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.

A tractable skew "t"-distribution on the real line is proposed. This includes as a special case the symmetric "t"-distribution, and otherwise provides skew extensions thereof. The distribution is potentially useful both for modelling data and in robustness studies. Properties of the new...

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...

Exact expressions for moments of ratios of quadratic forms in normal variables are considered. The emphasis is on their relationship with inverse moments of such quadratic forms. Multivariate extension of work of Cressie and Borkent (1986) is briefly explored.

Expectiles and M-quantiles are related to means and M-estimates of location in the same way as quantiles are related to the median. We show that expectiles and certain M-quantiles of a distribution F are precisely the ordinary quantiles of distributions G related by an explicit formula to F.

Let Z1,Z2 and W1,W2 be mutually independent random variables, each Zi following the standard normal distribution and Wi following the chi-squared distribution on ni degrees of freedom. Then, the pair of random variables , has the bivariate spherically symmetric t distribution; this has both...

Mean squared error properties of kernel estimates of regression quantiles, for both fixed and random design cases, are derived and discussed.