Empirical likelihood‐based inference for genetic mixture models

The authors show how the genetic effect of a quantitative trait locus can be estimated by a nonparametric empirical likelihood method when the phenotype distributions are completely unspecified. They use an empirical likelihood ratio statistic for testing the genetic effect and obtaining confidence intervals. In addition to studying the asymptotic properties of these procedures, the authors present simulation results and illustrate their approach with a study on breast cancer resistance genes.     

Semiparametric estimation in the normal variance-mean mixture model

In this paper we study the problem of statistical inference on the parameters of the semiparametric variance-mean mixtures. This class of mixtures has recently become rather popular in statistical and financial modelling. We design a semiparametric estimation procedure that first estimates the mean of the underlying normal distribution and then recovers non-parametrically the density of the corresponding mixing distribution. We illustrate the performance of our procedure on simulated and real data.     

Identification of target clusters by using the restricted normal mixture model

This paper addresses the problem of identifying groups that satisfy the specific conditions for the means of feature variables. In this study, we refer to the identified groups as “target clusters” (TCs). To identify TCs, we propose a method based on the normal mixture model (NMM) restricted by a linear combination of means. We provide an expectation–maximization (EM) algorithm to fit the restricted NMM by using the maximum-likelihood method. The convergence property of the EM algorithm and a reasonable set of initial estimates are presented. We demonstrate the method's usefulness and validity through a simulation study and two well-known data sets. The proposed method provides several types of useful clusters, which would be difficult to achieve with conventional clustering or exploratory data analysis methods based on the ordinary NMM. A simple comparison with another target clustering approach shows that the proposed method is promising in the identification.     

Bias reduction of the maximum-likelihood estimator for a conditional Gaussian MA(1) model

In this paper, we consider an estimation for the unknown parameters of a conditional Gaussian MA(1) model. In the majority of cases, a maximum-likelihood estimator is chosen because the estimator is consistent. However, for small sample sizes the error is large, because the estimator has a bias of O(n^{? 1}). Therefore, we provide a bias of O(n^{? 1}) for the maximum-likelihood estimator for the conditional Gaussian MA(1) model. Moreover, we propose new estimators for the unknown parameters of the conditional Gaussian MA(1) model based on the bias of O(n^{? 1}). We investigate the properties of the bias, as well as the asymptotical variance of the maximum-likelihood estimators for the unknown parameters, by performing some simulations. Finally, we demonstrate the validity of the new estimators through this simulation study.     

Detecting the number of components in a finite mixture having normal components

A procedure is described which detects the number of components in a mixture distribution having normal components. The test is based on the behavior of the sample order statistics near the center of the distribution. Numerical results are presented and comparisons with tests proposed by Shapiro and Francia (1972) and Baker (1958) are provided.     

Restricted tests for testing independence of time to failure and cause of failure in a competing-risks model

We consider the competing-risks problem without making any assumption concerning the independence of the risks. Maximum-likelihood estimates of the cause-specific hazard rates are obtained under the condition that their ratio is monotone. We also consider the likelihood-ratio test for testing the proportionality of two cause-specific hazard rates against the alternative that the ratio of these two hazard rates is monotonic. This testing problem is equivalent to testing independence against likelihood-ratio dependence of the time to failure and the cause of failure in the competing-risks setup. We allow for random censoring on the right. The asymptotic null distribution of the test statistic is obtained and is found to be of the chi-bar-square type. The problem is extended to the case of more than two risks. A numerical example is given to illustrate the procedure.     

Improved hypothesis testing in a general multivariate elliptical model

This paper investigates improved testing inferences under a general multivariate elliptical regression model. The model is very flexible in terms of the specification of the mean vector and the dispersion matrix, and of the choice of the error distribution. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal and Student-t distributions as special cases. We obtain Skovgaard's adjusted likelihood ratio (LR) statistics and Barndorff-Nielsen's adjusted signed LR statistics and we compare the methods through simulations. The simulations suggest that the proposed tests display superior finite sample behaviour as compared to the standard tests. Two applications are presented in order to illustrate the methods.     

Maximum-likelihood estimation of the random-clumped multinomial model as a prototype problem for large-scale statistical computing

Numerical methods are needed to obtain maximum-likelihood estimates (MLEs) in many problems. Computation time can be an issue for some likelihoods even with modern computing power. We consider one such problem where the assumed model is a random-clumped multinomial distribution. We compute MLEs for this model in parallel using the Toolkit for Advanced Optimization software library. The computations are performed on a distributed-memory cluster with low latency interconnect. We demonstrate that for larger problems, scaling the number of processes improves wall clock time significantly. An illustrative example shows how parallel MLE computation can be useful in a large data analysis. Our experience with a direct numerical approach indicates that more substantial gains may be obtained by making use of the specific structure of the random-clumped model.     

Testing homogeneity in a multivariate mixture model

Testing homogeneity is a fundamental problem in finite mixture models. It has been investigated by many researchers and most of the existing works have focused on the univariate case. In this article, the authors extend the use of the EM‐test for testing homogeneity to multivariate mixture models. They show that the EM‐test statistic asymptotically has the same distribution as a certain transformation of a single multivariate normal vector. On the basis of this result, they suggest a resampling procedure to approximate the P‐value of the EM‐test. Simulation studies show that the EM‐test has accurate type I errors and adequate power, and is more powerful and computationally efficient than the bootstrap likelihood ratio test. Two real data sets are analysed to illustrate the application of our theoretical results. The Canadian Journal of Statistics 39: 218–238; 2011 © 2011 Statistical Society of Canada     

Asymptotic test of mixture model and its applications to QTL interval mapping

Quantitative trait loci (QTL) mapping has been a standard means in identifying genetic regions harboring potential genes underlying complex traits. Likelihood ratio test (LRT) has been commonly applied to assess the significance of a genetic locus in a mixture model content. Given the time constraint in commonly used permutation tests to assess the significance of LRT in QTL mapping, we study the behavior of the LRT statistic in mixture model when the proportions of the distributions are unknown. We found that the asymptotic null distribution is stationary Gaussian process after suitable transformation. The result can be applied to one-parameter exponential family mixture model. Under certain condition, such as in a backcross mapping model, the tail probability of the supremum of the process is calculated and the threshold values can be determined by solving the distribution function. Simulation studies were performed to evaluate the asymptotic results.     

Learning a multivariate Gaussian mixture model with the reversible jump MCMC algorithm

This paper is a contribution to the methodology of fully Bayesian inference in a multivariate Gaussian mixture model using the reversible jump Markov chain Monte Carlo algorithm. To follow the constraints of preserving the first two moments before and after the split or combine moves, we concentrate on a simplified multivariate Gaussian mixture model, in which the covariance matrices of all components share a common eigenvector matrix. We then propose an approach to the construction of the reversible jump Markov chain Monte Carlo algorithm for this model. Experimental results on several data sets demonstrate the efficacy of our algorithm.     

Two new confidence intervals for the coefficient of variation in a normal distribution

In this article we introduce an approximately unbiased estimator for the population coefficient of variation, τ, in a normal distribution. The accuracy of this estimator is examined by several criteria. Using this estimator and its variance, two approximate confidence intervals for τ are introduced. The performance of the new confidence intervals is compared to those obtained by current methods.     

A model selection approach for the identification of quantitative trait loci in experimental crosses

Summary. We consider the problem of identifying the genetic loci (called quantitative trait loci (QTLs)) contributing to variation in a quantitative trait, with data on an experimental cross. A large number of different statistical approaches to this problem have been described; most make use of multiple tests of hypotheses, and many consider models allowing only a single QTL. We feel that the problem is best viewed as one of model selection. We discuss the use of model selection ideas to identify QTLs in experimental crosses. We focus on a back-cross experiment, with strictly additive QTLs, and concentrate on identifying QTLs, considering the estimation of their effects and precise locations of secondary importance. We present the results of a simulation study to compare the performances of the more prominent methods.     

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     

Moments of order statistics from a non-overlapping mixture model with applications to truncated laplace distribution

We employ two different approaches to derive single and product moments of order statistics from a truncated Laplace distribution. A direct evaluation method establishes recurrence relations whereas the more general non-overlapping mixture model incorporates the truncated Laplace distribution as a special case. The results are thereafter applied to estimate location and scale parameters of such distributions.     

A new test for the presence of a normal mixture distribution based on the posterior Bayes factor

We present a new test for the presence of a normal mixture distribution, based on the posterior Bayes factor of Aitkin (1991). The new test has slightly lower power than the likelihood ratio test. It does not require the computation of the MLEs of the parameters or a search for multiple maxima, but requires computations based on classification likelihood assignments of observations to mixture components.     

A competing risks derivation of a mixture model for the analysis of survival data

Boag (1949) and Berkson and Gage (1952) proposed a mixture model for the analysis of survival time data when aproportion of treated patients are cured. This paper presents a derivation of the Boag/Berkson-Gage mixture model as well as a eneralization of the model based on the theory of competing risks. The assumptions underlying the model are stated and discussed and a general likelihood function is obtained. Use of the model is illustrated ith data from the Stanford Heart Transplant Program.