Distribution Of Likelihood Ratio Statistic Research Articles

A reverse Gaussian correlation inequality by adding cones

Let γ denote any centered Gaussian measure on Rd. It is proved that for any closed convex sets A and B in Rd, and any closed convex cones C and D in Rd, if D⊇C∘, where C∘ is the polar cone of C, then γ((A+C)∩(B+D))≤γ(A+C)⋅γ(B+D), and γ((A+C)∩(B−D))≥γ(A+C)⋅γ(B−D). As an application, this new inequality is used to bound the asymptotic posterior distributions of likelihood ratio statistics for convex cones.

Forecast Comparison of Nonlinear Time Series Models of Us GDP: A Bayesian Approach

This thesis utilizes modern Bayesian tools to evaluate the forecasting performance of two of the most widely used nonlinear time series models of post-war US GDP, the Markov Switching (MS) model and the Self-Exciting threshold autoregressive (SETAR) model. We develop a clear, empirical ground for model selection, forecast comparison and forecast combination in the context of nonlinear economic time series. The use of the Bayesian methodology introduces an immediate framework to deal with nonlinearities. This holds not only in estimation, by treating nuisance parameters as unknown random variables, and in testing, by avoiding the approximation of unknown distributions of likelihood ratio statistics in order to find evidence of nonlinearity, but also in the choice of the best model according to its forecasting potential. To our knowledge, the literature avoids welcoming existing Bayesian methods in forecasting and comparison of nonlinear time series models; all known attempts from a Bayesian perspective are consumed only to the estimation part. Here we propose to use predictive likelihoods, an easy to adopt and to interpret criterion, as an alternative to classical forecast evaluation criterions like the root mean squared forecast error (RMSE). The advantage of our approach is very clear in the nonlinear time series framework that we examine. Preservation of computational resources, efficiency in manipulation of nuisance parameters, and parameter uncertainty are fully incorporated while we improve over individual forecasts by implementing Bayesian Model Averaging.

The likelihood ratio test for non-standard hypotheses near the boundary of the null – with application to the assessment of non-inferiority

Abstract We consider a class of testing problems where the null space is the union of k-1 subgraphs of the form h j (θ j )≤θ k , with j=1,…,k-1, (θ 1,…,θ k ) the unknown parameter, and h j given increasing functions. The data consist of k independent samples, assumed to be drawn from a distribution with parameter θ j , j=1,…,k, respectively. An important class of examples covered by this setting is that of non-inferiority hypotheses, which have recently become important in the evaluation of drugs or therapies. When the true parameter approaches the boundary at a 1/√n rate, we give the explicit form of the asymptotic distribution of the log-likelihood ratio statistic. This extends previous work on the distribution of likelihood ratio statistics to local alternatives. We consider the prominent example of binomial data and illustrate the theory for k=2 and 3 samples. We explain how this can be used for planning a non-inferiority trial. To this end we calculate the optimal sample ratios yielding the maximal power in a binomial non-inferiority trial.

Estimating the π* goodness of fit index for finite mixtures of item response models

Testing the fit of finite mixture models is a difficult task, since asymptotic results on the distribution of likelihood ratio statistics do not hold; for this reason, alternative statistics are needed. This paper applies the pi* goodness of fit statistic to finite mixture item response models. The pi* statistic assumes that the population is composed of two subpopulations - those that follow a parametric model and a residual group outside the model; pi* is defined as the proportion of population in the residual group. The population was divided into two or more groups, or classes. Several groups followed an item response model and there was also a residual group. The paper presents maximum likelihood algorithms for estimating item parameters, the probabilities of the groups and pi*. The paper also includes a simulation study on goodness of recovery for the two- and three-parameter logistic models and an example with real data from a multiple choice test.

A range of simple summary genome‐wide statistics for detecting genetic linkage using high density marker data

A simple approach to design and analysis of genome-wide linkage scans is described, based on an approximation to the joint distribution of likelihood ratio statistics at a large number of single nucleotide polymorphism (SNP) loci. The approximation is readily calculated and makes it feasible to study the test properties of a range of summary statistics for the entire sequence of point-wise test values. Both the null distribution in the absence of genetic effects and the alternative distribution under various models of single or multi-gene inheritance can readily be simulated. This allows the power of a variety of statistics to be evaluated. The most powerful statistics proved to be the "quantile statistics" defined as quantiles of the set of pointwise test statistics. As a proof of principle study, the method was applied to a small dataset of 40 individuals from 8 families known to be segregating mutations in one or other of the DNA mismatch repair genes, MLH1 and MSH2. These included seven affected sib pairs and 11 discordant sib pairs, who were genotyped at 57,429 autosomal SNPs using the Affymetrix GeneChip Human Mapping 50 k Xba 240 Array. While the sample size was not sufficient to detect the linkage, there was a clear signal at the MLH1 location indicating that relatively small sample sizes would have been adequate to detect linkage to this gene.

構造方程式モデリングにおける適合度指標とモデル改善について : 展望とシミュレーション研究による新たな知見

In this paper we surveyed several fit indices, model modification methods, and other related previous studies in structural equation modeling. Inconsistency in results was detected from the review of the literature. To obtain more reliable knowledge, we performed simulation studies some ten times larger in scale than previous studies to examine the soundness of Golden Rules and the relationship between indices. Also, bias in the distribution of likelihood ratio statistics which depends both sample size and model complexty was indicated as a result of another large simulation study. This could be one of the causes of inconsistency in former studies of fit indices. We also examined fit indices for structural equation modeling with mean structure, multiple groups and categorical indicators. Stepwise estimation methods, model modification methods, and Item Parceling Issues were also reviewed.

Asymptotic Expansion of the Null Distribution of LR Statistic for Multivariate Linear Hypothesis when the Dimension is Large

In this paper we derive an asymptotic expansion of the null distribution of likelihood ratio statistic for multivariate linear hypothesis when the dimension is comparable to the sample size. Our asymptotic approximations are numerically compared with some other approximations including the large sample approximation due to Box [Box, G. E. P. (1949). A general distribution theory for a class of likelihood criteria. Biometrika 36:317–346]. It is shown that the approximations proposed in this paper are good when the dimension is large and close to the sample size, and further our approximations are similar to the Box's approximation even for most of the usual large sample cases.

On the asymptotic distribution of likelihood ratio statistic for testing multisample spherically in the complex case

In this paper, asymptotic expansions of the distribution of the likelihood ratio statistic for testing multisample spherically in the complex case have been derived in the null and nonnull cases when the alternative are close to the null hypothesis. These expansions are obtained in the form of series of beta distributions.

Nonlinear Hypotheses, Inequality Restrictions, and Non-Nested Hypotheses: Exact Simultaneous Tests in Linear Regressions

In the context of the classical linear model, the problem of comparing two arbitrary hypotheses on the regression coefficients is considered. Problems involving nonlinear hypotheses, inequality restrictions, or non-nested hypotheses are included as special cases. Exact bounds on the null distribution of likelihood ratio statistics are derived. In an important special case, a bounds test similar to the Durbin-Watson test is proposed. Multiple testing problems are also studied

Asymptotic nonnull distribution of likelihood ratio statistic for testing homogeneity of complex multivariate gaussian populations

In this paper asymptotic expansion of the nonnull distribution of the likelihood ratio statistic for testing homogeneity of several multivariate complex Gaussian populations has been derived. Nonnull moments are derived under the assumption of equality of covariance matrices. The asymptotic nonnull distribution is derived under the sequence of local alternatives, using logarithmic expansion for gamma function and some results on zonal polynomials.

An Exact Confidence Interval for the Relative Potency Estimated from a Multivariate Bioassay

In their paper on the combination of results from independent univariate bioassays, Armitage, Bennett, and Finney (1976) show how likelihood ratio (LR) tests can be used to test for agreement between assays and to derive a confidence interval for the common relative potency. Both Finney (1978, ?13.3) and V0lund (1980) adapt the strategy of Armitage et al. (1976) to the very similar problem of combining the results from the correlated variables of a single multivariate assay, a problem which had previously been well explored by Rao (1954). More recently, the analysis of a single multivariate assay has been reconsidered by Carter and Hubert (1985) and Laska, Kushner, and Meisner (1985). The main purpose of this paper is to describe a new, distributionally exact, confidence interval for the relative potency in a single multivariate assay. The intervals already described in the literature were derived from the asymptotic distributions of LR statistics or maximum likelihood (ML) estimators. The new exact interval can be derived from a likelihood score (LS) statistic, though it is fundamentally identical to the exact interval for the relative potency from several independent univariate assays derived by the author (Williams, 1978) using an approach suggested by E. J. Williams (1973). Section 2 of this paper establishes some notation and describes the preliminary statistical analysis. Section 3 summarizes the established confidence intervals that derive from asymptotic distributions of LR statistics or ML estimators. An additional LR interval, employing a Bartlett correction factor, is also mentioned. The new exact LS interval is presented in Section 4, and a numerical example is given in Section 5. Finally, Section 6 compares the confidence intervals obtained by these different methods.

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 normal approximation to distributions of likelihood ratio statistics in life testing, reliability and multivariate analysis of variance : N. Singh. Microelectron. Reliab.24 (4) 697 (1984)

Experimental work to define 0.35 pm contact holes to size at i-line exposure (mercury arc at 365 nm) using embedded attenuated phase shift reticles is presented. The performance provided by two types of embedded reticle is compared to that using a ‘leaky chrome’ attenuated phase shift reticle. Results are supported by simulations, using ‘SOLID’, whereby focus/exposure matrices using binary and attenuated phase shift reticles are derived for contemporary and advanced i-line photoresists.

Distribution of Likelihood Ratio Statistic for Testing Equality of Covariance Matrices of Multivariate Gaussian Models

Distribution of Likelihood Ratio Statistic for Testing Equality of Covariance Matrices of Multivariate Gaussian Models

A normal approximation to distributions of likelihood ratio statistics in life testing, reliability and multivariate analysis of variance

A general approach to normal approximation of distributions of some likelihood ratio statistics in life testing, reliability and the multivariate analysis of variance (MANOVA) is discussed. A few special cases are considered for illustration.

Distribution of likelihood ratio statistic for testing equality of covariance matrices of multivariate Gaussian models

SUMMARY In this paper the exact distribution of the likelihood ratio statistic for testing equality of covariance matrices of multivariate Gaussian models has been derived. Percentage points of the test statistic have also been tabulated.

Necessary and Sufficient Conditions for Explicit Solutions in the Multivariate Normal Estimation Problem for Patterned Means and Covariances

The problem of finding maximum likelihood estimates for patterned means and covariance matrices in multivariate analysis is considered. Necessary and sufficient conditions are presented for the existence of explicit solutions and the obtaining of these explicit solutions in one iteration of the scoring equations from any positive definite starting point. Cases in which averaging yields the explicit maximum likelihood estimates are discussed. These results can be applied to the problems of finding maximum likelihood estimates for the parameters in the complete, compound and circular symmetry patterns; mixed models in the analysis of variance; and for finding asymptotic distributions of likelihood ratio statistics when the parameters under the null hypothesis have explicit maximum likelihood estimates.

Inference about the change-point in a sequence of random variables

Abstract : The report discusses the problem of making inference about the point in a sequence of zero-one variables at which the binomial parameter changes. The asymptotic distribution of the maximum likelihood estimate of the change-point is derived in computable form using random walk results. The asymptotic distributions of likelihood ratio statistics are obtained for testing hypotheses about the change-point. Some exact numerical results for these asymptotic distributions are given and their accuracy as finite sample approximations is discussed. (Author)

Inference about the change-point in a sequence of binomial variables

The problem of making inferences about the point in a sequence of zero-one variables at which the binomial parameter changes is considered. The asymptotic distribution of the maximum likelihood estimate of the change-point is derived in computable form using random walk- results. The asymptotic distributions of likelihood ratio statistics for testing hypothesis about the change-point are also obtained. Some exact numerical results for these asymptotic distributions are given and their accuracy as finite sample approximations is discussed.