Likelihood Ratio Test For Homogeneity Research Articles

Empirical likelihood ratio tests for homogeneity of component lifetime distributions based on system lifetime data

Empirical likelihood ratio tests for homogeneity of component lifetime distributions based on system lifetime data

On hypothesis testing in latent class and finite mixture stochastic frontier models, with application to a contaminated normal-half normal model

Latent class and finite mixture stochastic frontier models have been proposed as a means of allowing either for technological heterogeneity or more flexible distributions of noise and inefficiency. As in the wider literature on latent class and finite mixture models, we are interested in class enumeration, particularly testing against homogeneity. We apply a modified likelihood ratio test for homogeneity in a stochastic frontier setting, based on established results for non-Gaussian latent class and finite mixture models, and provide evidence from Monte Carlo experiments which suggest the applicability of results regarding a modified likelihood ratio test to the stochastic frontier setting. We demonstrate an application to testing a model with a contaminated normal noise term against a model with a normally distributed noise term, finding that the former is preferred, with significant implications for efficiency prediction.

Empirical Likelihood Ratio Tests for Homogeneity of Multiple Populations in the Presence of Auxiliary Information

The empirical likelihood ratio test (ELRT) statistic is constructed for testing the homogeneity of several nonparametric populations in the presence of some auxiliary information. It is shown—under some regularity conditions and under the null hypothesis that all distribution functions of the populations are equal—that the asymptotic distribution of the ELRT is a chi-squared distribution. The proposed ELRT could be more powerful than the Kruskal–Wallis test, as extra information can be efficiently employed by ELRT. The advantage of ELRT over T&P (2006) is that researchers do not need to select approximately normal statistics for inter-group comparisons, and ELRT is more suitable for the multi-population consistency test with a small sample size.

Standardized likelihood ratio test for homogeneity of variance of several normal populations

In this study, we propose a standardized likelihood ratio test for homogeneity of variances of several normal populations. We obtain the exact mean and variance of the likelihood ratio test. The proposed test, which does not require intensive simulation, is quite easy to implement especially for researchers. Furthermore, we derive the null distribution of the likelihood ratio test via Monte Carlo simulation. The results of our simulation study reveal clearly that the proposed test performs very well even for small sample sizes and a large number of groups.

Asymptotic null distribution of the modified likelihood ratio test for homogeneity in finite mixture models

Likelihood-based methods play a central role in statistical inference for parametric models. Among these, the modified likelihood ratio test is preferred in testing for homogeneity in finite mixture models. The test statistic is related to the maximum of a quadratic function under general regularity conditions. Re-parameterization is shown to have overcome the difficulty when linear independence is not satisfied. Models with parameter constraints are also considered. The asymptotic null distribution of the test statistic is shown to have a chi-bar-squared distribution in both constrained and unconstrained cases. We extend the result to linear models and demonstrate that the chi-bar-squared distribution is also applicable. The general asymptotic result provides a much simpler testing procedure with an exact form of the asymptotic distribution compared to re-sampling approach in the literature. It also offers accurate p-value as shown in simulation. The results are checked by extensive simulation and are supplemented by a breast cancer data example.

Statistical Analysis related to Exceptional Snow Loads

In the present paper the favorable fit for exceedances relate d to the accidental snow loads, described in a companion paper, is studied. In particular, the Hill, t-Hill and modified Hill estimators are used and the quality of their distributional fits is assessed. Also t-estimation methods are considered. Assuming that the data is driven by Pareto or exponential distribution, resp ectively, the exact likelihood ratio tests for homogeneity to validate the resu lts of regression fits are applied. Further, the statistical dependence between the exceedances and the corresponding altitudes is studied.

On Statistical Tests for Homogeneity of Two Bivariate Zero-Inflated Poisson Populations

The problem of testing treatment difference in the occurrence of a study endpoint in a randomized parallel-group comparative clinical trial with repeated responses under the assumption that the responses follow a bivariate zero-inflated Poisson (ZIP) distribution is considered. Likelihood ratio test for homogeneity of two bivariate ZIP populations is derived. Approximate formula for sample size calculation is also obtained, which achieves a desired power for detecting a clinically meaningful difference under an alternative hypothesis. An example concerning the comparison of treatment effect in an addictive clinical trial in terms of the number of days of illicit drug use during a month is given for illustrative purposes.

A Note on the Modified Likelihood Ratio Test for Homogeneity in Beta Mixture Models

This article presents a note on the modified likelihood ratio test for homogeneity in beta mixture models. Under consistency of the penalized maximum likelihood estimators, the limiting distribution of the test statistic converges to the chi-bar-squared distributions. The statistic degenerates to zero with a weight due to the negative definiteness of a complicated random matrix. The probability that this matrix is negative definite is related to the parameter values under the homogeneity hypothesis. The dependency pattern enables the introduction of an upper bound on the asymptotic null distribution. Simulation study is investigated to verify the accuracy of the results.

Comparative simulation study of likelihood ratio tests for homogeneity of the exponential distribution

The aim of this paper is to present and discuss the power of the exact likelihood ratio homogeneity testing procedure of the number of components k in the exponential mixture. First we present the likelihood ratio test for homogeneity (ELR), the likelihood ratio test for homogeneity against two-component exponential mixture (ELR2), and finally the likelihood ratio test for homogeneity against three-component exponential mixture (ELR3). Comparative power study of mentioned homogeneity tests against three-component subpopulation alternative is provided. Therein we concentrate on various setups of the scales and weights, which allow us to make conclusions for generic settings. The natural property is observed, namely increase of the power of exact likelihood ratio ELR, ELR2 and ELR3 tests with scale parameters considered in the alternative. We can state that the differences in power of ELR, ELR2 and ELR3 tests are small – therefore using of the computationally simpler ELR2 test is recommended for broad usage rather than computationally more expensive ELR3 test in the cases when unobserved heterogeneity is modelled. Anyhow caution should be taken before automatic usage of ELR3 in more informative settings, since the application of automatic methods hoping that the data will enforce its true structure is deceptive. Application of obtained results in reliability, finance or social sciences is straightforward.

Antedependence models for nonstationary categorical longitudinal data with ignorable missingness: likelihood‐based inference

Time index-ordered random variables are said to be antedependent (AD) of order (p1 ,p2 , … ,pn ) if the kth variable, conditioned on the pk immediately preceding variables, is independent of all further preceding variables. Inferential methods associated with AD models are well developed for continuous (primarily normal) longitudinal data, but not for categorical longitudinal data. In this article, we develop likelihood-based inferential procedures for unstructured AD models for categorical longitudinal data. Specifically, we derive maximum likelihood estimators (MLEs) of model parameters; penalized likelihood criteria and likelihood ratio tests for determining the order of antedependence; and likelihood ratio tests for homogeneity across groups, time invariance of transition probabilities, and strict stationarity. We give closed-form expressions for MLEs and test statistics, which allow for the possibility of empty cells and monotone missing data, for all cases save strict stationarity. For data with an arbitrary missingness pattern, we derive an efficient restricted expectation-maximization algorithm for obtaining MLEs. We evaluate the performance of the tests by simulation. We apply the methods to longitudinal studies of toenail infection severity (measured on a binary scale) and Alzheimer's disease severity (measured on an ordinal scale). The analysis of the toenail infection severity data reveals interesting nonstationary behavior of the transition probabilities and indicates that an unstructured first-order AD model is superior to stationary and other structured first-order AD models that have previously been fit to these data. The analysis of the Alzheimer's severity data indicates that the antedependence is second order with time-invariant transition probabilities, suggesting the use of a second-order autoregressive cumulative logit model.

Exact Likelihood Ratio Testing for Homogeneity of the Exponential Distribution

The aim of this article is to discuss homogeneity testing of the exponential distribution. We introduce the exact likelihood ratio test of homogeneity in the subpopulation model, ELR, and the exact likelihood ratio test of homogeneity against the two-components subpopulation alternative, ELR2. The ELR test is asymptotically optimal in the Bahadur sense when the alternative consists of sampling from a fixed number of components. Thus, in some setups the ELR is superior to frequently used tests for exponential homogeneity which are based on the EM algorithm (like the MLRT, ADDS, and D-tests). One important example of superiority of ELR and ELR2 tests is the case of lower contamination. We demonstrate this fact by both theoretical comparisons and simulations.

One-sided multiple comparisons for treatment means with a control mean

We study comparisons of several treatments with a common control when it is believed a priori that the treatment means, μ i , are at least as large as the control mean, μ 0 . In this setting, which is called a tree ordering, we study multiple comparisons that determine whether μ i > μ 0 or μ i = μ 0 for each treatment. The classical procedure by Dunnett (1955) and the step-down and step-up techniques by Dunnett and Tamhane (1991, 1992) are well known. The results in Marcus and Talpaz (1992) provide multiple comparisons based on the maximum likelihood estimates restricted by the tree ordering. We also study two-stage procedures that consist of the likelihood ratio test of homogeneity with the alternative constrained by the tree ordering followed by two-sample t comparisons with possibly different critical values for the two-sample comparisons. Marcus et al. (1976) discuss the use of closed tests in such situations and propose using a closed version of the restricted likelihood ratio test. We describe step-down versions of the Marcus–Talpaz, the two-stage, and the likelihood ratio procedures, as well as a closed version of the Marcus–Talpaz multiple comparison procedure. Using Monte Carlo techniques, we study the familywise errors and powers of these procedures and make some recommendations concerning techniques that perform well for all tree ordered mean vectors.

A modified likelihood ratio test for homogeneity in bivariate normal mixtures of two samples

This paper investigates the asymptotic properties of a modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models of two samples. The asymptotic null distribution of the modified likelihood ratio statistic is found to be χ22, where χ22 is a chi-squared distribution with 2 degrees of freedom.

Modified likelihood ratio test for homogeneity in bivariate normal mixtures with presence of a structural parameter

This paper investigates the asymptotic properties of the modified likelihood ratio statistic for testing homogeneity in bivariate normal mixture models with an unknown structural parameter. It is shown that the modified likelihood ratio statistic has χ22 null limiting distribution.

Modified likelihood ratio test for homogeneity in normal mixtures with two samples

This paper investigates the modified likelihood ratio test(LRT) for homogeneity in normal mixtures of two samples with mixing proportions unknown. It is proved that the limit distribution of the modified likelihood ratio test is χ(1)2.

Modified likelihood ratio test for homogeneity in a mixture of von Mises distributions

Directional data often arise in many sciences, including astronomy, biology, ecology, geology and medicine. One particular statistical problem of interest is whether the data are from a mixture of two von Mises distributions or one single von Mises distribution. Motivating examples include a DNA microarray experiment, where it is suggested that a proportion of circadian genes have systematically different phase/peak expressions in two different tissues. We study the use of the modified likelihood ratio test (MLRT) to this class of problems. The MLRT statistic is shown to have a simple χ 1 2 null limiting distribution. The result is extended to mixture models with general parametric kernels. The simulation study gives additional insight into the finite-sample performance of the test. Two real data examples are used to illustrate the proposed method.

Submarket Dynamics of Time to Sale

We argue that the rush to apply multiple regression estimation to time on the market (TOM) durations may have led to important details and idiosyncrasies in local housing market dynamics being overlooked. What is needed is a more careful examination of the fundamental properties of time to sale data. The approach promoted and presented here, therefore, is to provide an examination of housing sale dynamics using a step-by-step approach. We present three hypotheses about TOM: (i) there is nonmonotonic duration dependence in the hazard of sale, (ii) the hazard curve will vary both over time and across intra-urban areas providing evidence of the existence of submarkets and (iii) institutional idiosyncrasies can have a profound effect on the shape and position of the hazard curve. We apply life tables, kernel-smoothed hazard functions and likelihood ratio tests for homogeneity to a large Scottish data set to investigate these hypotheses. Our findings have important implications for TOM analysis.

The likelihood ratio test for homogeneity in bivariate normal mixtures

This paper investigates the asymptotic properties of the likelihood ratio statistic for testing homogeneity in a bivariate normal mixture model with known covariance. The asymptotic null distributions of the likelihood ratio statistic and a modified likelihood ratio statistic are obtained in explicit form. The distributions are identical. The results of a small simulation study to approximate the null distribution are presented.

The Likelihood Ratio Tests for Homogeneity of Inverse Gaussian Means Under Simple Order and Simple Tree Order

The inverse Gaussian (IG) family is now widely used for modeling non negative skewed measurements. In this article, we construct the likelihood-ratio tests (LRTs) for homogeneity of the order constrained IG means and study the null distributions for simple order and simple tree order cases. Interestingly, it is seen that the null distribution results for the normal case are applicable without modification to the IG case. This supplements the numerous well known and striking analogies between Gaussian and inverse Gaussian families

A test for homogeneity of ordered means of inverse Gaussian populations

The inference theory and methodology for the inverse Gaussian (IG) family of right skewed distributions with nonnegative support are well known to bear a striking resemblance to the normal theory and methods. The IG family is therefore increasingly used for modeling and analyzing nonnegative, right skewed data. In this note, we consider the problem of testing homogeneity of order-restricted IG means with a common known scale parameter using the approach used in Mudholkar and McDermott (Biometrika 76 (1989) 161). It is shown that the null hypothesis of equal means in the IG model admits a union–intersection decomposition and, the p-values of the one-tailed Z-like tests used for testing the component hypotheses are independently distributed as in the normal case. This allows testing homogeneity of IG means under a spectrum of order constraints using the classical methods for combining independent p-values. The power properties of this test, for the simple order-restricted case, are empirically examined and compared with the order-restricted likelihood ratio test (Tian and Mudholkar (The likelihood ratio tests for homogeneity of inverse Gaussian means under simple order and simple tree order, submitted for publication)), and the chi-square test which ignores the order restrictions.