The law of the iterated logarithm and maximal smoothing principle for the kernel distribution function estimator

Two new properties of the kernel distribution function estimator of diverse nature are derived. Firstly, a law of the iterated logarithm is proved for both the integrated absolute error and the integrated squared error of the estimator. Secondly, the maximal smoothing principle in kernel density estimation developed by Terrell is extended to kernel distribution function estimation, which allows, among others, the derivation of an alternative quick-and-simple bandwidth selector. In fact, there is a common link between the two topics: both problems are solved through the use of the same, not-so-standard, methodology. The results based on simulated data and a real data set are also presented.

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

[2]  B. B. Winter.  Strong uniform consistency of integrals of density estimators , 1973 .

[3]  Zhezhen Jin,et al.  On kernel estimation of a multivariate distribution function , 1999 .

[4]  G. Terrell.  The Maximal Smoothing Principle in Density Estimation , 1990 .

[5]  Integrated squared error of kernel-type estimator of distribution function , 1992 .

[6]  L. Devroye,et al.  Nonparametric density estimation : the L[1] view , 1987 .

[7]  Rolf-Dieter Reiss,et al.  Nonparametric Estimation of Smooth Distribution Functions , 2016 .

[8]  B. B. Winter.  Convergence rate of perturbed empirical distribution functions , 1979 .

[9]  Note on the minimum mean integrated squared error of kernel estimates of a distribution function and its derivatives , 1993 .

[10]  Nathaniel Schenker,et al.  Qualms about Bootstrap Confidence Intervals , 1985 .

[11]  B. Efron.  Better Bootstrap Confidence Intervals , 1987 .

[12]  R. Nickl,et al.  An exponential inequality for the distribution function of the kernel density estimator, with applications to adaptive estimation , 2009 .

[13]  Alan M. Polansky,et al.  Multistage plug—in bandwidth selection for kernel distribution function estimates , 2000 .

[14]  M. R. Leadbetter,et al.  HAZARD ANALYSIS II , 1964 .

[15]  S. Brendle,et al.  Calculus of Variations , 1927, Nature.

[16]  M. C. Jones.  The performance of kernel density functions in kernel distribution function estimation , 1990 .

[17]  J. Yukich.  A note on limit theorems for perturbed empirical processes , 1989 .

[18]  T. Gasser,et al.  Nonparametric estimates of distribution functions , 1983 .

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

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

[21]  A. Azzalini.  A note on the estimation of a distribution function and quantiles by a kernel method , 1981 .

[22]  Helen Finkelstein.  The Law of the Iterated Logarithm for Empirical Distribution , 1971 .

[23]  A. Polansky.  Bandwidth selection for the smoothed bootstrap percentile method , 2001 .

[24]  Jan W. H. Swanepoel,et al.  A general result on the uniform in bandwidth consistency of kernel-type function estimators , 2011 .

[25]  James Stephen Marron,et al.  Estimation of integrated squared density derivatives , 1987 .

[26]  D. W. Scott,et al.  Biased and Unbiased Cross-Validation in Density Estimation , 1987 .

[27]  Jiří Zelinka,et al.  Kernel Smoothing in MATLAB: Theory and Practice of Kernel Smoothing , 2012 .

[28]  P. J. Green,et al.  Density Estimation for Statistics and Data Analysis , 1987 .

[29]  On the asymptotic behaviour of the ISE for automatic kernel distribution estimators , 2003 .

[30]  David W. Scott,et al.  Frequency Polygons: Theory and Application , 1985 .

[31]  Lijian Yang,et al.  Kernel estimation of multivariate cumulative distribution function , 2008 .

[32]  D. Peter,et al.  Kernel estimation of a distribution function , 1985 .

[33]  P. Sarda.  Smoothing parameter selection for smooth distribution functions , 1993 .

[34]  Naomi S. Altman,et al.  Bandwidth selection for kernel distribution function estimation , 1995 .

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

[36]  Kernel Smoothing to Improve Bootstrap Confidence Intervals , 1997 .

[37]  J. Swanepoel.  Mean intergrated squared error properties and optimal kernels when estimating a diatribution function , 1988 .

[38]  Carlos Tenreiro,et al.  Asymptotic behaviour of multistage plug-in bandwidth selections for kernel distribution function estimators , 2006 .