zbMATH — the first resource for mathematics

Nonparametric comparison of several regression functions: Exact and asymptotic theory. (English) Zbl 0927.62040
Summary: A new test is proposed for the comparison of two regression curves $$f$$ and $$g$$. We prove an asymptotic normal law under fixed alternatives which can be applied for power calculations, for constructing confidence regions and for testing precise hypotheses of a weighted $$L^2$$ distance between $$f$$ and $$g$$. In particular, the problem of nonequal sample sizes is treated, which is related to a peculiar formula of the area between two step functions.
These results are extended in various directions, such as the comparison of $$k$$ regression functions or the optimal allocation of the sample sizes when the total sample size is fixed. The proposed pivot statistic is not based on a nonparametric estimator of the regression curves and therefore does not require the specification of any smoothing parameter.

MSC:
 62G08 Nonparametric regression and quantile regression 62G10 Nonparametric hypothesis testing 62G05 Nonparametric estimation 62G30 Order statistics; empirical distribution functions
Full Text:
References:
 [1] Azzalini, A. and Bowman, A. (1993). On the use of nonparametric regression for checking linear relationships. J. Roy. Statist. Soc. Ser. B 55 549-557. JSTOR: · Zbl 0800.62222 [2] Buckley, M. J., Eagleson, G. K. and Silverman, B. W. (1988). The estimation of residual variance in nonparametric regression. Biometrika 75 189-199. JSTOR: · Zbl 0639.62032 [3] Carter, C. K. and Eagleson, G. K. (1992). A comparison of variance estimators in nonparametric regression. J. Roy. Statist. Soc. Ser. B 54 773-780. JSTOR: [4] Cox, D., Koh, G., Wahba, G. and Yandell, B. S. (1988). Testing the (parametric) null model hy pothesis in (semiparametric) partial and generalized spline models. Ann. Statist. 16 113-119. · Zbl 0673.62017 [5] Delgado, M. A. (1993). Testing the equality of nonparametric regression curves. Statist. Probab. Lett. 17 199-204. · Zbl 0771.62034 [6] Dette, H., Munk, A. and Wagner, T. (1997). A review of variance estimators with applications to multivariate nonparametric regression models. In Multivariate Design and Sampling (S. Ghosh, ed.) Dekker, New York. · Zbl 0946.62056 [7] Dette, H., Munk, A. and Wagner, T. (1998). Estimating the variance in nonparametric regression by quadratic forms-what is a reasonable choice? J. Roy. Statist. Soc. Ser. B 60 751-764. JSTOR: · Zbl 0944.62041 [8] Eubank, R. L. and Spiegelman, C. H. (1990). Testing the goodness of fit of a linear model via nonparametric regression techniques. J. Amer. Statist. Assoc. 85 387-392. JSTOR: · Zbl 0702.62037 [9] Gasser, T., Sroka, L. and Jennen-Steinmetz, C. (1986). Residual variance and residual pattern in nonlinear regression. Biometrika 73 625-633. JSTOR: · Zbl 0649.62035 [10] Hall, P. and Hart, J. D. (1990). Bootstrap test for difference between means in nonparametric regression. J. Amer. Statist. Assoc. 85 1039-1049. JSTOR: · Zbl 0717.62037 [11] Hall, P. and Marron, J. S. (1990). On variance estimation in nonparametric regression. Biometrika 77 415-419. JSTOR: · Zbl 0711.62035 [12] Hall, P., Kay, J. W. and Titterington, D. M. (1990). Asy mptotically optimal difference-based estimation of variance in nonparametric regression. Biometrika 77 521-528. JSTOR: · Zbl 1377.62102 [13] Härdle, W. and Mammen, E. (1993). Comparing nonparametric versus parametric regression fits. Ann. Statist. 21 1926-1947. · Zbl 0795.62036 [14] Härdle, W. and Marron, J. S. (1990). Semiparametric comparison of regression curves. Ann. Statist. 18 83-89. · Zbl 0703.62053 [15] Hocking, R. R. (1985). The Analy sis of Linear Models. Brooks/Cole, Monterey, CA. · Zbl 0625.62054 [16] King, E. C., Hart, J. D. and Wehrly, T. E. (1991). Testing the equality of regression curves using linear smoothers. Statist. Probab. Lett. 12 239-247. [17] Orey, S. (1958). A central limit theorem for m-dependent random variables. Duke Math. J. 52 543-546. · Zbl 0107.13403 [18] Rice, J. (1984). Bandwidth choice for nonparametric regression. Ann. Statist. 12 1215-1230. · Zbl 0554.62035 [19] Sacks, J. and Ylvisaker, D. (1970). Designs for regression problems with correlated errors. III. Ann. Math. Statist. 41 2057-2074. · Zbl 0234.62025 [20] Ullah, A. and Zinde-Walsh, V. (1992). On the estimation of residual variance in nonparametric regression. J. Nonparametr. Statist. 1 263-265. · Zbl 1263.62062
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.