BARTLETT CORRECTED LIKELIHOOD RATIO TESTS IN LOCATION-SCALE NONLINEAR MODELS

The paper derives Bartlett corrections for improving the chisquare approximation to the likelihood ratio statistics in a class of location-scale family of distributions, which encompasses the elliptical family of distributions and also asymmetric distributions such as the extreme value distributions. We present, in matrix notation, a Bartlett corrected likelihood ratio statistic for testing that a subset of the nonlinear regression coefficients in this class of models equals a given vector of constants. The formulae derived are simple enough to be used analytically to obtain several Bartlett corrections in a variety of important models. We show that these formulae generalize a number of previously published results. We also present simulation results comparing the sizes and powers of the usual likelihood ratio tests and their Bartlett corrected versions when the scale parameter is considered known and when this parameter is uncorrectly specified.     

A K-SAMPLE EXTENSION TO FLIGNER'S CLASS OF TESTS FOR SCALE

The parameteric tests for equality of variance are well known. The classical F-test is typically used to test the hypothesis of equality of two variances, while tests such as those developed by Bartlett (1937) are commonly used for the k-sample hypothesis. These tests assume an underlying normal distribution and are quite sensitive to departures from normality (Box, 1953). Thus, when considering data that are from non-normal distributions, alternative nonparametric tests must be employed.
Fligner (1979) has proposed a class of two-sample distribution-free tests which possess very desirable properties and are attractive alternatives to other nonparametric tests for scale. The present paper extends the Fligner class of tests to the more general k-sample case.     

A MONTE CARLO COMPARISON OF ELLIPTICAL-THEORY AND OTHER TESTS FOR EQUALITY OF COVARIANCE MATRICES

A Monte Carlo study of the size and power of tests of equality of two covariance matrices is carried out. Tests based upon normality assumptions, elliptical distribution assumptions as well as distribution-free tests are compared. Samples are generated from normal, elliptical and non-elliptical populations. The elliptical-theory tests, in particular, have poor size properties for both elliptical distributions with moderate sample sizes and for non-elliptical distributions.     

On the distribution of the likelihood ratio test of equality of normal populations

It has recently been shown by Perlman (1980) that when testing the equality of several normal distributions it is the likelihood ratio test which is unbiased rather than a test based on a modified statistic in common use. This paper gives expansions for the null distribution of the likelihood ratio statistic as well as for the nonnull distribution in a special case.     

Level-specific correction for nonparametric likelihoods

The popular empirical likelihood method not only has a convenient chi-square limiting distribution but is also Bartlett correctable, leading to a high-order coverage precision of the resulting confidence regions. Meanwhile, it is one of many nonparametric likelihoods in the Cressie–Read power divergence family. The other likelihoods share many attractive properties but are not Bartlett correctable. In this paper, we develop a new technique to achieve the effect of being Bartlett correctable. Our technique is generally applicable to pivotal quantities with chi-square limiting distributions. Numerical experiments and an example reveal that the method is successful for several important nonparametric likelihoods.     

Asymptotic non-null distribution of a test connected with exponential populations

In this paper asymptotic expansions of the non-null distribution of the likelihood ratio criterion for testing the equality of several one parameter exponential distributions are obtained under local alternatives. These expansions are in terms of Chi-square distributions.     

Posterior Predictive Comparisons for the Two-sample Problem

The two-sample problem of inferring whether two random samples have equal underlying distributions is formulated within the Bayesian framework as a comparison of two posterior predictive inferences rather than as a problem of model selection. The suggested approach is argued to be particularly advantageous in problems where the objective is to evaluate evidence in support of equality, along with being robust to the priors used and being capable of handling improper priors. Our approach is contrasted with the Bayes factor in a normal setting and finally, an additional example is considered where the observed samples are realizations of Markov chains.     

Methods to test for equality of two normal distributions

Statistical tests for two independent samples under the assumption of normality are applied routinely by most practitioners of statistics. Likewise, presumably each introductory course in statistics treats some statistical procedures for two independent normal samples. Often, the classical two-sample model with equal variances is introduced, emphasizing that a test for equality of the expected values is a test for equality of both distributions as well, which is the actual goal. In a second step, usually the assumption of equal variances is discarded. The two-sample t test with Welch correction and the F test for equality of variances are introduced. The first test is solely treated as a test for the equality of central location, as well as the second as a test for the equality of scatter. Typically, there is no discussion if and to which extent testing for equality of the underlying normal distributions is possible, which is quite unsatisfactorily regarding the motivation and treatment of the situation with equal variances. It is the aim of this article to investigate the problem of testing for equality of two normal distributions, and to do so using knowledge and methods adequate to statistical practitioners as well as to students in an introductory statistics course. The power of the different tests discussed in the article is examined empirically. Finally, we apply the tests to several real data sets to illustrate their performance. In particular, we consider several data sets arising from intelligence tests since there is a large body of research supporting the existence of sex differences in mean scores or in variability in specific cognitive abilities.     

Tests for Paired Lifetime Data with Frailty Models

We consider the problem of hypothesis testing of the equality of marginal survival distributions observed from paired lifetime data. Usual procedures include the paired t-test, which may perform poor for certain types of data. We propose asymptotic tests based on gamma frailty models with Weibull conditional distributions, and investigate their theoretical properties using large sample theory. For finite samples, we conduct simulations to evaluate the powers of the associated tests. For moderate and less skewed data, the proposed tests are the most powerful among the commonly applied testing procedures. A data example is illustrated to demonstrate the methods.     

Generalised Smooth Tests of Goodness of Fit Utilising L‐moments

In this paper we present a semiparametric test of goodness of fit which is based on the method of L‐moments for the estimation of the nuisance parameters. This test is particularly useful for any distribution that has a convenient expression for its quantile function. The test proceeds by investigating equality of the first few L‐moments of the true and the hypothesised distributions. We provide details and undertake simulation studies for the logistic and the generalised Pareto distributions. Although for some distributions the method of L‐moments estimator is less efficient than the maximum likelihood estimator, the former method has the advantage that it may be used in semiparametric settings and that it requires weaker existence conditions. The new test is often more powerful than competitor tests for goodness of fit of the logistic and generalised Pareto distributions.     

C472. The expected value of a conditional variance: An upper bound

The article derives Bartlett corrections for improving the chi-square approximation to the likelihood ratio statistics in a class of symmetric nonlinear regression models. This is a wide class of models which encompasses the t model and several other symmetric distributions with longer-than normal tails. In this paper we present, in matrix notation, Bartlett corrections to likelihood ratio statistics in nonlinear regression models with errors that follow a symmetric distribution. We generalize the results obtained by Ferrari, S. L. P. and Arellano-Valle, R. B. (1996). Modified likelihood ratio and score tests in linear regression models using the t distribution. Braz. J. Prob. Statist., 10, 15–33, who considered a t distribution for the errors, and by Ferrari, S. L. P. and Uribe-Opazo, M. A. (2001). Corrected likelihood ratio tests in a class of symmetric linear regression models. Braz. J. Prob. Statist., 15, 49–67, who considered a symmetric linear regression model. The formulae derived are simple enough to be used analytically to obtain several Bartlett corrections in a variety of important models. We also present simulation results comparing the sizes and powers of the usual likelihood ratio tests and their Bartlett corrected versions.     

Report of the ASA Technical Panel on the Census Undercount

A new model is proposed for the joint distribution of paired survival times generated from clinical trials and certain reliability settings. The new model can be considered an extension to the bivariate exponential models studied in the literature. Here, a more flexible bivariate Weibull model will be derived, and two exact parametric tests for testing the equality of marginal survival distributions are developed.     

Testing for ordered failure rates under general progressive censoring

For exponentially distributed failure times under general progressive censoring schemes, testing procedures for ordered failure rates are proposed using the likelihood ratio principle. Constrained maximum likelihood estimators of the failure rates are found. The asymptotic distributions of the test statistics are shown to be mixtures of chi-square distributions. When testing the equality of the failure rates, a simulation study shows that the proposed test with restricted alternative has improved power over the usual chi-square statistic with an unrestricted alternative. The proposed methods are illustrated using data of survival times of patients with squamous carcinoma of the oropharynx.     

Improvement of the quality of the chi-square approximation for the ADF test on a covariance matrix with a linear structure

The asymptotically distribution-free (ADF) test statistic was proposed by Browne (1984). It is known that the null distribution of the ADF test statistic is asymptotically distributed according to the chi-square distribution. This asymptotic property is always satisfied, even under nonnormality, although the null distributions of other famous test statistics, e.g., the maximum likelihood test statistic and the generalized least square test statistic, do not converge to the chi-square distribution under nonnormality. However, many authors have reported numerical results which indicate that the quality of the chi-square approximation for the ADF test is very poor, even when the sample size is large and the population distribution is normal. In this paper, we try to improve the quality of the chi-square approximation to the ADF test for a covariance matrix with a linear structure by using the Bartlett correction applicable under the assumption of normality. By conducting numerical studies, we verify that the obtained Bartlett correction can perform well even when the assumption of normality is violated.     

Multi-Sample Likelihood Ratio Tests Based on Bipolar Watson Distributions Defined on the Hypersphere

We derive likelihood ratio tests for the equality of the directional parameters of k bipolar Watson distributions defined on the hypersphere with common concentration parameter. We analyze the power of these tests in the case of two distributions supposing in the alternative hypothesis two directional parameters forming an angle, which varies from 18° to 90°. We also compare the likelihood ratio tests with a high-concentration F-test.     

On Robust Analysis of a Normal Location Parameter

Pericchi and Smith considered a normal location parameter problem with double-exponential and Student t prior distributions. These two prior distributions both belong to the class of scale mixtures of normal distributions and are useful in providing a robust analysis of the normal location parameter problem. In this paper we extend the analysis to other scale mixtures of normal distributions, such as the exponential power and the symmetric stable distributions.     

Tests for the multivariate two-sample problem based on empirical probability measures

Nonparametric tests are proposed for the equality of two unknown p-variate distributions. Empirical probability measures are defined from samples from the two distributions and used to construct test statistics as the supremum of the absolute differences between empirical probabilities, the supremum being taken over all possible events. The test statistics are truly multivariate in not requiring the artificial ranking of multivariate observations, and they are distribution-free in the general p-variate case. Asymptotic null distributions are obtained. Powers of the proposed tests and a competitor are examined by Monte Carlo techniques.     

Prior-likelihood factorization and missing data

Marginal posterior distributions, when not available ana­lytically, can be at present numerically inaccessible if the number of parameters for intergration exeeeds 7 to 10. For the normal multivariate regression model, with data absent (missing)in a monotone pattern, some integrations have been accomplished analytically (Guttman and Menzefricke, 1983; Bartlett, 1983; for example).

In this note we show how monotely missing data support an extended prior-likelihood factorization and the needed posterior extended prior-likelihood factorization and the can be obtained directly using standard results.