Hanfeng Chen

A modified likelihood ratio test for homogeneity in finite mixture models

Testing for homogeneity in finite mixture models has been investigated by many researchers. The asymptotic null distribution of the likelihood ratio test (LRT) is very complex and difficult to use in practice. We propose a modified LRT for homogeneity in finite mixture models with a general parametric kernel distribution family. The modified LRT has a χ‐type of null limiting distribution and is asymptotically most powerful under local alternatives. Simulations show that it performs better than competing tests. They also reveal that the limiting distribution with some adjustment can satisfactorily approximate the quantiles of the test statistic, even for moderate sample sizes.

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.

Large sample distribution of the likelihood ratio test for normal mixtures

This article concerns with the problem of testing whether a mixture of two normal distributions with bounded means and specific variance is simply a pure normal. The large sample behavior of the likelihood ratio test for the problem is carefully investigated. In the case of one mean parameter, it is shown that the large sample null distribution of the likelihood ratio test statistic is the squared supremum of a Gaussian process with zero mean and explicitly given covariances. In the case of two mean parameters, both the simple and composite hypotheses of normality are considered. Under the simple null hypothesis, the large sample null distribution is found to be an independent sum of a chi-square variable and the squared supremum of another Gaussian process whose covariance structure is slightly different from the one mean parameter case, while under the composite null hypothesis, the chi-square term is absent.

The Accuracy of Approximate Intervals for a Binomial Parameter

Abstract This article considers the uniform convergence of the coverage probabilities of some approximate confidence intervals for the binomial parameter p, induced by central limit arguments. A uniform upper bound on the coverage probabilities of any such interval obtained by transformation of the sample proportion [pcirc] n is derived. The coverage probability of the interval induced by the arcsine transformation turns out to be very close to the upper bound. Replacing [pcirc] n by the Bayes estimate (X + β)/(n + 2β) for some β, however, gives even better uniform asymptotic properties; the choice β = z α 2/2 is recommended, where z α is the (1 − α)th quantile of the standard normal distribution. With such β, numerical results show that this interval is satisfactory, even in the uniform sense, since it possesses uniform asymptotic coefficients that are very close to the nominal ones. Furthermore, the best k for β of the form β = kz α 2 such that the uniform asymptotic coefficient is as close to the nomin...

A Logistic Regression Mixture Model for Interval Mapping of Genetic Trait Loci Affecting Binary Phenotypes

Often in genetic research, presence or absence of a disease is affected by not only the trait locus genotypes but also some covariates. The finite logistic regression mixture models and the methods under the models are developed for detection of a binary trait locus (BTL) through an interval-mapping procedure. The maximum-likelihood estimates (MLEs) of the logistic regression parameters are asymptotically unbiased. The null asymptotic distributions of the likelihood-ratio test (LRT) statistics for detection of a BTL are found to be given by the supremum of a χ2-process. The limiting null distributions are free of the null model parameters and are determined explicitly through only four (backcross case) or nine (intercross case) independent standard normal random variables. Therefore a threshold for detecting a BTL in a flanking marker interval can be approximated easily by using a Monte Carlo method. It is pointed out that use of a threshold incorrectly determined by reading off a χ2-probability table can result in an excessive false BTL detection rate much more severely than many researchers might anticipate. Simulation results show that the BTL detection procedures based on the thresholds determined by the limiting distributions perform quite well when the sample sizes are moderately large.

Bahadur representations of the empirical likelihood quantile processes

The empirical likelihood method was introduced by Art B. Owen about a decade ago to construct confidence intervals as a nonparametric technique. It is shown that the empirical likelihood method has very general application. One of the advantages of the empirical likelihood method is that it can readily take into account the model information available for statistical analysis. In this paper we investigate Bahadur representations of the empirical likelihood auantiles under the commonlv used constrained estimation model and the selection biis model. It is shown that tie additional model information improves the quantile estimation in large samples.

Concavity of Box-Cox log-likelihood function

In this article, we prove that, in the single sample case, the Box-Cox log-likelihood function for the power transformation parameter is strictly concave downward.

Estimating the Proportion of True Null Hypotheses in Nonparametric Exponential Mixture Model with Appication to the Leukemia Gene Expression Data

We revisit the problem of estimating the proportion π of true null hypotheses where a large scale of parallel hypothesis tests are performed independently. While the proportion is a quantity of interest in its own right in applications, the problem has arisen in assessing or controlling an overall false discovery rate. On the basis of a Bayes interpretation of the problem, the marginal distribution of the p-value is modeled in a mixture of the uniform distribution (null) and a non-uniform distribution (alternative), so that the parameter π of interest is characterized as the mixing proportion of the uniform component on the mixture. In this article, a nonparametric exponential mixture model is proposed to fit the p-values. As an alternative approach to the convex decreasing mixture model, the exponential mixture model has the advantages of identifiability, flexibility, and regularity. A computation algorithm is developed. The new approach is applied to a leukemia gene expression data set where multiple significance tests over 3,051 genes are performed. The new estimate for π with the leukemia gene expression data appears to be about 10% lower than the other three estimates that are known to be conservative. Simulation results also show that the new estimate is usually lower and has smaller bias than the other three estimates.

Simultaneous confidence bands for comparing diagnostic markers

Diagnostic markers are evaluated and compared by their sensitivities and specificities. In this article simultaneous confidence bands (CBs) for a portion of a difference of the receiver operating characteristic (ROC) curves for two diagnostic markers with paired data are established. When the variances of healthy group and diseased group are proportional, the coverage probabilities of the CBs are asymptotically correct. By the same approach asymptotic CBs for a portion of a single ROC curve are also derived. All results obtained are applicable to either actual measurements or rating data. The methods are illustrated with a set of real data arising from a cancer clinical trial.

Uniform robustness against nonnormality of the t and f tests

The size of the two-sample t test is generally thought to be robust against nonnormal distributions if the sample sizes are large. This belief is based on central limit theory, and asymptotic expansions of the moments of the t statistic suggest that robustness may be improved for moderate sample sizes if the variance, skewness, and kurtosis of the distributions are matched, particularly if the sample sizes are also equal. It is shown that asymptotic arguments such as these can be misleading and that, in fact, the size of the t test can be as large as unity if the distributions are allowed to be completely arbitrary. Restricting the distributions to be identical or symmetric (but otherwise arbitrary) does not guarantee that the size can be controlled either, but controlling the tail-heaviness of the distributions does. The last result is proved more generally for the k-sample F test.