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.     

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.     

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.     

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.     

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.     

The null distribution of the likelihood ratio test for a mixture of two normals after a restricted box-cox transformation

Maclean et al. (1976) applied a specific Box-Cox transformation to test for mixtures of distributions against a single distribution. Their null hypothesis is that a sample of n observations is from a normal distribution with unknown mean and variance after a restricted Box-Cox transformation. The alternative is that the sample is from a mixture of two normal distributions, each with unknown mean and unknown, but equal, variance after another restricted Box-Cox transformation. We developed a computer program that calculated the maximum likelihood estimates (MLEs) and likelihood ratio test (LRT) statistic for the above. Our algorithm for the calculation of the MLEs of the unknown parameters used multiple starting points to protect against convergence to a local rather than global maximum. We then simulated the distribution of the LRT for samples drawn from a normal distribution and five Box-Cox transformations of a normal distribution. The null distribution appeared to be the same for the Box-Cox transformations studied and appeared to be distributed as a chi-square random variable for samples of 25 or more. The degrees of freedom parameter appeared to be a monotonically decreasing function of the sample size. The null distribution of this LRT appeared to converge to a chi-square distribution with 2.5 degrees of freedom. We estimated the critical values for the 0.10, 0.05, and 0.01 levels of significance.     

Asymptotic power comparison of T2-type test and likelihood ratio test for a mean vector based on two-step monotone missing data

Abstract

In a 2-step monotone missing dataset drawn from a multivariate normal population, T²-type test statistic (similar to Hotelling’s T² test statistic) and likelihood ratio (LR) are often used for the test for a mean vector. In complete data, Hotelling’s T² test and LR test are equivalent, however T²-type test and LR test are not equivalent in the 2-step monotone missing dataset. Then we interest which statistic is reasonable with relation to power. In this paper, we derive asymptotic power function of both statistics under a local alternative and obtain an explicit form for difference in asymptotic power function. Furthermore, under several parameter settings, we compare LR and T²-type test numerically by using difference in empirical power and in asymptotic power function. Summarizing obtained results, we recommend applying LR test for testing a mean vector.     

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.     

The generalized likelihood ratio test for the Behrens-Fisher problem

A solution is suggested for the Behrens-Fisher problem of testing the equality of the means from two normal populations where variances are unknown and not assumed equal, by considering an adaption of the generalized likelihood ratio test. The test developed and called the adjusted likelihood ratio test has size close to the nominal significance level and compares favourably with regard to size and power to the Welch-Aspin test. An asymptotic result shows the connection between the generalized likelihood ratio test and the most commonly used test statistic for the Behrens-Fisher problem.     

On a gmanova model likelihood ratio test criterion

For the generalized MANOVA (GMANOVA) model of Potthoff and Roy (1964), X = BξA + E, Khatri (1966) derives the likelihood ratio test criterion for test-ing the composite double linear null hypothesis CξV = 0, C,V known. This criterion plays an important role in statistics, and several authors have recently studied its further properties. However, Khatri's (1966) de-reviation of the distribution of this criterion is involved. By noting that the GMANOVA model is re-stricted MANOVA model, this paper presents an alter-native simple derivation of the distribution of this criterion. The derivation is based on the generalized Sverdrup's lemma, Kabe (1965).     

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.     

A likelihood ratio test for interaction among variances

A likelihood ratio test for interaction in an analysis of variance model, when observations are variances is considered. Another statistic based on the logarithmic transformation is also considered and the null and non-null distributions of the test statistics are considered.     

Robust confidence intervals for contrasts based upon a likelihood ratio test

This paper describes how to compute robust confidence intervals for differences of the effects using the likelihood ratio testF _M in the two-way analysis of variance. The probability for the α-error and the average length of the confidence intervals withF _m and the quadratic formQ _M are investigated and compared with the classical confidence intervals fort-distributed and lognormal errors. We also give a warning of building confidence intervals withF _M andQ _M in the presence of heterogeneous scale parameters, because these tests which do not regard heteroscedasticity are then much too liberal.     

Likelihood ratio test for univariate Gaussian mixture

We consider Gaussian mixtures and particularly the problem of testing homogeneity, that is testing no mixture, against a mixture with two components. Seven distinct cases are addressed, corresponding to the possible restrictions on the parameters. For each case, we give a statistic that we claim to be the likelihood ratio test statistic. The proof is given in a simple case. With the help of a bound for the maximum of a Gaussian process we calculate the percentile points. The results are illustrated by simulation.     

Asymptotic properties of likelihood ratio test statistics in affected‐sib‐pair analysis

In an affected‐sib‐pair genetic linkage analysis, identical by descent data for affected sib pairs are routinely collected at a large number of markers along chromosomes. Under very general genetic assumptions, the IBD distribution at each marker satisfies the possible triangle constraint. Statistical analysis of IBD data should thus utilize this information to improve efficiency. At the same time, this constraint renders the usual regularity conditions for likelihood‐based statistical methods unsatisfied. In this paper, the authors study the asymptotic properties of the likelihood ratio test (LRT) under the possible triangle constraint. They derive the limiting distribution of the LRT statistic based on data from a single locus. They investigate the precision of the asymptotic distribution and the power of the test by simulation. They also study the test based on the supremum of the LRT statistics over the markers distributed throughout a chromosome. Instead of deriving a limiting distribution for this test, they use a mixture of chi‐squared distributions to approximate its true distribution. Their simulation results show that this approach has desirable simplicity and satisfactory precision.     

Empirical likelihood ratio test for or against a set of inequality constraints

We use the empirical likelihood ratio approach introduced by Owen (Biometrika 75 (1988), 237–249) to test for or against a set of inequality constraints when the parameters are defined by estimating functions. Our objective in this paper is to show that under fairly general conditions, the limiting distributions of the empirical likelihood ratio test statistics are of chi-bar square type (as in the parametric case) and give the expression of the weighting values. The results obtained here are similar to those in El Barmi and Dykstra (1995) where a full distributional model is assumed. This work presents also an extension of the results in Qin and Lawless (1995).     

The likelihood ratio test for high-dimensional linear regression model

The paper considers a significance test of regression variables in the high-dimensional linear regression model when the dimension of the regression variables p, together with the sample size n, tends to infinity. Under two sightly different cases, we proved that the likelihood ratio test statistic will converge in distribution to a Gaussian random variable, and the explicit expressions of the asymptotical mean and covariance are also obtained. The simulations demonstrate that our high-dimensional likelihood ratio test method outperforms those using the traditional methods in analyzing high-dimensional data.     

Asymptotic expansion of the nonnull distribution of likelihood ratio statistic for testing multisample sphericity

Asymptotic expansion of the nonnull distribution of the likelihood ratio statistic for testing muitisample sphericity in q multinormal populations is derived for the alternatives close to the null hypothesis.     

A generalized likelihood ratio test for monitoring profile data

Profile data emerges when the quality of a product or process is characterized by a functional relationship among (input and output) variables. In this paper, we focus on the case where each profile has one response variable Y, one explanatory variable x, and the functional relationship between these two variables can be rather arbitrary. The basic concept can be applied to a much wider case, however. We propose a general method based on the Generalized Likelihood Ratio Test (GLRT) for monitoring of profile data. The proposed method uses nonparametric regression to estimate the on-line profiles and thus does not require any functional form for the profiles. Both Shewhart-type and EWMA-type control charts are considered. The average run length (ARL) performance of the proposed method is studied. It is shown that the proposed GLRT-based control chart can efficiently detect both location and dispersion shifts of the on-line profiles from the baseline profile. An upper control limit (UCL) corresponding to a desired in-control ARL value is constructed.