臺灣博碩士論文加值系統

Many different methods have been proposed to construct nonparametric estimates of a smooth regression function and a density function. In this paper an improved method for nonparametric estimation is proposed. It is based on the exact equality for the regular function, which is an extended version of Kao (2004). The main goal of this article is to develop the above equality for nonparametric estimation of functions. According the Law of Large Numbers, the new estimator can be reduced as a kernel estimator. We apply the idea of bandwidth selection solve-the-equation rule to the new estimator and compared with kernel density estimator. Our results are applicable to nonparametric regression. A comparison of the local linear regression method and the proposed method with the same bandwidth selection rule as obtained from direct plug-in methodology described by Ruppert, Sheather, and Wand (1995) is also given in the simulations. It is shown that the new estimators should be particularly useful in some situations asymptotically. The idea given in this work is particularly useful for accurate extrapolation in regression and density estimations.

1. Introduction
2. Nonparametric Density Estimation
2.1 Survey of Existing Methods
2.2 Estimation
2.3 Properties of the New Estimate
2.4 Bandwidth Selection
2.5 Simulations
3. Nonparametric Regression
3.1 Survey of Existing Methods
2.2 Estimation
2.3 Properties of the New Estimate
2.4 Some Bandwidth Selection Rules
2.5 Simulations
4. Conclusion

1. Ahmad, I. A. and Ran, I. S. (2004), "Kernel contrasts: a data-based method of choosing smoothing parameters in nonparametric density estimation", Journal of Nonparametric Statistics, 16, 671-707.
2. Bowman, A. W. (1984), "An alternative method of cross-validation for the smoothing of density estimates", Biometrika, 71, 353-360.
3.Chiu, S. T. (1992), "An automatic bandwidth selector for kernel density estimate", Biometrika, 79, 177-182.
4. Cleveland, W. S. (1979), "Robust locally weighted regression and smoothing scatterplots", Journal of the American Statistical Association, 74, 829-836.
5.Eubank, R. L. (1988), Spline Smoothing and Nonparametric Regression, Marcel Dekker, New York.
6. Fan, J. and Gijbels, I. (1988), Local Polynomial Modeling and Its Application, Chapman and Hall, New York.
7. Hall, P. (1984), "Central limit theorem for itegrated square error of multivariate nonparametric density estimators", Journal of multivariate analysis, 14, 1-16.
8. Hall, P. and Marron, J. S. (1987), "Estimation of itegrated squared density derivatives", Statistics and Probability Letters, 6, 109-115.
9. Hall, P., Marron, J. S., and Park, B. U. (1992), "Smoothing cross-validation ", Probability Theory and Related Fields, 92, 1-20.
10. Hall, P., Sheather, S., Jones, M, C., and Marron, J. S. (1991). "On optimal data-based bandwidth selestion in kernel density estimation",Biometrika, 78, 263-269.
11. Hardle, W. (1990), Applied Nonparametric Regression, Cambridge University Press, Boston, MA.
12. Hardle, W., Hall, P., and Marron, J. S. (1988), "How far are automatically chosen regression smoothing parameters from their optimum?", Journal of the American Statistical Association, 83, 86-95.
13. Jones, M. C., Marron, J. S., and Sheather, S. J. (1996), "A brief survey of bandwidth selection for density estimation", Journal of the American Statistical Association, 91, 401-407.
14. Kao, C. S. (2004), "Error bounds for some new approximation forms of regular functions", Bulletin of the Institute of Mathmematics Academia Sinica, 32, 1-14.
15. Muller, H. G. (1984), "Smooth optimum kernel estimators of densities,regression curves and modes", The Annals of Statistics, 12, 766-774.
16. Muller, H. G. (1988), Nonparametric Regression Analysis of Longitudinal Data, Lectures Notes in Statistics, 46. Springer-Verlag, Berlin.
17. Park, B. U. and Marron,J. S. (1990), "Comparison of data driven bandwidth selectors", Journal of the American Statistical Association, 85, 66-72.
18. Rosenblattm, M. (1956), " Remarks on some nonparametric estimates of a density function", Annals of Mathematical Statistics, 27, 832-837.
19. Rudemo, M. (1982), "Empirical choice of histograms and kernel density estimators", Scandinavian Journal of Statistics, 9, 65-78.
20. Ruppert, D., Sheather, S. J., and Wand, M. P. (1995), "An effective bandwidth selector for local least squares regression", Journal of the American Statistical Association, 90, 1257-1270.
21. Scott, D. W. (1992), Multivariate Density Estimation: Theory, Practice, and Visualization, Wiley, New York.
22. Scott, D. W. and Terrell, G. R. (1987), "Biased and unbiased cross-validation in density estimation", Journal of the American Statistical Association, 82, 1131-1146.
23. Sheather, S. J. and Jones, M. C. (1991), "A reliable data-based bandwidth selection method for kernel density estimation", Journal of the Royal Statistical Society, Series B, 53. 683-690.
24. Silverman, B. W. (1986), Density Estimation for Statistics and Data Analysis, Chapman and Hall, London, UK.
25. Simonoff, J. S. (1996), Smoothing Methods ain Statistics, Springer-Verlag, New York.
26. Stone, C. J. (1977), "Consistent nonparametric regression (with discussion)", The Annals of Statistics, 5, 595-645.
27. Wand. M. P. and Jones, M. C. (1995), Kernel Smoothing, Chapman and Hall, London.