The Consistency of Automatic Kernel Density Estimates

We consider the Parzen-Rosenblatt kernel density estimate on Rd with data-dependent smoothing factor. Sufficient conditions on the asymptotic behavior of the smoothing factor are given under which the estimate is pointwise consistent almost everywhere for all densities f to be estimated. When the smoothing factor is a function only of the sample size n, it is shown that these conditions are also necessary, a generalization of results by Deheuvels. The consistency of various automatic kernel density estimates is a simple consequence of these theorems.

[1]  L. Devroye.  On arbitrarily slow rates of global convergence in density estimation , 1983 .

[2]  L. Devroye.  The Equivalence of Weak, Strong and Complete Convergence in $L_1$ for Kernel Density Estimates , 1983 .

[3]  L. Devroye.  THE EQUIVALENCE OF WEAK , STRONG AND COMPLETE CONVERGENCE IN Ll FOR KERNEL DENSITY ESTIMATES ' BY LUC DEVROYE McGill , 1983 .

[4]  Peter Hall,et al.  Cross-validation in density estimation , 1982 .

[5]  A. Krieger,et al.  Weak Convergence and Efficient Density Estimation at a Point , 1981 .

[6]  David W. Scott,et al.  Monte Carlo Study of Three Data-Based Nonparametric Probability Density Estimators , 1981 .

[7]  Jerome Sacks,et al.  ASYMPTOTICALLY OPTIMUM KERNELS FOR DENSITY ESTIMATION AT A POINT , 1981 .

[8]  Stuart Geman,et al.  Sieves for Nonparametric Estimation of Densities and Regressions. , 1981 .

[9]  E. F. Schuster,et al.  On the Nonconsistency of Maximum Likelihood Nonparametric Density Estimators , 1981 .

[10]  Paul Deheuvels,et al.  Estimation automatique de la densité , 1980 .

[11]  L. Devroye,et al.  The $L_1$ Convergence of Kernel Density Estimates , 1979 .

[12]  M. Rosenblatt.  Global measures of deviation for kernel and nearest neighbor density estimates , 1979 .

[13]  B. Silverman,et al.  Choosing the window width when estimating a density , 1978 .

[14]  R. Tapia,et al.  Nonparametric Probability Density Estimation , 1978 .

[15]  J. Bretagnolle,et al.  Estimation des densités: risque minimax , 1978 .

[16]  L. Breiman,et al.  Variable Kernel Estimates of Multivariate Densities , 1977 .

[17]  P. Deheuvels.  Estimation non paramétrique de la densité par histogrammes généralisés , 1977 .

[18]  D. W. Scott,et al.  Kernel density estimation revisited , 1977 .

[19]  Robert P. W. Duin,et al.  On the Choice of Smoothing Parameters for Parzen Estimators of Probability Density Functions , 1976, IEEE Transactions on Computers.

[20]  T. Wagner,et al.  Nonparametric estimates of probability densities , 1975, IEEE Trans. Inf. Theory.

[21]  E. Nadaraya,et al.  On the Integral Mean Square Error of Some Nonparametric Estimates for the Density Function , 1974 .

[22]  J. Habbema.  A stepwise discriminant analysis program using density estimetion , 1974 .

[23]  E. Wegman.  Nonparametric probability density estimation , 1972 .

[24]  M. Woodroofe.  On Choosing a Delta-Sequence , 1970 .

[25]  G. Bennett.  Probability Inequalities for the Sum of Independent Random Variables , 1962 .

[26]  J. Kiefer.  On large deviations of the empiric D. F. of vector chance variables and a law of the iterated logarithm. , 1961 .

[27]  T. Broadbent.  Measure and Integral , 1957, Nature.

[28]  M. Rosenblatt.  Remarks on Some Nonparametric Estimates of a Density Function , 1956 .

[29]  S.,et al.  CONSISTENT CROSS-VALIDATED DENSITY ESTIMATION , 2022 .