Ian R. Harris

Simulating multivariate distributions with specific correlations

The mixture approach for simulating bivariate distributions introduced by Michael, J. R. and Schucany, W. R. (2002). The mixture approach for simulating bivariate distributions with specific correlations. The American Statistician, 56, 48–54, is generalized to generate pseudo-random numbers from multivariate distributions. The simulated joint distributions have identical marginals and equal positive pairwise correlations. The approach is illustrated for the p-dimensional families of beta and gamma distributions. For these families the formulas for the correlations have simple closed forms and the computations are quite simple.

2 Minimum distance estimation: The approach using density-based distances

Publisher Summary This chapter discusses the concept of minimum distance estimation using density-based distances. Density-based minimum distance methods have proven to be valuable additions to the theory of statistics as demonstrated by the rich literature of the past two decades. In parametric models, the estimators often possess full asymptotic efficiency simultaneously with attractive robustness properties. The chapter also discusses minimum Hellinger distance estimation, including the Hellinger deviance test and penalized Hellinger distance estimation. In the chapter, General disparities, residual adjustment functions, and related inference are introduced and the negative exponential disparity and weighted likelihood estimators (including linear regression models) are described. A generalized divergence measure and the resulting estimators are also discussed in the chapter.

Hellinger distance as a penalized log likelihood

The present paper studies the minimum Hellinger distance estimator by recasting it as the maximum likelihood estimator in a data driven modification of the model density. In the process, the Hellinger distance itself is expressed as a penalized log likelihood function. The penalty is the sum of the model probabilities over the non-observed values of the sample space. A comparison of the modified model density with the original data provides insights into the robustness of the minimum Hellinger distance estimator. Adjustments of the amount of penalty leads to a class of minimum penalized Hellinger distance estimators, some members of which perform substantially better than the minimum Hellinger distance estimator at the model for small samples, without compromising the robustness properties of the latter.

A generalized divergence for statistical inference

The power divergence (PD) and the density power divergence (DPD) families have proven to be useful tools in the area of robust inference. In this paper, we consider a superfamily of divergences which contains both of these families as special cases. The role of this superfamily is studied in several statistical applications, and desirable properties are identified and discussed. In many cases, it is observed that the most preferred minimum divergence estimator within the above collection lies outside the class of minimum PD or minimum DPD estimators, indicating that this superfamily has real utility, rather than just being a routine generalization. The limitation of the usual first order influence function as an effective descriptor of the robustness of the estimator is also demonstrated in this connection.

Tests of hypotheses in discrete models based on the penalized Hellinger distance

Analogues of the likelihood ratio, Rao, and Wald tests are introduced in discrete parametric models based on the family of penalized Hellinger distances. It is shown that the tests based on a particular member of this family provide attractive alternatives to the tests based on the ordinary Hellinger distance. These tests share the robustness of the Hellinger distance test, but are often closer to the likelihood-based tests at the model, especially in small samples. The convergence of ordinary Hellinger distance tests to limiting [chi]2 distributions are quite slow. The proposed tests are improvements in this respect.

Bayesian estimators of the intraclass correlation coefficient in the one-way random effects model

In this paper Bayesian methods are used to estimate the intraclass correlation coefficient in the balanced one-way random effects model. An estimator associated with the likelihood function derived from a pivotal quantity along with estimators using reference priors are considered. In addition, an estimator based on a posterior median is examined. These estimators are compared to one aiiothei and to the REML (restricted maximum likelihood) estimator in terms of MSE (mean-squared error) A beta-type approximation to the pivotal likelihood is considered This can be combined with a beta prior to produce closed-form expressions that approximate the posterior mean and mode These approximations generally perform well as judged by Bayes risk. Of the estimators considered the authors recommend the one obtained from the pivotal approach. The authors indicate how the estimation procedures may be extended to the unbalanced one-way random effects model.

A blended estimator for a measure of agreement with a gold standard

St. Laurent developed a measure of agreement for method comparison studies in which an approximate method of measurement was compared to a gold standard method of measurement. The measure of agreement proposed was shown to be related to a population intraclass correlation coefficient. In this paper, the authors develop a family of estimators for the measure of agreement based on pivotal quantities. A blend of two particular members of the family is suggested as an estimator itself. In general, the blended estimator outperforms the maximum likelihood estimator in terms of bias and mean-squared error.

Pivotal estimation with applications for the intraclass correlation coefficient in the balanced one-way random effects model

In this paper pivotal quantities are used to obtain estimators of parameters. As an application of this technique, a family of estimators is presented for the intraclass correlation coefficient in the balanced one-way random effects model. The family is derived by equating a pivotal quantity to different values of the pivoting distribution, and includes the familiar REML and ML estimators. It is shown that under certain conditions, in terms of mean-squared error, most members of the family of estimators are admissible within the family. The authors provide guidance concerning the choice of an individual member of the family for estimation purposes and indicate how the method can be extended to unbalanced designs.

Optimal one-way random effects designs for the intraclass correlation based on confidence intervals

ABSTRACT Confidence intervals for the intraclass correlation coefficient (ρ) are used to determine the optimal allocation of experimental material in one-way random effects models. Designs that produce narrow intervals are preferred since they provide greater precision to estimate ρ. Assuming the total cost and the relative cost of the two stages of sampling are fixed, the authors investigate the number of classes and the number of individuals per class required to minimize the expected length of confidence intervals. We obtain results using asymptotic theory and compare these results to those obtained using exact calculations. The best design depends on the unknown value of ρ. Minimizing the maximum expected length of confidence intervals guards against worst-case scenarios. A good overall recommendation based on asymptotic results is to choose a design having classes of size 2 + √4 + 3r, where r is the relative cost of sampling at the class-level compared to the individual-level. If r = 0, then the overall cost is the sample size and the recommendation reduces to a design having classes of size 4.

Locally quadratic log likelihood and data-based transformations

The investigation of multi-parameter likelihood functions is simplified if the log likelihood is quadratic near the maximum, as then normal approximations to the likelihood can be accurately used to obtain quantities such as likelihood regions. This paper proposes that data-based transformations of the parameters can be employed to make the log likelihood more quadratic, and illustrates the method with one of the simplest bivariate likelihoods, the normal two-parameter likelihood.