Thomas P. Hettmansperger

Nonparametric methods in general linear models

Distribution theory of rank statistics: Distribution Theory of Linear Rank-Order Statistics Distribution Theory of Signed Rank Order Statistics Distribution Theory of Multivariate Linear Rank-Order Statistics Nonparametric inference in linear models: Distribution-Free Rank-Order Tests for Some Linear Hypotheses Rank-Order Estimation Theory in Some Linear Models Asymptotically Distribution-Free Aligned Rank- Order Tests for Some General Linear Hypotheses Rank-Order Tests for Miscellaneous Problems in Linear Models Appendix.

A Cautionary Note on the Method of Least Median Squares

Abstract This article describes and illustrates a local instability that may arise when using the method of least median squares (LMS) to fit models to data. This idea is contrary to the generally held belief that the method is highly resistant to perturbations in the data. In fact, slight changes in centrally located data can cause the LMS estimate to change by a large amount. The LMS method uses a criterion function that is calculated on half samples. If there are two or more half-samples with roughly the same value of the criterion function, then by slight changes in some of the data the LMS solution can be made to jump from one half sample to the other. An example of a data set from a standard text that exhibits this feature is presented. This suggests that some caution should be exercised when using this method. It does not automatically guarantee complete robustness to misspecifications in the data.

Inference for mixtures of symmetric distributions

This article discusses the problem of estimation of parameters in finite mixtures when the mixture components are assumed to be symmetric and to come from the same location family. We refer to these mixtures as semiparametric because no additional assumptions other than symmetry are made regarding the parametric form of the component distributions. Because the class of symmetric distributions is so broad, identifiability of parameters is a major issue in these mixtures. We develop a notion of identifiability of finite mixture models, which we call k-identifiability, where k denotes the number of components in the mixture. We give sufficient conditions for k-identifiability of location mixtures of symmetric components when k = 2 or 3. We propose a novel distance-based method for estimating the (location and mixing) parameters from a k-identifiable model and establish the strong consistency and asymptotic normality of the estimator. In the specific case of L 2 -distance, we show that our estimator generalizes the Hodges-Lehmann estimator. We discuss the numerical implementation of these procedures, along with an empirical estimate of the component distribution, in the two-component case. In comparisons with maximum likelihood estimation assuming normal components, our method produces somewhat higher standard error estimates in the case where the components are truly normal, but dramatically outperforms the normal method when the components are heavy-tailed.

Tests of Hypotheses Based on Ranks in the General Linear Model

A unified approach is developed for testing hypotheses in the general linear model based on the ranks of the residuals. It complements the nonparametric estimation procedures recently reported in the literature. The testing and estimation procedures together provide a robust alternative to least squares. The methods are similar in spirit to least squares so that results are simple to interpret. Hypotheses concerning a subset of specified parameters can be tested, while the remaining parameters are treated as nuisance parameters. Asymptotically, the test statistic is shown to have a chi-square distribution under the null hypothesis. This result is then extended to cover a sequence of contiguous alternatives from which the Pitman efficacy is derived. The general application of the test requires the consistent estimation of a functional of the underlying distribution and one such estimate is furnished.

A Robust Alternative Based on Ranks to Least Squares in Analyzing Linear Models

A unified approach based on the ranks of the residuals iS developed for testing and estimation in the linear model, The methods are robust and efficient relative to least squares methods. For ease of application, the strong analogy to least squares strategy is emphasized. The procedures are illustrated on examples from regression and analysis of covariance.

The ranked-set sample sign test

The method of ranked-set samples (RSS) as opposed to simple random sampling is discussed. In the ranked-set sampling method, k observations are ranked by an expert then a measurement is taken on the observation designated to have rank j for j = 1,…,k. This is repeated until we have k measurements on the order statistics.We observed k 2 items to get k measurements. This method is effective when observations are inexpensive and measurement is perhaps destructive orinvasive. The properties of the sign test along with the mediam and corresponding confidence interval for the RSS method are developed. It is shown that when it is appropriate, there can bea substantial efficiency gain when using ranked-set sampling. We consider a simple model to described theeffect of imperfect judgment. We also provide a brief comparison of the RSS median to the RSS mean.

Confidence intervals based on interpolated order statistics

Confidence intervals for the population median based on interpolating adjacent order statistics are presented. They are shown to depend only slightly on the underlying distribution. A simple, nonlinear interpolation formula is given which works well for a broad collection of underlying distributions.

A Geometric Interpretation of Inferences Based on Ranks in the Linear Model

Abstract Four different approaches, based on ranks, to testing hypotheses are unified through the geometry of the linear model. The various tests are identified with different but algebraically equivalent forms of the classical F test. Small sample differences are investigated via a Monte Carlo study using both rank and signed rank tests.

Regression Diagnostics for Rank-Based Methods

Abstract Residual plots and diagnostic techniques have become important tools in examining the least squares fit of a linear model. In this article we explore the properties of the residuals from a rank-based fit of the model. We present diagnostic techniques that detect outlying cases and cases that have an influential effect on the rank-based fit. We show that the residuals from this fit can be used to detect curvature not accounted for by the fitted model. Furthermore, our diagnostic techniques inherit the excellent efficiency properties of the rank-based fit over a wide class of error distributions, including asymmetric distributions. We illustrate these techniques with several examples.

Robust Bounded-Influence Tests in Linear Models

Abstract A robust test that we call an aligned generalized M test for testing subhypotheses in general linear models is developed, and its asymptotic properties are studied. The test is a robustification of the well known F test, and it is an elegant alternative to Ronchettis (1982) class of τ tests, p-values associated with it can be approximated readily using existing chi-square tables. The test is based on an appropriately constructed quadratic form and uses the generalized M estimators of the parameters in the reduced model. Under the null hypothesis the asymptotic distribution is a central chi square, and under contiguous alternatives it is a noncentral chi square with the same degrees of freedom. The test can be viewed as a generalization of Sens (1982) M test for linear models. The influence function of the test is bounded. The bound not only applies to the influence of residuals but to the influence of position in the factor space as well. On the other hand, Sens test has bounded influence only...