Confidence bands in nonparametric regression

Abstract New bias-corrected confidence bands are proposed for nonparametric kernal regression. These bands are constructed using only a kernel estimator of the regression curve and its data-selected bandwidth. They are shown to have asymptotically correct coverage properties and to behave well in a small-sample study. One consequence of the large-sample developments is that Bonferroni-type bands for the regression curve at the design points also have conservative asymptotic coverage behavior with no bias correction.

[1]  W. Härdle.  Applied Nonparametric Regression , 1992 .

[2]  Gordon J Johnston,et al.  Probabilities of maximal deviations for nonparametric regression function estimates , 1982 .

[3]  Ker-Chau Li,et al.  Honest Confidence Regions for Nonparametric Regression , 1989 .

[4]  J. Sacks,et al.  Model robust confidence intervals , 1982 .

[5]  J. Rice.  Bandwidth Choice for Nonparametric Regression , 1984 .

[6]  Douglas W. Nychka,et al.  The Average Posterior Variance of a Smoothing Spline and a Consistent Estimate of the Average Squared Error , 1990 .

[7]  Douglas Nychka,et al.  Bayesian Confidence Intervals for Smoothing Splines , 1988 .

[8]  J. Marron,et al.  On variance estimation in nonparametric regression , 1990 .

[9]  Wolfgang Karl Härdle,et al.  Asymptotic maximal deviation of M-smoothers , 1989 .

[10]  James Stephen Marron,et al.  BOOTSTRAP SIMULTANEOUS ERROR BARS FOR NONPARAMETRIC REGRESSION , 1991 .

[11]  Ulrich Stadtmüller,et al.  Estimation of Heteroscedasticity in Regression Analysis , 1987 .

[12]  P. Révész,et al.  Strong approximations in probability and statistics , 1981 .

[13]  G. Wahba.  Bayesian "Confidence Intervals" for the Cross-validated Smoothing Spline , 1983 .

[14]  W. Härdle,et al.  How Far are Automatically Chosen Regression Smoothing Parameters from their Optimum , 1988 .

[15]  H. Müller,et al.  Kernels for Nonparametric Curve Estimation , 1985 .

[16]  B. Silverman,et al.  Some Aspects of the Spline Smoothing Approach to Non‐Parametric Regression Curve Fitting , 1985 .

[17]  D. Andrews,et al.  Asymptotic optimality of generalized CL, cross-validation, and generalized cross-validation in regression with heteroskedastic errors , 1991 .

[18]  Jerome Sacks,et al.  Confidence Bands for Regression Functions , 1985 .

[19]  W. Härdle.  Applied Nonparametric Regression , 1991 .

[20]  W. Härdle,et al.  Bootstrapping in Nonparametric Regression: Local Adaptive Smoothing and Confidence Bands , 1988 .

[21]  D. M. Titterington,et al.  On confidence bands in nonparametric density estimation and regression , 1988 .

[22]  T. Gasser,et al.  Residual variance and residual pattern in nonlinear regression , 1986 .

[23]  P. Bickel,et al.  On Some Global Measures of the Deviations of Density Function Estimates , 1973 .

[24]  P. Hall,et al.  Asymptotically optimal difference-based estimation of variance in nonparametric regression , 1990 .

[25]  U. Stadtmüller,et al.  Asymptotic properties of nonparametric curve estimates , 1986 .

[26]  Kjell A. Doksum,et al.  Uniform Confidence Bounds for Regression Based on a Simple Moving Average , 1985 .

[27]  H. Müller.  Nonparametric regression analysis of longitudinal data , 1988 .

[28]  D. Cox,et al.  Asymptotic Analysis of Penalized Likelihood and Related Estimators , 1990 .

[29]  P. Whittle,et al.  Bounds for the Moments of Linear and Quadratic Forms in Independent Variables , 1960 .