On the Relationship between Directional and Omnibus Statistical Tests

Abstract.  A common statistical problem involves the testing of a K -dimensional parameter vector. In both parametric and semiparametric settings, two types of directional tests – linear combination and constrained tests – are frequently used instead of omnibus tests in hopes of achieving greater power for specific alternatives. In this paper, we consider the relationship between these directional tests, as well as their relationship to omnibus tests. Every constrained directional test is shown to be asymptotically equivalent to a specific linear combination test under a sequence of contiguous alternatives and vice versa. Even when the direction of the alternative is known, the constrained test in general will not be optimal unless the objective function used to derive it is efficient. For an arbitrary alternative, insight into the power characteristics of directional tests in comparison to omnibus tests can be gained by a chi-square partition of the omnibus test.     

The Asymptotic Distribution of the S–Gini Index

Several generalizations of the classical Gini index, placing smaller or greater weights on various portions of income distribution, have been proposed by a number of authors. For purposes of statistical inference, the large sample distribution theory of the estimators of those measures of economic inequality is required. The present paper was stimulated by the use of bootstrap by Xu (2000) to estimate the variance of the estimator of the S –Gini index. It shows that the theory of L –statistics (Chernoff, Gastwirth & Johns, 1967; Shorack & Wellner, 1986) makes possible the construction of a consistent estimator for the S –Gini index and proof of its asymptotic normality. The paper also presents an explicit formula for the asymptotic variance. The formula should be helpful in planning the size of samples from which the S –Gini index can be estimated with a prescribed margin of error.     

On empirical bayes sequential estimation

We study the empirical Bayes approach to the sequential estimation problem. An empirical Bayes sequential decision procedure, which consists of a stopping rule and a terminal decision rule, is constructed for use in the component. Asymptotic behaviors of the empirical Bayes risk and the empirical Bayes stopping times are investigated as the number of components increase.     

On robust empirical bayes analysis of means and variances from stratified samples

Ghosh and Lahiri (1987a,b) considered simultaneous estimation of several strata means and variances where each stratum contains a finite number of elements, under the assumption that the posterior expectation of any stratum mean is a linear function of the sample observations - the so called“posterior linearity” property. In this paper we extend their result by retaining the “posterior linearity“ property of each stratum mean but allowing the superpopulation model whose mean as well as the variance-covariance structure changes from stratum to stratum. The performance of the proposed empirical Bayes estimators are found to be satisfactory both in terms of “asymptotic optimality” (Robbins (1955)) and “relative savings loss” (Efron and Morris (1973)).     

Estimation of parameters of 1-out-of-2:g repairable system

Moment estimators of l-out-of-2:G repairable system are supplied under four sampling schemes assuming that the failure and repair time distribution of the units are exponential with unknown parameters λ, μ respectively. Information metrices of the estimators are supplied. Also it is shown that the estimators are asymptotically normally distributed in every sampling scheme.     

On the merging of quoted survival rates to obtain an overall rate

Frequently in a review paper on treatment or management of a disease, various papers, each quoting different measures of survival, are reviewed. Assuming a common underlying distribution an overall death rate can be estimated. This paper describes the maximum likelihood estimation of the rate and an asymptotic test for its homogeneity among different investigations. An example is given of the methodfs use in analysis of published results on survival from salivary gland tumours.     

The limiting distribution of the F-statistic from nonnormal universes

We consider a linear regression model with an unbalanced 1-fold nested error structure, where group effect and error are from nonnormal universes. The limiting distribution of the F-statistic in this model is derived, as the sample size is large and group sizes take values from a finite set of distinct integers. The result is used to approximate the F-distribution quantile and to test the significance of the random effect variance component. Results are also applicable to the F-statistic in the one-way random-effects model. The effects of departure from normality on the F-statistic distribution are given.     

Some asymptotic results in the multivariate lognormal estimation theory

An asymptotic expansion of the variance of the uniformly minimum variance unbiased estimator of a class of parameters of the multivariate lognormal distribution is considered. It is obtained by some calculations of the zonal polynomials. Then it is compared with an asymptotic mean square error of the maximum likelihood estimator of the same parameter.     

Asymptotic distributions for quadratic forms with applications to censored data tests of fit

We obtain the asymptotic distributions of certain forms in observations that are possible Type I and Type II censored. This result is directly applicable to the study of asympototic distributions for censored data versions of the Shapiro- wilk test for normality. Moreove, it applies more generally than just to the null hypothesis of normality.     

The variance and covariance of a generalized index of similarity especially for a generalizaition on an index of hellinger and bhattacharyya

Formulae are derived for approximating the variance or a general similarity measlire and the covariance between two similarity measures for multinomial populations.