## The Annals of Statistics

### Nonparametric comparison of regression curves: an empirical process approach

#### Abstract

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 Kulasekera [J. Amer. Statist. Assoc. 90 (1995) 1085-1093] and Kulasekera and Wang [J. Amer. Statist. Assoc. 92 (1997) 500-511] 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 is analyzed in order to demonstrate the benefits of the new test compared to the so far available procedures which require homoscedasticity.

#### Article information

Source
Ann. Statist., Volume 31, Number 3 (2003), 880-920.

Dates
First available in Project Euclid: 25 June 2003

https://projecteuclid.org/euclid.aos/1056562466

Digital Object Identifier
doi:10.1214/aos/1056562466

Mathematical Reviews number (MathSciNet)
MR1994734

Zentralblatt MATH identifier
1032.62037

Subjects
Primary: 62G05: Estimation
Secondary: 60F15: Strong theorems 62F17

#### Citation

Neumeyer, Natalie; Dette, Holger. Nonparametric comparison of regression curves: an empirical process approach. Ann. Statist. 31 (2003), no. 3, 880--920. doi:10.1214/aos/1056562466. https://projecteuclid.org/euclid.aos/1056562466

#### References

• 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.
• BILLINGSLEY, P. (1968). Convergence of Probability Measures. Wiley, New York.
• BOWMAN, A. and YOUNG, S. (1996). Graphical comparison of nonparametric curves. Appl. Statist. 45 83-98.
• CABUS, P. (2000). Testing for the comparison of non-parametric regression curves. Preprint 99-29,
• IRMAR, Univ. Rennes, France.
• DELGADO, M. A. (1993). Testing the equality of nonparametric regression curves. Statist. Probab. Lett. 17 199-204.
• DELGADO, M. A. and GONZÁLEZ-MANTEIGA, W. (2001). Significance testing in nonparametric regression based on the bootstrap. Ann. Statist. 29 1469-1507.
• DETTE, H. and NEUMEy ER, N. (2001). Nonparametric analysis of covariance. Ann. Statist. 29 1361-1400.
• FAN, J. (1992). Design-adaptive nonparametric regression. J. Amer. Statist. Soc. 87 998-1004.
• FAN, J. and GIJBELS, I. (1996). Local Poly nomial Modelling and Its Applications. Chapman and Hall, London.
• GASSER, T., KNEIP, A. and KÖHLER, W. (1991). A flexible and fast method for automatic smoothing. J. Amer. Statist. Assoc. 86 643-652.
• GASSER, T., MÜLLER, H.-G. and MAMMITZSCH, V. (1985). Kernels for nonparametric curve estimation. J. Roy. Statist. Soc. Ser. B 47 238-252.
• HALL, P. and HART, J. D. (1990). Bootstrap test for difference between means in nonparametric regression. J. Amer. Statist. Assoc. 85 1039-1049.
• 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.
• HÄRDLE, W. and MARRON, J. S. (1990). Semiparametric comparison of regression curves. Ann. Statist. 18 63-89.
• HJELLVIK, V. and TJØSTHEIM, D. (1995). Nonparametric tests of linearity for time series. Biometrika 77 351-368.
• 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.
• KULASEKERA, K. B. (1995). Comparison of regression curves using quasi-residuals. J. Amer. Statist. Assoc. 90 1085-1093.
• 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.
• MUNK, A. and DETTE, H. (1998). Nonparametric comparison of several regression functions: Exact and asy mptotic theory. Ann. Statist. 26 2339-2368.
• NADARAy A, E. A. (1964). On estimating regression. Theory Probab. Appl. 9 141-142.
• NOLAN, D. and POLLARD, D. (1987). U-processes: Rates of convergence. Ann. Statist. 15 780-799.
• NOLAN, D. and POLLARD, D. (1988). Functional limit theorems for U-processes. Ann. Probab. 16 1291-1298.
• POLLARD, D. (1984). Convergence of Stochastic Processes. Springer, New York.
• RICE, J. A. (1984). Bandwidth choice for nonparametric regression. Ann. Statist. 12 1215-1230.
• SACKS, J. and YLVISAKER, D. (1970). Designs for regression problems with correlated errors. III. Ann. Math. Statist. 41 2057-2074.
• SHORACK, G. R. and WELLNER, J. A. (1986). Empirical Processes with Applications to Statistics. Wiley, New York.
• STUTE, W. (1997). Nonparametric model checks for regression. Ann. Statist. 25 613-641.
• VAN DER VAART, A. W. and WELLNER, J. A. (1996). Weak Convergence and Empirical Processes. Springer, New York.
• WATSON, G. S. (1964). Smooth regression analysis. Sankhy¯a Ser. A 26 359-372.
• WU, C. F. Y. (1986). Jacknife, bootstrap and other resampling methods in regression analysis (with discussion). Ann. Statist. 14 1261-1350.
• YOUNG, S. G. and BOWMAN, A. W. (1995). Nonparametric analysis of covariance. Biometrics 51 920-931.
• ZHENG, J. X. (1996). A consistent test of functional form via nonparametric estimation techniques. J. Econometrics 75 263-289.