The asymptotic distribution of multivariate M-estimates is studied. It is shown that, in general, consistency leads to asymptotic normality and a Law of the Iterated Logarithm. The results are used to compute via matrix derivatives the asymptotic distribution of a class of estimates due to Maronna.

Motivated by problems of frontier estimation in productivity analysis, and boundary estimation in scatter-point image analysis, we consider polynomial-based estimators of the edge of a distribution. Our aim is to develop methods for correcting polynomial-type estimators of bias, and for...

We propose nonparametric methods for estimating the support curve of a bivariate density, when the density decreases at a rate which might vary along the curve. Attention is focused on cases where the rate of decrease is relatively fast, this being the most difficult setting. It demands the use...

Much of the practical interest attached to curves and surfaces derives from features of roughness, rather than smoothness. For example, considerable attention has been paid to fractal models of curves and surfaces, for which the notions of a normal and curvature are usually not well defined....

We describe a unified approach to the construction of confidence bands in nonparametric density estimation and regression. Our techniques are based on interpolation formulae in numerical differentiation, and our arguments generate a variety of bands depending on the assumptions one is prepared...

Some types of density estimators, particularly those based on trigonometric series, converge reasonably quickly to their limit except in the neighbourhood of one or two singularities. In this situation the mean integrated square error, the traditional measure of the efficiency of a density...

Performance of the bootstrap for estimating tail probabilities is usually explained by saying that the bootstrap provides a one-term Edgeworth correction. However, simulation studies show that the bootstrap usually performs better than explicit Edgeworth correction. We present a theory which...

In a range of practical problems the boundary of the support of a bivariate distribution is of interest, for example where it describes a limit to efficiency or performance, or where it determines the physical extremities of a spatially distributed population in forestry, marine science,...

This paper is devoted to tests for uniformity based on sum-functions of m-spacings, where m diverges to infinity as the sample size, n, increases. It is shown that if m diverges at a slower rate than n1/2 then the commonly used sum-function will detect alternatives distant (mn)-1/4 from the...

Let Y be an absolutely continuous random variable and W a nonnegative variable independent of Y. It is to be expected that when W is close to 1 in some sense, the distribution of the scale mixture YW will be close to Y. This notion has been investigated by a number of workers, who have provided...