Useful generalized properties of L1 estimators

Recent results by G. Appa and C. Smith, as well as I. Barrodale and F. D. K. Roberts, underscore several properties exhibited for fitting a linear model to a set of observation points under the criterion of least sum of absolute deviations(commonly denoted as the L₁ criterion). This paper will generalize these properties to the non-full rank case and relax in a natural way some assumptions given by Appa and Smith.     

A transformation for estimating the trinomial logit model with grouped data

For logit models where the outcome variables are the proportions of individuals falling into each of three categories, this paper develops a data transformation through which GLS estimates can be obtained by running OLS on the transformed data.     

Estimating parameters in autoregressive models in non-normal situations: symmetric innovations

The estimation of coefficients in a simple regression model with autocorrelated errors is considered. The underlying distribution is assumed to be symmetric, one of Student's t family for illustration. Closed form estimators are obtained and shown to be remarkably efficient and robust. Skew distributions will be considered in a future paper.     

Bias of the lse estimator of the first order autoregressive model under tukey contamination

Results of an exhaustive study of the bias of the least square estimator (LSE) of an first order autoregression coefficient α in a contaminated Gaussian model are presented. The model describes the following situation. The process is defined as X_t = α X_t-1 + Y_t . Until a specified time T, Y_t are iid normal N(0, 1). At the moment T we start our observations and since then the distribution of Y_t, t≥T, is a Tukey mixture T(εσ) = (1 – ε)N(0,1) + εN(0, σ²). Bias of LSE as a function of α and ε, and σ² is considered. A rather unexpected fact is revealed: given α and ε, the bias does not change montonically with σ (“the magnitude of the contaminant”), and similarly, given α and σ, the bias is not growing with ε (“the amount of contaminants”).     

MORE ON THE PRE-TEST ESTIMATOR IN RIDGE REGRESSION

ABSTRACT

The problem of estimation of the regression coefficients in a multiple regression model is considered under a multicollinearity situation when it is suspected that the regression coefficients may be restricted to a subspace. The objective of this paper is to compare the usual preliminary test estimator and the preliminary test ridge regression estimator in the sense of the dispersion matrix of one dominating that of the other. In particular we proved two results giving necessary and sufficient conditions for the superiority of the preliminary test ridge regression estimator over the preliminary test estimator associated with the δ = 0 (or Δ = 0) and δ ≠ 0 (or Δ ≠ 0).     

A simple least squares method for estimating a change in mean

A simple least squares method for estimating a change in mean of a sequence of independent random variables is studied. The method first tests for a change in mean based on the regression principle of constrained and unconstrained sums of squares. Conditionally on a decision by this test that a change has occurred, least squares estimates are used to estimate the change point, the initial mean level (prior to the change point) and the change itself. The estimates of the initial level and change are functions of the change point estimate. All estimates are shown to be consistent, and those for the initial level and change are shown to be asymptotically jointly normal. The method performs well for moderately large shifts (one standard deviation or more), but the estimates of the initial level and change are biased in a predictable way for small shifts. The large sample theory is helpful in understanding this problem. The asymptotic distribution of the change point estimator is obtained for local shifts in mean, but the case of non-local shifts appears analytically intractable.     

Concerning several methods for estimating crop acreages using remote sensing data

The objective of this paper is to present a comprehensive survey of the proportion estimation methods so far found in the literature and a newly proposed method based on the concept of statistically equivalent blocks. All methods are restricted to the case of normal mixtures only. Graphical and semigraphical techniques are excluded. Several proportion estimation methods have been described and their properties discussed.     

Properties of Designs in Experiments on Spatial Lattices

ABSTRACT

When spatial variation is present in experiments, it is clearly sensible to use designs with favorable properties under both generalized and ordinary least squares. This will make the statistical analysis more robust to misspecification of the spatial model than would be the case if designs were based solely on generalized least squares. In this article, treatment information is introduced as a way of studying the ordinary least squares properties of designs. The treatment information is separated into orthogonal frequency or polynomial components which are assumed to be independent under the spatial model. The well-known trend-resistant designs are those with no treatment information at the very low order frequency or polynomial components which tend to have the higher variances under the spatial model. Ideally, designs would be chosen with all the treatment information distributed at the higher-order components. However, the results in this article show that there are limits on how much trend resistance can be achieved as there are many constraints on the treatment information. In addition, appropriately chosen Williams squares designs are shown to have favorable properties under both ordinary and generalized least squares. At all times, the ordinary least squares properties of the designs are balanced against the generalized least squares objectives of optimizing neighbor balance.     

multiple and conditional deletion diagnostics for general linear models

In this paper we develop multiple case deletion statistics for the general linear model so that a residual vector and a leverage matrix are identified which have roles analogous to residuals and leverage for ordinary least squares models. We extend the notion of the conditional deletion diagnostic to general linear models. The residuals, leverage and deletion diagnostics are illustrated with data modelled by a linear growth curve.     

Extensions of multiple linear regression

Many statistical methods are linked together through their connection with weighted least squares and hence regression. This article reviews these connections, emphasising the iteratively weighted least squares algorithm.