Testing equality of two normal covariance matrices with monotone missing data

Abstract

The problem of testing equality of two multivariate normal covariance matrices is considered. Assuming that the incomplete data are of monotone pattern, a quantity similar to the Likelihood Ratio Test Statistic is proposed. A satisfactory approximation to the distribution of the quantity is derived. Hypothesis testing based on the approximate distribution is outlined. The merits of the test are investigated using Monte Carlo simulation. Monte Carlo studies indicate that the test is very satisfactory even for moderately small samples. The proposed methods are illustrated using an example.     

Analysis of a continuous-time proportional hazard model using discrete duration data

Many economic duration variables are often available only up to intervals, and not up to exact points. However, continuous time duration models are conceptually superior to discrete ones. Hence, in duration analyses, one faces a situation with discrete data and a continuous model. This paper discusses (i) the asymptotic bias of a conventional approximation procedure in which a discrete duration is treated as an exact observation; and (ii) the efficiency of a correct maximum likelihood estimator which appropriately accounts for the discrete nature of the data.     

Testing symmetry under a skew Laplace model

We develop tests of hypothesis about symmetry based on samples from possibly asymmetric Laplace distributions and present exact and limiting distribution of the test statistics. We postulate that the test statistic derived under the Laplace model is a rational choice as a measure of skewness and can be used in testing symmetry for other, quite general classes of skew distributions. Our results are applied to foreign exchange rates for 15 currencies.     

Influence Diagnostics on Testing Linear Hypothesis in Linear Models with Correlated Errors

To assess the influence of observations on the parameter estimates, case deletion diagnostics are commonly used in linear regression models. For linear models with correlated errors we study the influence of observations on testing a linear hypothesis using single and multiple case deletions. The change in likelihood ratio test and F test theoretically is derived and it is shown these tests to be completely determined by two proposed generalized externally studentized residuals. An illustrative example of a real data set is also reported.     

Testing the Equality of Two Exponential Distributions

This paper proposes an overlapping-based test statistic for testing the equality of two exponential distributions with different scale and location parameters. The test statistic is defined as the maximum likelihood estimate of the Weitzman's overlapping coefficient, which estimates the agreement of two densities. The proposed test statistic is derived in closed form. Simulated critical points are generated for the proposed test statistic for various sample sizes and significance levels via Monte Carlo Simulations. Statistical powers of the proposed test are computed via simulation studies and compared to those of the existing Log likelihood ratio test.     

Bayesian Estimation and Model Choice in Item Response Models

Item response models are essential tools for analyzing results from many educational and psychological tests. Such models are used to quantify the probability of correct response as a function of unobserved examinee ability and other parameters explaining the difficulty and the discriminatory power of the questions in the test. Some of these models also incorporate a threshold parameter for the probability of the correct response to account for the effect of guessing the correct answer in multiple choice type tests. In this article we consider fitting of such models using the Gibbs sampler. A data augmentation method to analyze a normal-ogive model incorporating a threshold guessing parameter is introduced and compared with a Metropolis-Hastings sampling method. The proposed method is an order of magnitude more efficient than the existing method. Another objective of this paper is to develop Bayesian model choice techniques for model discrimination. A predictive approach based on a variant of the Bayes factor is used and compared with another decision theoretic method which minimizes an expected loss function on the predictive space. A classical model choice technique based on a modified likelihood ratio test statistic is shown as one component of the second criterion. As a consequence the Bayesian methods proposed in this paper are contrasted with the classical approach based on the likelihood ratio test. Several examples are given to illustrate the methods.     

To be or not to be valid in testing the significance of the slope in simple quantitative linear models with autocorrelated errors

In this article, the validity of procedures for testing the significance of the slope in quantitative linear models with one explanatory variable and first-order autoregressive [AR(1)] errors is analyzed in a Monte Carlo study conducted in the time domain. Two cases are considered for the regressor: fixed and trended versus random and AR(1). In addition to the classical t -test using the Ordinary Least Squares (OLS) estimator of the slope and its standard error, we consider seven t -tests with n-2\,\hbox{df} built on the Generalized Least Squares (GLS) estimator or an estimated GLS estimator, three variants of the classical t -test with different variances of the OLS estimator, two asymptotic tests built on the Maximum Likelihood (ML) estimator, the F -test for fixed effects based on the Restricted Maximum Likelihood (REML) estimator in the mixed-model approach, two t -tests with n - 2 df based on first differences (FD) and first-difference ratios (FDR), and four modified t -tests using various corrections of the number of degrees of freedom. The FDR t -test, the REML F -test and the modified t -test using Dutilleul's effective sample size are the most valid among the testing procedures that do not assume the complete knowledge of the covariance matrix of the errors. However, modified t -tests are not applicable and the FDR t -test suffers from a lack of power when the regressor is fixed and trended ( i.e. , FDR is the same as FD in this case when observations are equally spaced), whereas the REML algorithm fails to converge at small sample sizes. The classical t -test is valid when the regressor is fixed and trended and autocorrelation among errors is predominantly negative, and when the regressor is random and AR(1), like the errors, and autocorrelation is moderately negative or positive. We discuss the results graphically, in terms of the circularity condition defined in repeated measures ANOVA and of the effective sample size used in correlation analysis with autocorrelated sample data. An example with environmental data is presented.     

On the accuracy and cost of numerical integration in several variables ∗

Identical numerical integration experiments are performed on a CYBER 205 and an IBM 3081 in order to gauge the relative performance of several methods of integration. The methods employed are the general methods of Gauss-Legendre, iterated Gauss-Legendre, Newton-Cotes, Romberg and Monte Carlo as well as three methods, due to Owen, Dutt, and Clark respectively, for integrating the normal density. The bi- and trivariate normal densities and four other functions are integrated; the latter four have integrals expressible in closed form and some of them can be parameterized to exhibit singularities or highly periodic behavior. The various Gauss-Legendre methods tend to be most accurate (when applied to the normal density they are even more accurate than the special purpose methods designed for the normal) and while they are not the fastest, they are at least competitive. In scalar mode the CYBER is about 2-6 times faster than the IBM 3081 and the speed advantage of vectorised to scalar mode ranges from 6 to 15. Large scale econometric problems of the probit type should now be routinely soluble.     

Some properties of Lin–Wong divergence on the past lifetime data

Measures of statistical divergence are used to assess mutual similarities between distributions of multiple variables through a variety of methodologies including Shannon entropy and Csiszar divergence. Modified measures of statistical divergence are introduced throughout the present article. Those modified measures are related to the Lin–Wong (LW) divergence applied on the past lifetime data. Accordingly, the relationship between Fisher information and the LW divergence measure was explored when applied on the past lifetime data. Throughout this study, a number of relations are proposed between various assessment methods which implement the Jensen–Shannon, Jeffreys, and Hellinger divergence measures. Also, relations between the LW measure and the Kullback–Leibler (KL) measures for past lifetime data were examined. Furthermore, the present study discusses the relationship between the proposed ordering scheme and the distance interval between LW and KL measures under certain conditions.     

Likelihood ratio tests for multivariate normality

This paper presents some powerful omnibus tests for multivariate normality based on the likelihood ratio and the characterizations of the multivariate normal distribution. The power of the proposed tests is studied against various alternatives via Monte Carlo simulations. Simulation studies show our tests compare well with other powerful tests including multivariate versions of the Shapiro–Wilk test and the Anderson–Darling test.