Functional measures of skewness and kurtosis, called asymmetry and gradient asymmetry functions, are described for continuous univariate unimodal distributions. They are defined and interpreted directly in terms of the density function and its derivative. Asymmetry is defined by comparing...

A class of local linear kernel density estimators based on weighted least squares kernel estimation is considered within the framework of Aalen's multiplicative intensity model. This model includes the filtered data model that, in turn, allows for truncation and/or censoring in addition to...

We note that the uniform is the optimal kernel density for kernel estimation of the distribution function, or its inverse, and that several other popular kernel densities perform virtually as well. Parallels with the kernel density estimation case are made.

Some theoretical results on the performance of the identity reproducing nonparametric regression estimator of Müller and Song (1993) are presented. Compared with Müller and Song's analysis, we develop (a) better asymptotic mean squared error (MSE) results in the 'interior' of design space, and...

Improved kernel-based estimates of integrated squared density derivatives are obtained by reinstating non-stochastic terms that have previously been omitted, and using the bandwidth to (approximately) cancel these positive quantities with the leading smoothing bias terms which are negative. Such...

The local dependence function is constant for the bivariate normal distribution. Here we identify all other distributions which also have constant local dependence. The key property is exponential family conditional distributions and a linear conditional mean. When given two marginal...

We consider a simple general construct, that of marginal replacement in a multivariate distribution, that provides, in particular, an interesting way of producing distributions that have a skew marginal from spherically symmetric starting points. Particular examples include a new multivariate...

Recently, much progress has been made on understanding the bandwidth selection problem in kernel density estimation. Here, analogous questions are considered for extensions to the basic problem, namely, for estimating derivatives, using 'better' kernel estimators, and for the multivariate case....