The Cgmanova Model

The Completely General MANOVA (CGMANOVA) model may be used to analyze many complex designs including the GMANOVA, EGMANOVA, MSUR models, the multivariate seemingly unrelated growth curve model, and numerous other designs that do not have closed form solutions using the likelihood ratio method. In this paper we review the theory of the CGMANOVA model and compare 11near model likelihood test results with the Wald statistic.     

A note on the manova model with double linear restrictions

In this note we show how one may construct goodness-of-fit tests to test hypotheses for the restricted MANOVA and GMANOVA models using the multivariate seemingly unrelated regression (MSUR) model.     

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).     

A note on connectedness in fixed effects manova and gmanova models withmissing cells

Murray and Smith (1985) and Hocking (1985) give a generalized definition and test of connectedness in the case of missing cells using the univariate cell-means model with linear restrictions on the cell-means. The test of connectedness is here extended to multivariate fixed effects models, including the usual MANOVA model with linear restrictions, the MANOVA model with double linear restrictions, and the GMANOVA model.     

Consistent estimators of the variance-covariance matrix of the gmanova model with missing data

A common problem in multivariate general linear models is partially missing response data. The simplest method of analysis in the presence of missing data has been to delete all observations on any individual with any missing data(listwise deletion) and utilize a traditional complete data approach. However: this can result in a great loss of information: and perhaps inconsistencies in the estimation of the variance-covariance matrix. In the generalized multivariate analysis of variance(GMANOVA) model with missing data: Kleinbaum(1973) proposed an estimated generalized least squares approach. In order to apply this: however: a consistent estimate of the variance-covariance matrix is needed. Kleinbaum proposed an estimator which is unbiased and consistent: but it does not take advantage of the fact that the underlying model is GMANOVA and not MANOVA. Using the fact that the underlying model is GMANOVA we have constructed four other con¬sistent estimators. A Monte Carlo simulation experiment is conducted tto further examine how well these estimators compare to the estimator proposed by Kleinbaum.     

On Comparisons of Exact Powers of Bivariate GMANOVA Tests

Exact powers of four classical tests in a GMANOVA model are compared numerically when the order of the error sum of square matrix is 2. The four tests are likelihood ratio (=LR), Pillai's V, Hotelling's T ², and Roy's largest root tests. It turns out that for small sizes, there are a few cases in which Rothenberg's condition for the relative magnitude of asymptotic powers of three standard tests does not hold.     

Modified likelihood ratio statistics for inflated beta regressions

The class of inflated beta regression models generalizes that of beta regressions [S.L.P. Ferrari and F. Cribari-Neto, Beta regression for modelling rates and proportions, J. Appl. Stat. 31 (2004), pp. 799–815] by incorporating a discrete component that allows practitioners to model data on rates and proportions with observations that equal an interval limit. For instance, one can model responses that assume values in (0, 1]. The likelihood ratio test tends to be quite oversized (liberal, anticonservative) in inflated beta regressions estimated with a small number of observations. Indeed, our numerical results show that its null rejection rate can be almost twice the nominal level. It is thus important to develop alternative testing strategies. This paper develops small-sample adjustments to the likelihood ratio and signed likelihood ratio test statistics in inflated beta regression models. The adjustments do not require orthogonality between the parameters of interest and the nuisance parameters and are fairly simple since they only require first- and second-order log-likelihood cumulants. Simulation results show that the modified likelihood ratio tests deliver much accurate inference in small samples. An empirical application is presented and discussed.     

Restricted likelihood ratio lack-of-fit tests using mixed spline models

Summary.  Penalized regression spline models afford a simple mixed model representation in which variance components control the degree of non-linearity in the smooth function estimates. This motivates the study of lack-of-fit tests based on the restricted maximum likelihood ratio statistic which tests whether variance components are 0 against the alternative of taking on positive values. For this one-sided testing problem a further complication is that the variance component belongs to the boundary of the parameter space under the null hypothesis. Conditions are obtained on the design of the regression spline models under which asymptotic distribution theory applies, and finite sample approximations to the asymptotic distribution are provided. Test statistics are studied for simple as well as multiple-regression models.     

Bartlett-corrected tests for normal linear models when the error covariance matrix is nonscaiar

This paper provides Bartlett corrections to improve likelihood ratio tests for heteroskedastic normal linear models when the error covariance matrix is nonscaiar and depends on a set of unknown parameters. The Bartlett corrections are simple enough to be used algebraically to obtain several closed-form expressions in special cases. The corrections have also advantages for numerical purposes because they involve only simple operations on matrices and vectors.     

Maximum likelihood estimation for the negative multinomial log-linear model

A log-linear model is defined for multiway contingency tables with negative multinomial frequency counts. The maximum likelihood estimator of the model parameters and the estimator covariance matrix is given. The likelihood ratio test for the general log-linear hypothesis also is presented.     

Statistical Inference and Applications of Mixture of Varying Coefficient Models

In this paper, we consider a new mixture of varying coefficient models, in which each mixture component follows a varying coefficient model and the mixing proportions and dispersion parameters are also allowed to be unknown smooth functions. We systematically study the identifiability, estimation and inference for the new mixture model. The proposed new mixture model is rather general, encompassing many mixture models as its special cases such as mixtures of linear regression models, mixtures of generalized linear models, mixtures of partially linear models and mixtures of generalized additive models, some of which are new mixture models by themselves and have not been investigated before. The new mixture of varying coefficient model is shown to be identifiable under mild conditions. We develop a local likelihood procedure and a modified expectation–maximization algorithm for the estimation of the unknown non‐parametric functions. Asymptotic normality is established for the proposed estimator. A generalized likelihood ratio test is further developed for testing whether some of the unknown functions are constants. We derive the asymptotic distribution of the proposed generalized likelihood ratio test statistics and prove that the Wilks phenomenon holds. The proposed methodology is illustrated by Monte Carlo simulations and an analysis of a CO₂‐GDP data set.     

Some aspects of maximum likelihood estimation of multistage cancer dose-response odels with application

Quantitative cancer dose-response models play an important role in cancer risk assessment. They also play a role in regulatory processes associated with potential occupational or environmental exposures. The multistage model is currently the most widely used cancer dose-response model. This paper describes the construction of the likelihood function in the special case of the multistage cancer dose-response models. The concavity of the likelihood function is also established. A criterion is developed to determine the degree of the polynomial portion of the multistage model. Finally, the restricted and unrestricted maximum likelihood estimators are considered and applied to some experimental data sets.     

Generalized manova double linear hypothesis with double linear restrictions

For 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.     

Inference from PseudoLikelihoods with Plug‐In Estimates

Effective implementation of likelihood inference in models for high‐dimensional data often requires a simplified treatment of nuisance parameters, with these having to be replaced by handy estimates. In addition, the likelihood function may have been simplified by means of a partial specification of the model, as is the case when composite likelihood is used. In such circumstances tests and confidence regions for the parameter of interest may be constructed using Wald type and score type statistics, defined so as to account for nuisance parameter estimation or partial specification of the likelihood. In this paper a general analytical expression for the required asymptotic covariance matrices is derived, and suggestions for obtaining Monte Carlo approximations are presented. The same matrices are involved in a rescaling adjustment of the log likelihood ratio type statistic that we propose. This adjustment restores the usual chi‐squared asymptotic distribution, which is generally invalid after the simplifications considered. The practical implication is that, for a wide variety of likelihoods and nuisance parameter estimates, confidence regions for the parameters of interest are readily computable from the rescaled log likelihood ratio type statistic as well as from the Wald type and score type statistics. Two examples, a measurement error model with full likelihood and a spatial correlation model with pairwise likelihood, illustrate and compare the procedures. Wald type and score type statistics may give rise to confidence regions with unsatisfactory shape in small and moderate samples. In addition to having satisfactory shape, regions based on the rescaled log likelihood ratio type statistic show empirical coverage in reasonable agreement with nominal confidence levels.     

Robust variance estimators for fixed-effect estimates with hierarchical-likelihood

In longitudinal studies, robust sandwich variance estimators are often used, and are especially useful when model assumptions are in doubt. However, the usual sandwich estimator does not allow for models with crossed random effects. The hierarchical likelihood extends the idea of the sandwich estimator to models not currently covered. By simulation studies, we show that the new sandwich estimator is robust against heteroscedastic errors and against misspecification of overdispersion in the y | v component.     

Profile likelihood in directed graphical models from BUGS output

We present a method for using posterior samples produced by the computer program BUGS (Bayesian inference Using Gibbs Sampling) to obtain approximate profile likelihood functions of parameters or functions of parameters in directed graphical models with incomplete data. The method can also be used to approximate integrated likelihood functions. It is easily implemented and it performs a good approximation. The profile likelihood represents an aspect of the parameter uncertainty which does not depend on the specification of prior distributions, and it can be used as a worthwhile supplement to BUGS that enable us to do both Bayesian and likelihood based analyses in directed graphical models.     

Computational algorithms for the factor model

Algorithms for computing the maximum likelihood estimators and the estimated covariance matrix of the estimators of the factor model are derived. The algorithms are particularly suitable for large matrices and for samples that give zero estimates of some error variances. A method of constructing estimators for reduced models is presented. The algorithms can also be used for the multivariate errors-in-variables model with known error covariance matrix.     

The power piecewise exponential model

In this paper an extension of the piecewise exponential distribution based on the distribution of the maximum of a random sample is considered. Properties of its density and hazard function are investigated. Maximum likelihood inference is discussed and the Fisher information matrix is identified. Results of two real data applications are reported, where model fitting is implemented by using maximum likelihood. The applications illustrate the better performance of the new distribution when compared with other recently proposed alternative models.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.     

Asymptotic properties of a quast-maximum likelihood estimator in truncated regression model with serial correlation

Robinson (1982a) presented a general approach to serial correlation in limited dependent variable models and proved the strong consistency and asymptotic normality of the quasi-maximum likelihood estimator (QMLE) for the Tobit model with serial correlation, obtained under the assumption of independent errors. This paper proves the strong consistency and asymptotic normality of the QMLE based on independent errors for the truncated regression model with serial correlation and gives consistent estimators for the limiting covariance matrix of the QMLE.