## Nonparametric comparison of regression curves: An empirical process approach.(English)Zbl 1032.62037

Summary: We propose a new test for the comparison of two regression curves that is based on a difference of two marked empirical processes based on residuals. The large sample behavior of the corresponding statistic is studied to provide a full nonparametric comparison of regression curves. In contrast to most procedures suggested in the literature, the new procedure is applicable in the case of different design points and heteroscedasticity.
Moreover, it is demonstrated that the proposed test detects continuous alternatives converging to the null at a rate $$N^{1/2}$$ and that, in contrast to all other available procedures based on marked empirical processes, the new test allows the optimal choice of bandwidths for curve estimation (e.g., $$N^{-1/5}$$ in the case of twice differentiable regression functions). As a by-product we explain the problems of a related test proposed by K. B. Kulasekera [J. Am. Stat. Assoc. 90, No. 431, 1085-1093 (1995; Zbl 0841.62039)] and K. B. Kulasekera and J. Wang [ibid. 92, No. 438, 500-511 (1997; Zbl 0894.62047)] with respect to accuracy in the approximation of the level. These difficulties mainly originate from the comparison with the quantiles of an inappropriate limit distribution.
A simulation study is conducted to investigate the finite sample properties of a wild bootstrap version of the new test and to compare it with the so far available procedures. Finally, heteroscedastic data are analyzed in order to demonstrate the benefits of the new test compared to the so far available procedures which require homoscedasticity.

### MSC:

 62G08 Nonparametric regression and quantile regression 62G30 Order statistics; empirical distribution functions 60F17 Functional limit theorems; invariance principles 60F15 Strong limit theorems

### Citations:

Zbl 0841.62039; Zbl 0894.62047
Full Text:

### References:

 [1] AN, H.-Z. and BING, C. (1991). A Kolmogorov-Smirnov ty pe statistic with application to test for nonlinearity in time series. Internat. Statist. Rev. 59 287-307. · Zbl 0748.62049 [2] BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York. · Zbl 0172.21201 [3] BOWMAN, A. and YOUNG, S. (1996). Graphical comparison of nonparametric curves. Appl. Statist. 45 83-98. · Zbl 0858.62003 [4] CABUS, P. (2000). Testing for the comparison of non-parametric regression curves. Preprint 99-29, [5] IRMAR, Univ. Rennes, France. [6] DELGADO, M. A. (1993). Testing the equality of nonparametric regression curves. Statist. Probab. Lett. 17 199-204. · Zbl 0771.62034 [7] DELGADO, M. A. and GONZÁLEZ-MANTEIGA, W. (2001). Significance testing in nonparametric regression based on the bootstrap. Ann. Statist. 29 1469-1507. · Zbl 1043.62032 [8] DETTE, H. and NEUMEy ER, N. (2001). Nonparametric analysis of covariance. Ann. Statist. 29 1361-1400. · Zbl 1043.62033 [9] FAN, J. (1992). Design-adaptive nonparametric regression. J. Amer. Statist. Soc. 87 998-1004. JSTOR: · Zbl 0850.62354 [10] FAN, J. and GIJBELS, I. (1996). Local Poly nomial Modelling and Its Applications. Chapman and Hall, London. · Zbl 0873.62037 [11] GASSER, T., KNEIP, A. and KÖHLER, W. (1991). A flexible and fast method for automatic smoothing. J. Amer. Statist. Assoc. 86 643-652. JSTOR: · Zbl 0733.62047 [12] GASSER, T., MÜLLER, H.-G. and MAMMITZSCH, V. (1985). Kernels for nonparametric curve estimation. J. Roy. Statist. Soc. Ser. B 47 238-252. JSTOR: · Zbl 0574.62042 [13] 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 [14] HALL, P., HUBER, C. and SPECKMAN, P. L. (1997). Covariate-matched one-sided tests for the difference between functional means. J. Amer. Statist. Assoc. 92 1074-1083. JSTOR: · Zbl 0889.62033 [15] HÄRDLE, W. and MARRON, J. S. (1990). Semiparametric comparison of regression curves. Ann. Statist. 18 63-89. · Zbl 0703.62053 [16] HJELLVIK, V. and TJØSTHEIM, D. (1995). Nonparametric tests of linearity for time series. Biometrika 77 351-368. JSTOR: · Zbl 0823.62044 [17] KING, E. C., HART, J. D. and WEHRLY, T. E. (1991). Testing the equality of two regression curves using linear smoothers. Statist. Probab. Lett. 12 239-247. [18] KULASEKERA, K. B. (1995). Comparison of regression curves using quasi-residuals. J. Amer. Statist. Assoc. 90 1085-1093. JSTOR: · Zbl 0841.62039 [19] KULASEKERA, K. B. and WANG, J. (1997). Smoothing parameter selection for power optimality in testing of regression curves. J. Amer. Statist. Assoc. 92 500-511. JSTOR: · Zbl 0894.62047 [20] MUNK, A. and DETTE, H. (1998). Nonparametric comparison of several regression functions: Exact and asy mptotic theory. Ann. Statist. 26 2339-2368. · Zbl 0927.62040 [21] NADARAy A, E. A. (1964). On estimating regression. Theory Probab. Appl. 9 141-142. [22] NOLAN, D. and POLLARD, D. (1987). U-processes: Rates of convergence. Ann. Statist. 15 780-799. · Zbl 0624.60048 [23] NOLAN, D. and POLLARD, D. (1988). Functional limit theorems for U-processes. Ann. Probab. 16 1291-1298. · Zbl 0665.60037 [24] POLLARD, D. (1984). Convergence of Stochastic Processes. Springer, New York. · Zbl 0544.60045 [25] RICE, J. A. (1984). Bandwidth choice for nonparametric regression. Ann. Statist. 12 1215-1230. · Zbl 0554.62035 [26] SACKS, J. and YLVISAKER, D. (1970). Designs for regression problems with correlated errors. III. Ann. Math. Statist. 41 2057-2074. · Zbl 0234.62025 [27] SHORACK, G. R. and WELLNER, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York. · Zbl 1170.62365 [28] STUTE, W. (1997). Nonparametric model checks for regression. Ann. Statist. 25 613-641. · Zbl 0926.62035 [29] VAN DER VAART, A. W. and WELLNER, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York. · Zbl 0862.60002 [30] WATSON, G. S. (1964). Smooth regression analysis. Sankhy\?a Ser. A 26 359-372. · Zbl 0137.13002 [31] WU, C. F. Y. (1986). Jacknife, bootstrap and other resampling methods in regression analysis (with discussion). Ann. Statist. 14 1261-1350. · Zbl 0618.62072 [32] YOUNG, S. G. and BOWMAN, A. W. (1995). Nonparametric analysis of covariance. Biometrics 51 920-931. · Zbl 0875.62312 [33] ZHENG, J. X. (1996). A consistent test of functional form via nonparametric estimation techniques. J. Econometrics 75 263-289. · Zbl 0865.62030
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.