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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.

##### MSC:
 62G10 Nonparametric hypothesis testing 62J02 General nonlinear regression 62G07 Density estimation 62E20 Asymptotic distribution theory in statistics
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##### References:
 [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
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