zbMATH — the first resource for mathematics

Testing goodness of fit of polynomial models via spline smoothing techniques. (English) Zbl 0793.62026
Summary: A new test statistic is proposed for testing goodness of fit of an \(m\)-th order polynomial regression model. The test statistic is \[ \int^ 1_ 0 \biggl[ \mu_ \lambda^{(m)} (t) \biggr]^ 2 \text{d}t, \] where \(\mu_ \lambda^{(m)}\) is the \(m\)-th order derivative of a \(2m\)-th order smoothing spline estimator for the regression function \(\mu\) and \(\lambda\) is its associated smoothing parameter. The large sample properties of the test statistic are derived under both the null hypothesis and local alternatives. A numerical example is included that illustrates the technique.

62G10 Nonparametric hypothesis testing
62J02 General nonlinear regression
62G07 Density estimation
62E20 Asymptotic distribution theory in statistics
Full Text: DOI
[1] Buckley, M.J., Detecting a smooth signal: optimality of cusum based procedures, Biometrika, 78, 253-262, (1991)
[2] Buckley, M.J.; Eagleson, G.K., An approximation to the distribution of quadratic forms in normal random variables, Austral. J. statist., 30A, 149-163, (1988) · Zbl 0652.62017
[3] Chen, J.C., Testing goodness of fit of polynomial models via spline smoothing techniques, Ph.D. dissertation, (1992), Dept. of Statist., Texas A & M Univ College Station, TX
[4] Cox, D.D.; Koh, E., A smoothing spline based test of model adequacy in polynomial regression, Ann. inst. statist. math., 41, 383-400, (1989) · Zbl 0692.62019
[5] Cox, D.D.; Koh, E.; Wahba, G.; Yandell, B., Testing the (parametric) null model hypothesis in (semiparametric) partial and generalized spline models, Ann statist., 16, 113-119, (1988) · Zbl 0673.62017
[6] De Jong, P., A central limit theorem for generalized quadratic forms, Probab. theory rel. fields, 25, 261-277, (1987) · Zbl 0596.60022
[7] Demmler, A.; Reinsch, C., Oscillation matrices with spline smoothing, Numer. math., 24, 375-382, (1975) · Zbl 0297.65002
[8] Eubank, R.L., Spline smoothing and nonparametric regression, (1988), Dekker New York · Zbl 0702.62036
[9] Eubank, R.L.; Hart, J.D., Testing goodness-of-fit in regression via order selection criteria, Ann. statist., 20, 1412-1425, (1992) · Zbl 0776.62045
[10] Eubank, R.L.; Hart, J.D., Commonality of cusum, von Neumann and smoothing-based goodness-of-fit tests, Biometrika, (1993), to appear in · Zbl 0792.62042
[11] Eubank, R.L.; LaRiccia, V.N., Testing for no effect in nonparametric regression, J. statist. plann. inference, 36, 1-14, (1993) · Zbl 0771.62035
[12] Eubank, R.L.; Spiegelman, C.H., Testing the goodness of fit of a linear model via nonparametric regression techniques, J. amer. statist. assoc., 85, 387-392, (1990) · Zbl 0702.62037
[13] Gasser, T.; Sroka, L.; Jennen-Steinmetz, C., Residual variance and residual pattern in nonlinear regression, Biometrika, 73, 625-633, (1986) · Zbl 0649.62035
[14] Graybill, F.A., Theory and application of the linear model, (1976), Duxbury Press Boston, MA · Zbl 0121.35605
[15] Hall, P.; Hart, J.D., Bootstrap test for difference between means in nonparametric regression, J. amer. statist. assoc., 85, 1039-1049, (1990) · Zbl 0717.62037
[16] Hall, P.; Kay, J.W.; Titterington, D.M., Asymptotically optimal difference-based estimation of variance in nonparametric regression, Biometrika, 77, 521-528, (1990) · Zbl 1377.62102
[17] Härdle, W.; Mammen, E., Comparing nonparametric versus parametric regression fits, Ann. statist., (1988), to appear in · Zbl 0795.62036
[18] Hart, J.D.; Wehrly, T.E., Kernel regression when the boundary region is large, with an application to testing the adequacy of polynomial models, J. amer. statist. assoc., 87, 1018-1024, (1992) · Zbl 0764.62036
[19] Jayasuriya, B.R., Testing for polynomial regression using nonparametric regression techniques, Ph.D. dissertation, (1990), Dept. of Statist., Texas A & M Univ College Station, TX
[20] King, E.; Hart, J.D.; Wehrly, T.E., Testing the equality of two regression curves using linear smoothers, Statist. probab. lett., 12, 239-247, (1991)
[21] Kleinbaum, D.G.; Kupper, L.L.; Muller, K.E., Application regression analysis and other multivariable methods, (1988), PWS-KENT Boston, MA
[22] Montgomery, D.C.; Peck, E.A., Introduction to linear regression analysis, (1982), Wiley New York · Zbl 0587.62134
[23] Müller, H.G., Goodness-of-fit diagnostics for regression models, Scand. J. statist., 19, 157-172, (1992) · Zbl 0760.62037
[24] Neumann, J.von, Distribution of the ratio of the Mean squared successive difference to the variance, Ann. math. statist., 12, 367-395, (1941) · Zbl 0060.29911
[25] Nychka, D., Smoothing splines as locally weighted averages, (1989), Dept. of Statist., North Carolina State Univ Raleigh, NC, unpublished manuscript
[26] Raz, J., Testing for no effect when estimating a smooth function by nonparametric regression: a randomization approach, J. amer. statist. assoc., 85, 132-138, (1990)
[27] Speckman, P., The asymptotic integrated Mean square error for smoothing noisy data by spline functions, (1981), Dept. of Statist., Univ. of Missouri Columbia, MO, unpublished manuscript
[28] Staniswalis, J.G.; Severini, T.A., Diagnostics for assessing regression models, J. amer. statist. assoc., 86, 684-692, (1991) · Zbl 0736.62063
[29] Wahba, G., Spline models for observational data, (1990), SIAM Philadelphia, PA · Zbl 0813.62001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.