Locally optimal window widths for kernel density estimation with large samples

The smoothing parameter or window width for a kernel estimator of a probability density at a point has been previously specified to minimize either asymptotic mean square error or asymptotic mean absolute error. In this note the ratio of these two widths is shown to be a constant for all kernels and density functions that satisfy the usual smoothness conditions. The fact that this ratio equals 0.985 supports recent comment that in this context these two error criteria do not yield large-sample results that differ by any meaningful amount. Isolated points at which the dominant term of the conventional bias expansion vanishes are examined. Consideration of additional terms and continuity leads to the conclusion that bias is adequately modeled by a multiple of a single rate in all large but finite sample sizes. In practice, for instance, at inflection points with a second-order kernel the abrupt change in exponent from 1/5 to 1/9 is not necessarily a good representation.bandwidth bias mean absolute error mean square error smoothing parameter

A local cross-validation algorithm

The usuall form of cross-validation is global in character, and is designed to estimate a density in some "average" sense over its entire support. In this paper we present a local version of squared-error cross-validation, suitable for estimating a probability density at a given point. It is shown theoretically to be asymptotically optimal in the sense of minimizing mean squared error. Numerical examples illustrate finite sample characteristic, and show that local cross-validation is a practical algorithm.cross-validation density estimation kernel method local estimation window size

On combining independent nonparametric regression estimators

Three estimators are investigated for linearly combining independent nonparametric regression estimators. Assuming fixed designs, the asymptotic mean squared errors and asymptotically optimal bandwidths are given for each estimator and compared. One estimator essentially ignores the differences in the sources and naively pools all of the data. The second utilizes individually optimized bandwidths and then estimates the best weights to combine them. The third estimator solves a general minimization problem and employs equal bandwidths and weights similar to those for combining unbiased estimators with unequal variance. It is found to be superior to the other two in most situations that would be encountered in practice.Asymptotic optimality Bandwidth Kernel Local linear

On the correlation of a group of rankings with an external ordering relative to the internal concordance

A method is presented for comparing the strength of agreement of a group rankings with an external ordering to the corresponding measure of concordance within the group. While the procedure is not model dependent, we illustrate the characteristics of interest using an existing model for a nonnull distribution for a population of rankings. U-statistics and a jackknife with adjusted degrees of freedom are employed to set approximate confidence intervals on the contrast between the two measures of rank order agreement.Concordance jackknife rankings U-statistics