Likelihood Ratio Test Criterion Research Articles

Nonlinear Mixed Effects Modeling for Slash Pine Dominant Height Growth Following Intensive Silvicultural Treatments

Abstract A modified Richard's growth model with nonlinear mixed effects is developed for modeling slash pine (Pinus elliottii Engelm.) dominant height growth in conjunction with different silvicultural treatments. All three parameters in the model turn out to have both fixed and random individual plot or silvicultural treatments effects. Moving average correlation with 2° was identified as within-plot error structure. The advantages of the mixed effects model in prediction for new responses are demonstrated in detail by formulations and examples. The modified Richards model has a form that combines dominant height growth and site index into one model form, so the incompatibility between height growth and site index model can be avoided. The general methodologies of nonlinear mixed effects model building, such as which parameters in the model should be considered to be random and which should be purely fixed, how to determine appropriate within-plot variance covariance structure, and how to specify between-plot variation via appropriate covariate modeling, are addressed in detail. Likelihood ratio test and Akaike information criterion (AIC) are used in model performance evaluation. Some useful graphical model diagnosis tools are also presented. FOR. SCI. 47(3):287–300.

Elliptically contoured model and factorization of Wilks' $ \Lambda$: noncentral case

Kshirsagar in a series of papers, see e.g., Kshirsagar (1964, 1971), McHenry and Kshirsagar (1977), factorizes Wilks' $ \Lambda$ into a number of factors and finds the independent central multivariate beta densities of these factors. These factors are the Wilks' likelihood ratio test criteria for testing goodness of fit of certain canonical variables. Essentially the factors of Wilks' $ \Lambda$ are the factors of the determinants of certain multivariate beta distributed matrices. The Bartlett decompositions of the underlying multivariate beta distribution into independent factors, determine the distributions of these factors. The present paper generalizes Kshirsagar's (1971) normal central distribution theory to elliptically contoured model noncentral distribution theory, showing that Kshirsagar's (1971) nonnull normal theory is nonnull robust for elliptically contoured model.

Elliptically contoured model and factorization of wllks1 A

In a series of papers, Kshirsagar (1964, 1971) and McHenry and Kshirsagar (1977), factorize Wilks' A into a number of factors and find the independent null multivariate beta densities of these factors. These factors are the likelihood ratio test criteria for testing the goodness of fit of certain assigned discriminant functions or canonical variables either in the space of independent or dependent variables. Essentially the factors of Wilks' A are the factors of certain multivariate beta distributed matrix or its determinant. The Bartlett decomposition of the underlying multivariate beta distribution into independent factors determines the distribution of these factors. The present paper generalizes Kshirsagar's (1971) normal theory to the elliptically contoured model, and shows that his results are null robust for the elliptically contoured model.

Determination of the number of factors in the presence of correlated error terms

The problem of the determination of the number of latent factors is analysed within a generalized factor model, where nonzero elements out of the diagonal of the variance-covariance matrix of error terms are admitted. In this framework, it is proved that classical methods for the selection of the factor model such as likelihood ratio test and information criteria are biased and systematically overestimate the number of factors. A Monte Carlo experiment shows the different behavior of classical methods and of two alternative criteria which have been developed for the generalized model.

On wilks' a factorization noncentral distribution theory

Kshirsagar in a series of papers, see e.g., Kshirsagar (1964), (1971), (1972), McHenry and Kshirsagar (1977), factorizes Wilks A into a number of factors and finds the independent central multivariate beta densities of these factors. These factors are the likelihood ratio test criteria for testing the goodness of fit of certain assigned discriminant functions or canonical variables either in the space of dependent or independent variables. Essentially the factors of Wilks A are the factors of a certain multivariate beta distributed matrix or its determinant. The Bartlett decomposition of the underlying multivariate beta distribution into independent factors determines the distributions of these factors. In the noncentral case the distributions of these factors are not independent. However, the marginal noncentral distribution of each factor can be obtained. The present paper studies a typical paper of Kshirsagar's (1971), and generalizes Kshirsagar's Wilks A factorization central distribution theory to th...

Testing of homogeneity of diagonal blocks with blockwise independence

In this paper the likelihood ratio test criterion for testing the equality of block covariance matrices for the multivariate multisamplesphericity model has been derived. The distribution of the test statistic, its moments and percentage points are also given.

Testing for the mean of the inverse gaussian distribution and adaptive estimation of parameters

This paper provides the modified likelihood ratio criterion for testing whether the mean of the inverse Gaussian distribution can be set to unity giving rise to Standard form of the Wald distribution. Estimates of probability of correct selection has been obtained on the basis of a Monte Carlo study of 1,000 samples. Finally a set of adaptive estimators for the parameters are proposed and studied on the basis of data generated from the two distributions.

Generalised Likelihood Ratio Tests for Two-Parameter Exponentals Under Type I Censoring

The paper considers generalized likelihood ratio tests for the equality of the location, parameters and⁄or the failure rates of k independent location and scale parameter exponentials when observations are censored in time. For testing the equality of the failure rates, asymptotic null distributions of the generalized likelihood ratio test (GLRT) criteria are obtained. Also, asymptotic distributions of GLRT criteria are obtained under local alternatives. For testing the equality of the location parameters, asymptotic null distributions of the GLRT criteria are obtained.

A Test for Equality of Exponential Distributions Based on Type-II Censored Samples

Moments of the likelihood ratio test (LRT) criterion for testing the equality of several exponential distributions based on type-II censored samples of unequal size are obtained. They are used to obtain the distribution in a computational form for the first time. A table of selected significe points of LRT is presented.

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).

Extensions of Wilks' integral equations and distributions of test statistics

Wilks [26] introduced two integral equations in connection with distribution problems in statistics. He called them Type A and Type B equations. Tretter and Walster ([22], [24]) solved the Type B equation and obtained the null and non-null distributions of the likelihood ratio criterion for testing linear hypotheses in the multinormal case. In this article we present several types of solutions of these equations along with new equations called Types C, D, E and F with their solutions. These include the integral equations satisfied by the density of a random variable which is (a) product of independent real gamma variates; (b) products of independent real beta variates; (c) ratio of products of independent beta and gamma variates; (d) arbitrary powers of products of gamma and beta variates; (e) arbitrary powers of products and ratios of beta and gamma variates, and more general cases.

CHANGE‐OVER DESIGNS WITH ERRORS FOLLOWING A FIRST ORDER AUTOREGRESSIVE PROCESS

SummaryThis paper deals with the problem of analysing the change over design in the context of a first order autoregressive process for the error terms. The method of maximum likelihood has been adopted for estimating treatment effects. The conditions derived for obtaining a balanced change over design show that a change over design balanced in the absence of autocorrelation is not necessarily balanced in the presence of autocorrelation. Also, it is observed that the autocorrelation co‐efficient and the treatment effect when p≠0 can be tested as usual with the likelihood ratio test criterion.

Some Results for the Normal Manova Linear Functional Relationships Model.

Given the normal multivariate linear regression model Y = BX + E, with B subjected to the linear restrictions H BJ = W A, J known, W and H unknown, A known, the maximum likelihood estimates of H, B, W, are obtained. A likelihood ratio test criterion for testing H = H0 , W = W0 is provided. The results are extended to the GMANOVA model. All results are obtained in terms of the original variates directly, unlike Healy (1980) who obtains the results for the MANOVA model in terms of the canonical transformations of the original variates.

Manova double linear hypothesis with double linear restrictions

Given the usual normal multivariate linear regression model Y = BX + E, with B subjected to double linear restrictions GBF' = T, a likelihood ratio test criterion for testing the composite linear null hypothesis HBJ' = U; G, F, T, H, J, U specified, is provided. The applications of such tests are discussed by Timm (1980).

On the exact distribution of the likelihood ratio criterion for testing

The exact null distribution of the likelihood ratio criter- 2 ion for testing the hypothesis H: y = y~; z = a I, a unknown and UQ a given known vector against the alternative A =f H in a p-vari- ate normal population N (y,z) has been derived in the form of Meijer's G-function using mellin integral transform and also in a chisquare series form. Asymptotic behavior of the distribution of -2 log L has also been discussed. Percentage points for p=2(l)10for various level of significance and various degrees of freedom have been computed, but only selected tables have been presented in this paper.

Generalized manova double linear hypothesis with double linear restrictions

AbstractFor the generalized MANOVA model of Potthoff and Roy [7], Gleser and Olkin [3] give a likelihood ratio test criterion for testing double linear parametric functions of the regression parameters. Their theory is extended in this paper to the testing of double linear parametric functions with double linear restrictions on the parameters. The theory is presented in terms of the original variates unlike Gleser and Olkin [3] who resort to canonical transformations of the original variates.

Distribution of the likelihood ratio criterion for testing Σ = Σ 0, μ = μ 0

The exact null distribution of the likelihood ratio criterion for testing H 0: Σ = Σ 0 and μ = μ 0 against alternatives H 1: Σ ≠ Σ 0 or μ ≠ μ 0 in N p(μ, Σ) has been obtained as (a) a chi-square series and (b) a beta series. Percentage points have been tabulated for p = 2(1) 6, α = .005, .01, .025, .05, .1, and .25 and various values of sample size N.

Likelihood analysis of recombinational disequilibrium in multiple-locus gametic frequencies.

Likelihood methods are developed for the estimation and testing of multiple-locus gametic disequilibria, using log-linear models of parametric effects. The estimates of disquilibrium are related to Kimura's Z-measure, and may be extended to multiple alleles and multiple loci. Likelihood ratio test criteria are constructed, which are asymptotically distributed as chi(2). The analysis is partitioned into various components corresponding to two-locus, residual three-locus, and higher order disequilibria. A four-locus example from Hordeum vulgare L. is utilized to illustrate the analysis. Most of the multiple-locus disequilibrium is accounted for by two-locus effects, and closely linked loci show considerably more disequilibrium than unlinked loci. It is shown that all possible pairwise comparisons are not statistically independent.

Maximum likelihood analysis of population differences in allelic frequencies.

Statistical techniques are presented for the analysis of geographic variation in allelic frequencies. Likelihood ratio test criteria are derived from a multinominal sampling distribution, and are used to answer three questions. (1) Are there geographic differences in allelic frequencies? (2) Are population differences in allelic frequencies associated with environmental differences? (3) Is there any residual "lack of fit" variation among populations, after accounting for that variation associated with environmental differences? The two- and three-allele cases are explicitly treated, and the extension to more alleles is indicated.

Order of Dependence in a Stationary Normally Distributed Two-Way Series

Order of dependence is defined in a normally distributed two-way series. Under certain stationarity and symmetry conditions it is shown that when the extents of the series in both directions are large in comparison with the order of dependence, the joint density function reduces to a simple form with a small number of parameters after some adjustments. In this form a central role is played by the Kronecker products of some matrices having common eigenvectors. Maximum likelihood estimates of the parameters and likelihood ratio test criteria for certain hypotheses on order of dependence are derived.