Local polynomial fitting in semivarying coefficient model. (English) Zbl 0995.62038
Summary: Varying coefficient models are useful extensions of the classical linear models. Under the condition that the coefficient functions possess about the same degrees of smoothness, the model can easily be estimated via simple local regression. This leads to the one-step estimation procedure.
We consider a semivarying coefficient model which is an extension of the varying coefficient model, which is called the semivarying-coefficient model. Procedures for estimation of the linear part and the nonparametric part are developed and their associated statistical properties are studied. The proposed methods are illustrated by some simulation studies and a real example.
We consider a semivarying coefficient model which is an extension of the varying coefficient model, which is called the semivarying-coefficient model. Procedures for estimation of the linear part and the nonparametric part are developed and their associated statistical properties are studied. The proposed methods are illustrated by some simulation studies and a real example.
MSC:
| 62G08 | Nonparametric regression and quantile regression |
| 62G20 | Asymptotic properties of nonparametric inference |
| 62G05 | Nonparametric estimation |
Keywords:
varying-coefficient models; local polynomial fit; one-step method; two-step method; optimal rate of convergence; mean squared errorsSoftware:
KernSmoothReferences:
| [1] | Breiman, L.; Friedman, J. H., Estimating optimal transformations for multiple regression and correlation (with discussion), J. Amer. Stat. Assoc., 80, 580-619 (1985) · Zbl 0594.62044 |
| [2] | Cai, Z.; Fan, J.; Yao, Q., Functional-coefficient regression models for nonlinear time series, J. Amer. Statis. Assoc., 98, 941-956 (2000) · Zbl 0996.62078 |
| [3] | Carroll, R. J.; Fan, J.; Gijbels, I.; Wand, M. P., Generalized partially linear single-index models, J. Amer. Statist. Assoc., 92, 477-489 (1997) · Zbl 0890.62053 |
| [4] | Chen, R.; Tsay, R. S., Functional-coefficient autoregressive models, J. Amer. Statist. Assoc., 88, 298-308 (1993) · Zbl 0776.62066 |
| [5] | Cleveland, W. S., Robust locally weighted regression and smoothing scatterplots, J. Amer. Statist. Assoc., 74, 829-836 (1979) · Zbl 0423.62029 |
| [6] | Cleveland, W. S.; Grosse, E.; Shyu, W. M., Local regression models, (Chambers, J. M.; Hastie, T. J., Statistical Models in S (1991), Wadsworth & Brooks: Wadsworth & Brooks Pacific Grove), 309-376 |
| [7] | Fan, J., Design-adaptive nonparametric regression, J. Amer. Stat. Assoc., 87, 998-1004 (1992) · Zbl 0850.62354 |
| [8] | Fan, J.; Gijbel, I., Data-driven bandwidth selection in local polynomial fitting: variable bandwidth and spatial adaptation, J. Roy. Statist. Soc. Ser. B., 57, 371-394 (1995) · Zbl 0813.62033 |
| [9] | Fan, J.; Gijbels, I., Local Polynomial Modeling and Its Applications (1996), Chapman and Hall: Chapman and Hall London · Zbl 0873.62037 |
| [10] | J. Fan, Q. Yao, and, Z. Cai, Varying-coefficient linear models with unknown indices, manuscript submitted for publication, 2000.; J. Fan, Q. Yao, and, Z. Cai, Varying-coefficient linear models with unknown indices, manuscript submitted for publication, 2000. |
| [11] | Fan, J.; Zhang, J., Comments on “Smoothing spline models for the analysis of nested and crossed samples of curves”, J. Amer. Statist. Assoc., 93, 980-984 (1998) |
| [12] | Fan, J.; Zhang, W., Statistical estimation in varying coefficient models, Ann. Statist., 27, 1491-1518 (1999) · Zbl 0977.62039 |
| [13] | Fan, J.; Zhang, W., Simultaneous confidence bands are and hypothesis testing in varying-coefficient models, Scand. J. Statist., 27, 715-731 (2000) · Zbl 0962.62032 |
| [14] | Friedman, J. H., Multivariate adaptive regression splines (with discussion), Ann. Statist., 19, 1-141 (1991) · Zbl 0765.62064 |
| [15] | Green, P. J.; Silverman, B. W., Nonparametric Regression and Generalized Linear Models: a Roughness Penalty Approach (1994), Chapman and Hall: Chapman and Hall London · Zbl 0832.62032 |
| [16] | Gu, C.; Wahba, G., Smoothing spline ANOVA with component-wise Bayesian “confidence intervals”, J. Comput. Graph. Statist., 2, 97-117 (1993) |
| [17] | Härdle, W.; Stoker, T. M., Investigating smooth multiple regression by the method of average derivatives, J. Amer. Statist. Assoc., 84, 986-995 (1989) · Zbl 0703.62052 |
| [18] | Hastie, T. J.; Tibshirani, R., Generalized Additive Models (1990), Chapman and Hall: Chapman and Hall London · Zbl 0747.62061 |
| [19] | Hastie, T. J.; Tibshirani, R. J., Varying-coefficient models, J. Roy. Statist. Soc. Ser. B., 55, 757-796 (1993) · Zbl 0796.62060 |
| [20] | Hoover, D. R.; Rice, J. A.; Wu, C. O.; Yang, L. P., Nonparametric smoothing estimates of time-varying coefficient models with longitudinal data, Biometrika, 85, 809-822 (1998) · Zbl 0921.62045 |
| [21] | Linhart, H.; Zucchini, W., Model Selection (1986), Wiley: Wiley New York · Zbl 0665.62003 |
| [22] | Mack, Y. P.; Silverman, B. W., Weak and strong uniform consistency of kernel regression estimates, Z. Wahr. Gebiete, 61, 405-415 (1982) · Zbl 0495.62046 |
| [23] | Ruppert, D., Empirical-bias bandwidths for local polynomial nonparametric regression and density estimation, J. Amer. Statist. Assoc., 92, 1049-1062 (1997) · Zbl 1067.62531 |
| [24] | Ruppert, D.; Sheather, S. J.; Wand, M. P., An effective bandwidth selector for local least squares regression, J. Amer. Statist. Assoc., 90, 1257-1270 (1995) · Zbl 0868.62034 |
| [25] | Stone, C. J.; Hansen, M.; Kooperberg, C.; Truong, Y. K., Polynomial splines and their tensor products in extended linear modeling, Ann. Statist., 25, 1371-1470 (1997) · Zbl 0924.62036 |
| [26] | Stone, M., Cross-validatory choice and assessment of statistical predictions (with discussion), J. Roy. Statist. Soc. Ser. B., 36, 111-147 (1974) · Zbl 0308.62063 |
| [27] | Wahba, G., Partial spline models for semiparametric estimation of functions of several variables, Statistical Analysis of Time Series, Proceedings of the Japan U.S. Joint Seminar, Tokyo (1984), Institute of Statistical Mathematics: Institute of Statistical Mathematics Tokyo, p. 319-329 |
| [28] | Wahba, G., Spline Models for Observing Data (1990), Society for Industrial and Applied Mathematics: Society for Industrial and Applied Mathematics Philadelphia · Zbl 0813.62001 |
| [29] | Wand, M. P.; Jones, M. C., Kernel Smoothing (1995), Chapman and Hall: Chapman and Hall London · Zbl 0854.62043 |
| [30] | West, M.; Harrison, P. J., Bayesian Forecasting and Dynamic Models (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0697.62029 |
| [31] | West, M.; Harrison, P. J.; Migon, H. S., Dynamic generalized linear models and Bayesian forecasting (with discussion), J. Amer. Statist. Assoc., 80, 73-97 (1985) · Zbl 0568.62032 |
| [32] | W. Zhang, and, S. Y. Lee, On local polynomial fitting of varying coefficient models, manuscript submitted for publication, 1998.; W. Zhang, and, S. Y. Lee, On local polynomial fitting of varying coefficient models, manuscript submitted for publication, 1998. |
| [33] | Zhang, W.; Lee, S. Y., Variable bandwidth selection in varying coefficient models, J. Multivariate Anal., 74, 116-134 (2000) · Zbl 0969.62032 |
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
