Brenton Clarke

The most popular and perhaps universal estimator of location and scale in robust estimation, where one accepts that ideally we have a normal population, but wish to guard against possible small departures from such, is Huber’s Proposal-2 M-estimator. We outline the first order small sample bias correction for the scale estimator, which has been verified both through theory and simulation. While there may be other ways of reducing small sample bias, say as in jackknifing or bootstrapping, these can be computationally intensive, and would not be routinely used with this iteratively derived estimator. It is suggested that bias reduced estimates of scale are most useful when forming confidence intervals for location and or scale based on the asymptotic distribution. In this paper we expand on the results of an earlier work by the authors to include Hampel’s three part re-descending psi function (with a three part re-descender for scale).

A representation of sums of squares in two way layouts deriving from the history of the discussion of the introduction of the ANOVA method of R.A.Fisher by J.O.Irwin was introduced recently by the first listed author of this paper (see Clarke (2002)). Partitions of Helmert matrices and Kronecker products were used to easily derive the distribution theory of component sums of squares in fixed effect models to do with the two way layout. In this paper we show how the derivation of distribution theory to do with mixed models and variance component models can easily follow from the same representation.

In literature several estimates have been proposed for unimodal densities. They are typically derived from the Grenander estimate (see Grenander (1956)) for decreasing densities which can be easily extended to the case of unimodal densities with a known mode at θ. The resulting estimator is the nonparametric maximum likelihood estimator (NPMLE). If the mode is unknown, the NPMLE does not exist anymore. One possible solution that has been proposed by Wegman (1969, 1970a,b) is to add the additional constraint of a modal interval of length ε, where ε has to be chosen by the statistician. More recently Bickel and Fan (1996) and Birge (1997) proposed methods that are based on an initial mode estimate θ and an application of the NPMLE with mode θ. The initial mode estimate requires the calculation of the MLE of ∫ for O(n) candidate points for the mode, where n denotes the sample size. They showed their methods to provide good estimates both for nonsmooth densities and for the mode of nonsmooth densities.