997) · Zbl 0882.62038
[49] Linton, O.B.; Nielsen, J.P., A kernel method of estimating structured nonparametric regression based on marginal integration, Biometrika, 82, 93-100, (1995) · Zbl 0823.62036
[50] Mammen, E., Estimating a smooth monotone regression function, Annals of statistics, 19, 724-740, (1991) · Zbl 0737.62038
[51] Mammen, E.; Thomas-Agnan, C., Smoothing splines and shape restrictions, Scandinavian journal of statistics, 26, 239-252, (1999) · Zbl 0932.62051
[52] Pagan, A.; Ullah, A., Nonparametric econometrics, (1999), Cambridge University Press Cambridge
[53] Powell, J.L.; Stock, J.H.; Stoker, T.M., Semiparametric estimation of index coefficients, Econometrica, 57, 1403-1430, (1989) · Zbl 0683.62070
[54] Ramsay, J.O., Estimating smooth monotone functions, Journal of the royal statistical society series B, 60, 365-375, (1998) · Zbl 0909.62041
[55] Robinson, P.M., Root-\(n\) consistent semiparametric regression, Econometrica, 56, 931-954, (1988) · Zbl 0647.62100
[56] Schlee, W., Nonparametric tests of the monotony and convexity of regression, (), 823-836 · Zbl 0525.62049
[57] Speckman, P., Kernel smoothing in partial linear models, Journal of the royal statistical society series B, 50, 413-446, (1988) · Zbl 0671.62045
[58] Spokoiny, V., Adaptive hypothesis testing using wavelets, Annals of statistics, 24, 2477-2498, (1996) · Zbl 0898.62056
[59] Stone, C., Additive regression and other nonparametric models, Annals of statistics, 13, 685-705, (1985) · Zbl 0605.62065
[60] Villalobos, M.; Wahba, G., Inequality-constrained multivariate smoothing splines with application to the estimation of posterior probabilities, Journal of the American statistical association, 82, 239-248, (1987) · Zbl 0614.62047
[61] Wand, M.P.; Jones, M.C., Kernel smoothing, (1995), Chapman & Hall London · Zbl 0854.62043
[62] Wooldridge, J., A test for functional form against nonparametric alternatives, Econometric theory, 8, 452-475, (1992)
[63] Wright, I.; Wegman, E., Isotonic, convex and related splines, Annals of statistics, 8, 1023-1035, (1980) · Zbl 0453.62036
[64] Wu, C., Jackknife bootstrap and other resampling methods in regression analysis, Annals of statistics, 14, 1261-1351, (1986) · Zbl 0618.62072
[65] Yatchew, A., Nonparametric regression model tests based on least squares, Econometric theory, 8, 435-451, (1992)
[66] Yatchew, A., Nonparametric regression techniques in economics, Journal of economic literature, XXXVI, 669-721, (1998)
[67] Yatchew, A., An elementary nonparametric differencing test of equality of regression functions, Economics letters, 62, 271-278, (1999) · Zbl 0917.90065
[68] Yatchew, A.; Bos, L., Nonparametric regression and testing in economic models, Journal of quantitative economics, 13, 81-131, (1997)
[69] Yatchew, A.; Sun, Y.; Deri, C., Efficient estimation of semiparametric equivalence scales with evidence from south africa, Journal of business and economics statistics, 21, 247-257, (2003)
[70] Zheng, J.X., A consistent test of functional form via nonparametric estimation techniques, Journal of econometrics, 75, 263-289, (1996) · Zbl 0865.62030
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