On construction of asymptotically correct confidence intervals

In this paper we discuss constructing confidence intervals based on asymptotic generalized pivotal quantities (AGPQs). An AGPQ associates a distribution with the corresponding parameter, and then an asymptotically correct confidence interval can be derived directly from this distribution like Bayesian or fiducial interval estimates. We provide two general procedures for constructing AGPQs. We also present several examples to show that AGPQs can yield new confidence intervals with better finite-sample behaviors than traditional methods.     

Testing for a change in covariance operator

The paper considers a problem of equality of two covariance operators. Using functional principal component analysis, a method for testing equality of K largest eigenvalues and the corresponding eigenfunctions, together with its generalization to a corresponding change point problem is suggested. Asymptotic distributions of the test statistics are presented.     

Consistency of maximum likelihood estimators in a large class of deconvolution models

We consider the maximum likelihood estimator $\hat{F}_n$ of a distribution function in a class of deconvolution models where the known density of the noise variable is of bounded variation. This class of noise densities contains in particular bounded, decreasing densities. The estimator $\hat{F}_n$ is defined, characterized in terms of Fenchel optimality conditions and computed. Under appropriate conditions, various consistency results for $\hat{F}_n$ are derived, including uniform strong consistency. The Canadian Journal of Statistics 41: 98–110; 2013 © 2012 Statistical Society of Canada     

Multinomial Selection Problem: A Study of BEM and AVC Algorithms

The two well-known and widely used multinomial selection procedures Bechhofor, Elmaghraby, and Morse (BEM) and all vector comparison (AVC) are critically compared in applications related to simulation optimization problems.

Two configurations of population probability distributions in which the best system has the greatest probability p _i of yielding the largest value of the performance measure and has or does not have the largest expected performance measure were studied.

The numbers achieved by our simulations clearly show that none of the studied procedures outperform the other in all situations. The user must take into consideration the complexity of the simulations and the performance measure probability distribution properties when deciding which procedure to employ.

An important discovery was that the AVC does not work in populations in which the best system has the greatest probability p _i of yielding the largest value of the performance measure but does not have the largest expected performance measure.     

The Planning Stage in Statistical Reasoning

An important step in the statistical problem-solving process is the selection of the appropriate statistical procedure for the real-world situation under analysis. A decision-tree term project has been found to be an effective teaching device to help MBA students understand this step. The project requires the students to construct a decision-tree structure, which, through a series of questions and responses, will lead from the statement of a statistical question to the appropriate sampling distribution to use in addressing the question.     

Asymptotically Optimal Nonparametric Empirical Bayes Via Predictive Recursion

An empirical Bayes problem has an unknown prior to be estimated from data. The predictive recursion (PR) algorithm provides fast nonparametric estimation of mixing distributions and is ideally suited for empirical Bayes applications. This article presents a general notion of empirical Bayes asymptotic optimality, and it is shown that PR-based procedures satisfy this property under certain conditions. As an application, the problem of in-season prediction of baseball batting averages is considered. There the PR-based empirical Bayes rule performs well in terms of prediction error and ability to capture the distribution of the latent features.     

Sums of powers of binomial coefficients

An asymptotic series for sums of powers of binomial coefficients is derived, the general term being defined and usable with a computer symbolic language. Sums of squares of coefficients in the symmetric case are shown to have a link with classical moment problems, but this property breaks down for cubes and higher powers. Problems of remainders for the asymptotic series are mentioned. Using the reflection formula for I'(.), a continuous form for a binomial function is set up, and this becomes oscillatory outstde the usual range. A new contmued fraction emerges for the logarithm of an adjusted sum of binomial squares. The note is a contribution to the problem of the interpretation of asymptotic series and processes for their convergence acceleration.     

Exact and approximate distributions for linear contrasts of means and variances

A Bayesian analysis is presented for the K-group Behrens-Fisher problem. Both exact posterior distributions and approximations were developed for both a general linear contrast of the K means and the K variances, given either proper diffuse or informative conjugate priors. The contrast of variances is a unique feature of the heterogeneous variance model that enables investigators to test specific effects of experimental manipulations on variance. Finally, important-differences were observed between the heterogeneous variance model and the homogeneous model.     

Robust test for means when population variances are unequal

We consider the problem of testing the equality of two population means when the population variances are not necessarily equal. We propose a Welch-type statistic, say T^* _c, based on Tiku!s ‘1967, 1980’ modified maximum likelihood estimators, and show that this statistic is robust to symmetric and moderately skew distributions. We investigate the power properties of the statistic T^* _c; T^* _c clearly seems to be more powerful than Yuen's ‘1974’ Welch-type robust statistic based on the trimmed sample means and the matching sample variances. We show that the analogous statistics based on the ‘adaptive’ robust estimators give misleading Type I errors. We generalize the results to testing linear contrasts among k population means