The likelihood ratio test for homogeneity in finite mixture models

The authors study the asymptotic behaviour of the likelihood ratio statistic for testing homogeneity in the finite mixture models of a general parametric distribution family. They prove that the limiting distribution of this statistic is the squared supremum of a truncated standard Gaussian process. The autocorrelation function of the Gaussian process is explicitly presented. A re‐sampling procedure is recommended to obtain the asymptotic p‐value. Three kernel functions, normal, binomial and Poisson, are used in a simulation study which illustrates the procedure.     

Maximum likelihood estimators in finite mixture models with censored data

The consistency of estimators in finite mixture models has been discussed under the topology of the quotient space obtained by collapsing the true parameter set into a single point. In this paper, we extend the results of Cheng and Liu (2001) to give conditions under which the maximum likelihood estimator (MLE) is strongly consistent in such a sense in finite mixture models with censored data. We also show that the fitted model tends to the true model under a weak condition as the sample size tends to infinity.     

Penalized likelihood-ratio test for finite mixture models with multinomial observations

Due to the irregularity of finite mixture models, the commonly used likelihood-ratio statistics often have complicated limiting distributions. We propose to add a particular type of penalty function to the log-likelihood function. The resulting penalized likelihood-ratio statistics have simple limiting distributions when applied to finite mixture models with multinomial observations. The method is especially effective in addressing the problems discussed by Chernoff and Lander (1995). The theory developed and simulations conducted show that the penalized likelihood method can give very good results, better than the well-known C(α) procedure, for example. The paper does not, however, fully explore the choice of penalty function and weight. The full potential of the new procedure is to be explored in the future.     

Testing for a finite mixture model with two components

Summary.  We consider a finite mixture model with k components and a kernel distribution from a general one-parameter family. The problem of testing the hypothesis k =2 versus k 3 is studied. There has been no general statistical testing procedure for this problem. We propose a modified likelihood ratio statistic where under the null and the alternative hypotheses the estimates of the parameters are obtained from a modified likelihood function. It is shown that estimators of the support points are consistent. The asymptotic null distribution of the modified likelihood ratio test proposed is derived and found to be relatively simple and easily applied. Simulation studies for the asymptotic modified likelihood ratio test based on finite mixture models with normal, binomial and Poisson kernels suggest that the test proposed performs well. Simulation studies are also conducted for a bootstrap method with normal kernels. An example involving foetal movement data from a medical study illustrates the testing procedure.     

Bootstrap likelihood ratio test for Weibull mixture models fitted to grouped data

Abstract

Weibull mixture models are widely used in a variety of fields for modeling phenomena caused by heterogeneous sources. We focus on circumstances in which original observations are not available, and instead the data comes in the form of a grouping of the original observations. We illustrate EM algorithm for fitting Weibull mixture models for grouped data and propose a bootstrap likelihood ratio test (LRT) for determining the number of subpopulations in a mixture model. The effectiveness of the LRT methods are investigated via simulation. We illustrate the utility of these methods by applying them to two grouped data applications.     

A likelihood ratio test for bimodality in two-component mixtures with application to regional income distribution in the EU

We propose a parametric test for bimodality based on the likelihood principle by using two-component mixtures. The test uses explicit characterizations of the modal structure of such mixtures in terms of their parameters. Examples include the univariate and multivariate normal distributions and the von Mises distribution. We present the asymptotic distribution of the proposed test and analyze its finite sample performance in a simulation study. To illustrate our method, we use mixtures to investigate the modal structure of the cross-sectional distribution of per capita log GDP across EU regions from 1977 to 1993. Although these mixtures clearly have two components over the whole time period, the resulting distributions evolve from bimodality toward unimodality at the end of the 1970s.     

Penalized minimum-distance estimates in finite mixture models

When finite mixture models are used to fit data, it is sometimes important to estimate the number of mixture components. A nonparametric maximum-likelihood approach may result in too many support points and, in general, does not yield a consistent estimator. A penalized likelihood approach tends to produce a fit with fewer components, but it is not known whether that approach produces a consistent estimate of the number of mixture components. We suggest the use of a penalized minimum-distance method. It is shown that the estimator obtained is consistent for both the mixing distribution and the number of mixture components.     

A likelihood ratio test of a homoscedastic normal mixture against a heteroscedastic normal mixture

It is generally assumed that the likelihood ratio statistic for testing the null hypothesis that data arise from a homoscedastic normal mixture distribution versus the alternative hypothesis that data arise from a heteroscedastic normal mixture distribution has an asymptotic χ ² reference distribution with degrees of freedom equal to the difference in the number of parameters being estimated under the alternative and null models under some regularity conditions. Simulations show that the χ ² reference distribution will give a reasonable approximation for the likelihood ratio test only when the sample size is 2000 or more and the mixture components are well separated when the restrictions suggested by Hathaway (Ann. Stat. 13:795–800, 1985) are imposed on the component variances to ensure that the likelihood is bounded under the alternative distribution. For small and medium sample sizes, parametric bootstrap tests appear to work well for determining whether data arise from a normal mixture with equal variances or a normal mixture with unequal variances.     

A modified signed likelihood ratio test in elliptical structural models

In this paper we deal with the issue of performing accurate testing inference on a scalar parameter of interest in structural errors-in-variables models. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal distribution as special case. We derive a modified signed likelihood ratio statistic that follows a standard normal distribution with a high degree of accuracy. Our Monte Carlo results show that the modified test is much less size distorted than its unmodified counterpart. An application is presented.     

Near-exact distributions for the sphericity likelihood ratio test statistic

In this paper three near-exact distributions are developed for the sphericity test statistic. The exact probability density function of this statistic is usually represented through the use of the Meijer G function, which renders the computation of quantiles impossible even for a moderately large number of variables. The main purpose of this paper is to obtain near-exact distributions that lie closer to the exact distribution than the asymptotic distributions while, at the same time, correspond to density and cumulative distribution functions practical to use, allowing for an easy determination of quantiles. In addition to this, two asymptotic distributions that lie closer to the exact distribution than the existing ones were developed. Two measures are considered to evaluate the proximity between the exact and the asymptotic and near-exact distributions developed. As a reference we use the saddlepoint approximations developed by Butler et al. [1993. Saddlepoint approximations for tests of block independence, sphericity and equal variances and covariances. J. Roy. Statist. Soc., Ser. B 55, 171–183] as well as the asymptotic distribution proposed by Box.     

Asymptotic optimality of double sequential mixture likelihood ratio test

In this article, we propose to use the weighted expected sample size (WESS) to evaluate the overall performance of sequential test plans on a finite set of parameters. Motivated by minimizing the WESS to control the expected sample sizes, we develop the method of double sequential mixture likelihood ratio test (2-SMLRT) for one-sided composite hypotheses. It is proved that the 2-SMLRT is asymptotically optimal on and its stopping time is finite under some conditions. The 2-SMLRT is general and includes the sequential probability ratio test (SPRT) and the double sequential probability ratio test (2-SPRT) as special cases. Simulation studies show that compared with the SPRT and 2-SPRT, the 2-SMLRT has smaller WESS and relative mean index with less or comparable expected sample sizes when the null hypothesis or alternative hypothesis holds.     

Variable selection in finite mixture of semi-parametric regression models

Abstract

In this paper we are concerned with variable selection in finite mixture of semiparametric regression models. This task consists of model selection for non parametric component and variable selection for parametric part. Thus, we encountered separate model selections for every non parametric component of each sub model. To overcome this computational burden, we introduced a class of variable selection procedures for finite mixture of semiparametric regression models using penalized approach for variable selection. It is shown that the new method is consistent for variable selection. Simulations show that the performance of proposed method is good, and it consequently improves pervious works in this area and also requires much less computing power than existing methods.     

A simple root selection method for univariate finite normal mixture models

It is well known that there exist multiple roots of the likelihood equations for finite normal mixture models. Selecting a consistent root for finite normal mixture models has long been a challenging problem. Simply using the root with the largest likelihood will not work because of the spurious roots. In addition, the likelihood of normal mixture models with unequal variance is unbounded and thus its maximum likelihood estimate (MLE) is not well defined. In this paper, we propose a simple root selection method for univariate normal mixture models by incorporating the idea of goodness of fit test. Our new method inherits both the consistency properties of distance estimators and the efficiency of the MLE. The new method is simple to use and its computation can be easily done using existing R packages for mixture models. In addition, the proposed root selection method is very general and can be also applied to other univariate mixture models. We demonstrate the effectiveness of the proposed method and compare it with some other existing methods through simulation studies and a real data application.     

The approximate null distribution of the likelihood ratio test for a mixture of two bivariate normal distributions with equal co variance

We define the mixture likelihood approach to clustering by discussing the sampling distribution of the likelihood ratio test of the null hypothesis that we have observed a sample of observations of a variable having the bivariate normal distribution versus the alternative that the variable has the bivariate normal mixture with unequal means and common within component covariance matrix. The empirical distribution of the likelihood ratio test indicates that convergence to the chi-squared distribution with 2 df is at best very slow, that the sample size should be 5000 or more for the chi-squared result to hold, and that for correlations between 0.1 and 0.9 there is little, if any, dependence of the null distribution on the correlation. Our simulation study suggests a heuristic function based on the gamma.     

Maximum log-likelihood ratio test for a change in three parameter Weibull distribution

Asymptotic behavior of a log-likelihood ratio statistic for testing a change in a three parameter Weibull distribution is studied. It is shown that if a shape parameter α>2$\alpha >2$ the law of iterated logarithm for maximum-likelihood estimators is still valid and the log-likelihood testing statistic is asymptotically distributed (after an appropriate normalization) according to a Gumbel distribution.     

Estimating the cure fraction in population-based cancer studies by using finite mixture models

Summary.  The cure fraction (the proportion of patients who are cured of disease) is of interest to both patients and clinicians and is a useful measure to monitor trends in survival of curable disease. The paper extends the non-mixture and mixture cure fraction models to estimate the proportion cured of disease in population-based cancer studies by incorporating a finite mixture of two Weibull distributions to provide more flexibility in the shape of the estimated relative survival or excess mortality functions. The methods are illustrated by using public use data from England and Wales on survival following diagnosis of cancer of the colon where interest lies in differences between age and deprivation groups. We show that the finite mixture approach leads to improved model fit and estimates of the cure fraction that are closer to the empirical estimates. This is particularly so in the oldest age group where the cure fraction is notably lower. The cure fraction is broadly similar in each deprivation group, but the median survival of the 'uncured' is lower in the more deprived groups. The finite mixture approach overcomes some of the limitations of the more simplistic cure models and has the potential to model the complex excess hazard functions that are seen in real data.     

Testing homogeneity in a mixture of von mises distributions with a structural parameter

The modified likelihood ratio statistic can be used to test the homogeneity in a variety of mixture models. Here, the authors propose the use of the modified and the iterative modified likelihood ratio for testing homogeneity against a two‐component von Mises mixture with a structural parameter. They derive the limiting distributions of the test statistics and propose methods to improve the accuracy of the asymptotic approximation in finite samples. Their simulations show that the tests maintain their nominal level and that they have adequate power. Data on movements of turtles are used as an illustration