On the distribution of a second degree polynomial statistic in singular normal var1aies

This note provides an alternative proof of a basic theorem on the distribution of a second degree polynomial statistic in noncentral singular normal variates originally proven by Rayner and Livingstone (1965)     

A monte carlo study of the power of alternative tests under the generalized manova model

Monte Carlo simulations are performed for a broad range of conditions. These simulations indicate that the powers of alternative tests under the generalized MANOVA model for small samples differ significantly, if a large reduction of the number of polynomial parameters is applied. The results show that, if the response covariance matrix ∑ is known, the best alternative is to use ∑. If, however, ∑ is unknown, substitution of an identity matrix for ∑ is recommended. This alternative usually results in a test with more power than the test with the usual estimate of ∑ employing covariates or the test with an estimate of E obtained from another sample.     

A confidence region with multiple intervals in multivariate bioassay

The exact confidence region for log relative potency resulting from likelihood score methods (Williams (1988) An exact confidence interval for the relative potency estimated from a multivariate bioassay, Biometrics, 44:861-868) will very likely consist of two disjoint confidence intervals. The two methods proposed by Williams which aim to select just one (the same) confidence interval from the confidence region are nearly – but not completely – consistent. The likelihood score interval and likelihood ratio interval are asymptotically equivalent. Williams's very strong claim concerning the confidence coefficient in the second selection method is still theoretically unproved; yet, simulations show that it is true for a wide range of practical experimental situations.     

Simultaneous prediction intervals for all patrwise differences among several future sample means

Hahn (1977) suggested a procedure for constructing prediction intervals for the difference between the means of two future samples from normal populations having equal variance, based on past samples selected from both populations. In this paper, we extend Hahn's work by constructing simultaneous prediction intervals for all pairwise differences among the means of k ≥ 2 future samples from normal populations with equal variances, using past samples taken from each of the k populations. For K = 2, this generalization reduces to Hahn's special case. These prediction intervals may be used when one has sampled the performance of several products and wishes to simultaneously as- sess the differences in future sample mean performance of these products with a predetermined overall coverage probability. The use of the new procedure is demonstrated with a numerical example.     

A Multivariate Extension of Inverse Gaussian Distribution Derived from Inverse Relationship

Abstract

We propose a new multivariate extension of the inverse Gaussian distribution derived from a certain multivariate inverse relationship. First we define a multivariate extension of the inverse relationship between two sets of multivariate distributions, then define a reduced inverse relationship between two multivariate distributions. We derive the multivariate continuous distribution that has the reduced multivariate inverse relationship with a multivariate normal distribution and call it a multivariate inverse Gaussian distribution. This distribution is also characterized as the distribution of the location of a multivariate Brownian motion at some stopping time. The marginal distribution in one direction is the inverse Gaussian distribution, and the conditional distribution in the space perpendicular to this direction is a multivariate normal distribution. Mean, variance, and higher order cumulants are derived from the multivariate inverse relationship with a multivariate normal distribution. Other properties such as reproductivity and infinite divisibility are also given.     

Nonparametric test for the homogeneity of the overall variability

In this paper, we propose a nonparametric test for homogeneity of overall variabilities for two multi-dimensional populations. Comparisons between the proposed nonparametric procedure and the asymptotic parametric procedure and a permutation test based on standardized generalized variances are made when the underlying populations are multivariate normal. We also study the performance of these test procedures when the underlying populations are non-normal. We observe that the nonparametric procedure and the permutation test based on standardized generalized variances are not as powerful as the asymptotic parametric test under normality. However, they are reliable and powerful tests for comparing overall variability under other multivariate distributions such as the multivariate Cauchy, the multivariate Pareto and the multivariate exponential distributions, even with small sample sizes. A Monte Carlo simulation study is used to evaluate the performance of the proposed procedures. An example from an educational study is used to illustrate the proposed nonparametric test.     

Bayesian modeling of autoregressive partial linear models with scale mixture of normal errors

Normality and independence of error terms are typical assumptions for partial linear models. However, these assumptions may be unrealistic in many fields, such as economics, finance and biostatistics. In this paper, a Bayesian analysis for partial linear model with first-order autoregressive errors belonging to the class of the scale mixtures of normal distributions is studied in detail. The proposed model provides a useful generalization of the symmetrical linear regression model with independent errors, since the distribution of the error term covers both correlated and thick-tailed distributions, and has a convenient hierarchical representation allowing easy implementation of a Markov chain Monte Carlo scheme. In order to examine the robustness of the model against outlying and influential observations, a Bayesian case deletion influence diagnostics based on the Kullback–Leibler (K–L) divergence is presented. The proposed method is applied to monthly and daily returns of two Chilean companies.     

Tables of percentage points of the k-variate normal distribution for large values of k

This paper gives tabulations of the upper α percentage points of the maximum absolute value of the k-variate normal distribution with common correlation ρ for values of k as high as 500. The tables are useful for performing multiple comparison procedures in experiments with large numbers of treatments.     

Investigating the role of orthogonal and non – orthogonal rotation in multivariate factor analysis,in regard to the repeatability of the extracted factors: A simulation study

ABSTRACT

Factor analysis (FA) is the most commonly used pattern recognition methodology in social and health research. A technique that may help to better retrieve true information from FA is the rotation of the information axes. The main goal is to test the reliability of the results derived through FA and to reveal the best rotation method under various scenarios. Based on the results of the simulations, it was observed that when applying non-orthogonal rotation, the results were more repeatable as compared to the orthogonal rotation, and, when no rotation was applied.     

Acceptance sampling plans by variables for a class of symmetric distributions

The estimation of percentage defectives using a normal sampling plan will not be appropriate when the assumption of normality is violated. In this paper, we propose a sampling plan based on a more general symmetric family of distributions with the parameters estimated using the modified maximum likelihood (MML) procedures introduced by Tiku and Suresh . This sampling plan works well for most of the symmetric non-normal distributions. Some numerical study has also been carried out to show the superiority of the proposed plan.