Dennis D. Boos

P values maximized over a confidence set for the nuisance parameter

Abstract For testing problems of the form H 0: v = v 0 with unknown nuisance parameter θ, various methods are used to deal with θ. The simplest approach is exemplified by the t test where the unknown variance is replaced by the sample variance and the t distribution accounts for estimation of the variance. In other problems, such as the 2 × 2 contingency table, one conditions on a sufficient statistic for 0 and proceeds as in Fishers exact test. Because neither of these standard methods is appropriate for all situations, this article suggests a new method for handling the unknown θ. This new method is a simple modification of the formal definition of a p value that involves taking a maximum over the nuisance parameter space of a p value obtained for the case when θ is known. The suggested modification is to restrict the maximization to a confidence set for the nuisance parameter. After giving a brief justification, we give various examples to show how this new method gives improved results for 2 × 2 tabl...

On Generalized Score Tests

Abstract Generalizations of Raos score test are receiving increased attention, especially in the econometrics and biostatistics literature. These generalizations are able to account for certain model inadequacies or lack of knowledge by use of empirical variance estimates. This article shows how the various forms of the generalized test statistic arise from Taylor expansion of the estimating equations. The general estimating equations structure unifies a variety of applications and helps suggest new areas of application.

The calculus of M-estimation

Since the seminal papers by Huber in the 1960s, M-estimation methods (also known as estimating equation methods) have been increasingly important for asymptotic analysis and approximate inference. This article illustrates the breadth and generality of the M-estimation approach, thereby facilitating its use inpractice and in the classroom as a unifying approach to the study of large-sample inference.

Minimum Hellinger Distance Estimation for Multivariate Location and Covariance

Abstract The Hellinger distance between a nonparametric density estimator and a model family is minimized to produce estimates of location and covariance in multivariate data. With suitable restrictions on the density estimators and the model family, these minimum Hellinger distance estimators (MHDEs) are shown to be affine invariant, consistent, and asymptotically normal. The robustness of the MHDE as measured by the breakdown point compares favorably against the previously studied M-estimators. Monte Carlo results suggest that the MHDEs are an attractive robust alternative to the usual sample means and covariance matrix.

Bootstrap Methods for Testing Homogeneity of Variances

This article describes the use of bootstrap methods for the problem of testing homogeneity of variances when means are not assumed equal or known. The methods are new in this context and allow the use of normal-theory test statistics such as F = s 2 1/s 2 2 without the normality assumption that is crucial for validity of critical values obtained from the F distribution. Both asymptotic analysis and Monte Carlo sampling show that the new resampling procedures compare favorably with older methods in terms of test validity and power.

Using Extreme Value Theory to Estimate Large Percentiles

Weissman (1978) suggested percentile estimators based on the joint limiting distribution of the k largest order statistics. The present work identifies situations where Weissmans estimators are a significant improvement over the usual sample percentile estimators and gives practical advice on how to use these new estimators effectively. In particular, large reductions in mean squared error can be made when the tails of the distributions are approximately exponential and p ≥ .95.

Minimum Distance Estimators for Location and Goodness of Fit

Abstract A weighted Cramer-von Mises distance between the empirical distribution function and the assumed model F0(x - θ) is minimized to produce estimators θ n that are asymptotically normal. If the weight function is taken proportional to (- ln f 0)″/f 0 , then θ n is asymptotically efficient and the minimized distance has the appropriate loss of one degree of freedom. Special attention is focused on the limiting distribution of this latter goodness-of-fit statistic in both null and alternative situations.

Controlling Variable Selection by the Addition of Pseudovariables

We propose a new approach to variable selection designed to control the false selection rate (FSR), defined as the proportion of uninformative variables included in selected models. The method works by adding a known number of pseudovariables to the real dataset, running a variable selection procedure, and monitoring the proportion of pseudovariables falsely selected. Information obtained from bootstrap-like replications of this process is used to estimate the proportion of falsely selected real variables and to tune the selection procedure to control the FSR.

Comparing Variances and Other Measures of Dispersion

Testing hypotheses about variance parameters arises in contexts where uniformity is important and also in relation to checking assumptions as a preliminary to analysis of variance (ANOVA), dose-response modeling, discriminant analysis and so forth. In contrast to procedures for tests on means, tests for variances derived assuming normality of the parent populations are highly nonrobust to nonnormality. Procedures that aim to achieve robustness follow three types of strategies: (1) adjusting a normal-theory test procedure using an estimate of kurtosis, (2) carrying out an ANOVA on a spread variable computed for each observation and (3) using resampling of residuals to determine p values for a given statistic. We review these three approaches, comparing properties of procedures both in terms of the theoretical basis and by presenting examples. Equality of variances is first considered in the two-sample problem followed by the k-sample problem (one-way design).

Monte Carlo Evaluation of Resampling-Based Hypothesis Tests

Abstract Monte Carlo estimation of the power of tests that require resampling can be very computationally intensive. It is possible to reduce the size of the inner resampling loop as long as the resulting estimator of power can be corrected for bias. A simple linear extrapolation method is shown to perform well in correcting for bias and thus reduces computation time in Monte Carlo power studies.