REGRESSION MODELING FOR NONPARAMETRIC ESTIMATION OF DISTRIBUTION AND QUANTILE FUNCTIONS

We propose a local linear estimator of a smooth distribution function. This estimator applies local linear techniques to observations from a regression model in which the value of the empirical distribution function equals the value of true distribution plus an error term. We show that, for most commonly used ker- nel functions, our local linear estimator has a smaller asymptotic mean integrated squared error than the conventional kernel distribution estimator. Importantly, since this MISEreduction occurs through a constant factor of a second order term, any bandwidth selection procedures for kernel distribution estimator can be easily adapted for our estimator. For the estimation of a smooth quantile function, we establish a regression model of the empirical quantile function and obtain a local quadratic estimator. It has better asymptotic performance than the kernel quantile estimator in both interior and boundary cases.

[1]  Jianqing Fan.  Design-adaptive Nonparametric Regression , 1992 .

[2]  E. Nadaraya.  On Estimating Regression , 1964 .

[3]  E. Parzen,et al.  Unified estimators of smooth quantile and quantile density functions , 1997 .

[4]  C. Z. Wei,et al.  A regression point of view toward density estimation , 1994 .

[5]  James Stephen Marron,et al.  Kernel Quantile Estimators , 1990 .

[6]  M. Lejeune,et al.  Smooth estimators of distribution and density functions , 1992 .

[7]  P. Hall,et al.  Bandwidth selection for the smoothing of distribution functions , 1998 .

[8]  K. Zou,et al.  Smooth non-parametric receiver operating characteristic (ROC) curves for continuous diagnostic tests. , 1997, Statistics in medicine.

[9]  E. Nadaraya,et al.  Some New Estimates for Distribution Functions , 1964 .

[10]  Hajime Yamato,et al.  UNIFORM CONVERGENCE OF AN ESTIMATOR OF A DISTRIBUTION FUNCTION , 1973 .

[11]  M. Falk.  Relative Deficiency of Kernel Type Estimators of Quantiles , 1984 .

[12]  Jianqing Fan.  Local Linear Regression Smoothers and Their Minimax Efficiencies , 1993 .

[13]  Jianqing Fan,et al.  Local polynomial modelling and its applications , 1994 .

[14]  Jianqing Fan,et al.  On automatic boundary corrections , 1997 .

[15]  M. Falk.  Relative efficiency and deficiency of kernel type estimators of smooth distribution functions , 1983 .

[16]  E. Mammen,et al.  A General Projection Framework for Constrained Smoothing , 2001 .

[17]  D. Mason,et al.  Weighted Empirical and Quantile Processes , 1986 .

[18]  E. Parzen.  Nonparametric Statistical Data Modeling , 1979 .

[19]  W. González-Manteiga,et al.  Edgeworth expansions for nonparametric distribution estimation with applications , 1997 .

[20]  Smooth nonparametric estimation of the quantile function , 1990 .

[21]  Shie-Shien Yang.  A Smooth Nonparametric Estimator of a Quantile Function , 1985 .