Ratio Test Criterion Research Articles

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.

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).

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).

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.

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.

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.

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.

The distribution of a statistic used for testing sphericity of normal distributions

SUMMARY The joint distribution of the sum of the rth powers (r = 1, ..., p - 1) and the product of all the latent roots of a p x p Wishart matrix is obtained and used to derive the null distributions of the likelihood ratio test criterion and the locally most powerful invariant test criterion for detecting deviations of the variances and covariances of a p-variate normal distribution from proportionality to specified numbers. The sphericity of the distribution is a special case. Explicit expressions are given for the null distributions in the trivariate case. In the bivariate case the two test criteria coincide and their null distribution has been known. The distribution of the locally most powerful test criterion being complicated for values of p larger than three, some approximations are fitted by the method of moments and compared.

On the Distribution of the Sphericity Test Criterion in Classical and Complex Normal Populations Having Unknown Covariance Matrices

Let $\mathbf{x}: p \times 1$ be distributed $N(\mathbf{\mu}, \mathbf{\Sigma})$ where $\mathbf{\mu}$ and $\mathbf{\Sigma}$ are both unknown. Let $\mathbf{S}$ be the sum of product matrix of a sample of size $N$. To test the hypothesis of sphericity, namely, $H_0:\mathbf{\Sigma} = \sigma^2\mathbf{I}_p$, where $\sigma^2 > 0$ is unknown, against $H_1:\mathbf{\Sigma} \neq \sigma^2\mathbf{I}_p$, Mauchly [10] obtained the likelihood ratio test criterion for $H_0$ in the form $W = |\mathbf{S}|/\lbrack(\operatorname{tr} \mathbf{S})/p\rbrack^p$. Thus the criterion $W$ is a power of the ratio of the geometric mean and the arithmetic mean of the roots $\theta_1, \theta_2, \cdots, \theta_p$ of $|\mathbf{S} - \theta\mathbf{I}| = \mathbf{0}$ (see Anderson [1]). In the null case, Machly [10] gave the density of $W$ for $p = 2$ and Consul [3], [4] for any $p$ in terms of Meijer's $G$-function defined in the next section. In this paper we have obtained the general moments of $W$ both in real and complex cases for arbitrary covariance matrices, and also the corresponding distributions of $W$ in terms of the $G$-function.