Homogeneity of variances in normal linear regression with a change point

Two methods for testing the equality of variances in straight lines regression with a change point are considered. One is likelihood ratio test and the other is Bayesian confidence interval, based on the highest posterior density for the ratio of variances, using non-informative priors. Methods are applied to the renal transplant data analyzed by Smith and Cook(1980) and Stephens(1994).     

On f-tests and linear hypothesis in a general linear model

A simple method of setting linear hypotheses testable by F-tests in a general linear model when the covariance matrix has a general form and is completely unknown, is provided. With some additional conditions imposed on the covariance matrix, there exist the UMP invariant tests of certain linear hypotheses. We derive them to compare the powers with those of F-tests obtained under no restrictions on the covariance matrix. The results are illustrated in a multiple regression model with some examples.     

Testing variance components in balanced linear growth curve models

It is well known that the testing of zero variance components is a non-standard problem since the null hypothesis is on the boundary of the parameter space. The usual asymptotic chi-square distribution of the likelihood ratio and score statistics under the null does not necessarily hold because of this null hypothesis. To circumvent this difficulty in balanced linear growth curve models, we introduce an appropriate test statistic and suggest a permutation procedure to approximate its finite-sample distribution. The proposed test alleviates the necessity of any distributional assumptions for the random effects and errors and can easily be applied for testing multiple variance components. Our simulation studies show that the proposed test has Type I error rate close to the nominal level. The power of the proposed test is also compared with the likelihood ratio test in the simulations. An application on data from an orthodontic study is presented and discussed.     

A mixture of the manova and gmanova models

A mixture of the MANOVA and GMANOVA models is presented. The expected value of the response matrix in this model is the sum of two matrix components. The first component represents the GMANOVA portion and the second component represents the MANOVA portion. Maximum likelihood estimators are derived for the parameters in this model, and goodness-of-fit tests are constructed for fuller models via the likelihood ration criterion. Finally, likelihood ration tests for general liinear hypotheses are developed and a numerical example is presented.     

Identification of the variance components in the general two-variance linear model

Bayesian analyses frequently employ two-stage hierarchical models involving two-variance parameters: one controlling measurement error and the other controlling the degree of smoothing implied by the model's higher level. These analyses can be hampered by poorly identified variances which may lead to difficulty in computing and in choosing reference priors for these parameters. In this paper, we introduce the class of two-variance hierarchical linear models and characterize the aspects of these models that lead to well-identified or poorly identified variances. These ideas are illustrated with a spatial analysis of a periodontal data set and examined in some generality for specific two-variance models including the conditionally autoregressive (CAR) and one-way random effect models. We also connect this theory with other constrained regression methods and suggest a diagnostic that can be used to search for missing spatially varying fixed effects in the CAR model.     

Estimation of parameters of a polynomial model under intra class correlation structure for incomplete longitudinal data

The problem of analyzing and modeling incomplete longitudinal data arising from clinical and epidemiological studies are discussed, A method for handling arbitrarily missing observations under the intra class correlation structure and a polynomial model is developed. Explicit expressions for likelihood equations and information matrix for a second degree polynomial model are provided. The method is illustrated through an example.     

Improved hypothesis testing in a general multivariate elliptical model

This paper investigates improved testing inferences under a general multivariate elliptical regression model. The model is very flexible in terms of the specification of the mean vector and the dispersion matrix, and of the choice of the error distribution. The error terms are allowed to follow a multivariate distribution in the class of the elliptical distributions, which has the multivariate normal and Student-t distributions as special cases. We obtain Skovgaard's adjusted likelihood ratio (LR) statistics and Barndorff-Nielsen's adjusted signed LR statistics and we compare the methods through simulations. The simulations suggest that the proposed tests display superior finite sample behaviour as compared to the standard tests. Two applications are presented in order to illustrate the methods.     

On some aspects of a general class of Yule distribution and its applications

Abstract

A wide class of Yule distribution is introduced here as a generalization of the extended Yule distribution of Martinez-Rodríguez et al. and the generalized Yule distribution of Mishra and investigate some of its properties. Various methods of estimation are employed for estimating the parameters of the distribution and generalized likelihood ratio test procedures are suggested for testing the significance of the parameters of the distribution. All these procedures are illustrated with the help of real data sets.     

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.     

On the best linear unbiased estimator and the linear sufficiency of a general growth curve model

Growth curve models are used to analyze repeated measures data (longitudinal data), which are functions of time. In this paper, some necessary and sufficient conditions for linear function B₁YB₂ to be the best linear unbiased estimator (BLUE) of estimable functions X₁ΘX₂ (or K₁ΘK₂) under the general growth curve model were established. In addition, the representations of BLUE(K₁ΘK₂) (or BLUE(X₁ΘX₂)) were derived when the conditions are satisfied. Two special notions of linear sufficiency with respect to the general growth curve model are given in the end. The findings of this paper enrich some known results in the literature.     

On the choice of the prior distribution in hypergeometric sampling

Information in a statistical procedure arising from sources other than sampling is called prior information, and its incorporation into the procedure forms the basis of the Bayesian approach to statistics. Under hypergeometric sampling, methodology is developed which quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information embedded in any prior distribution. The most conservative prior distribution from a specified class (each member of which carries the available prior information) is that prior distribution within the class over which the likelihood function has the greatest average domination. Four different families of prior distributions are developed by considering a Bayesian approach to the formation of lots. The most conservative prior distribution from each of the four families of prior distributions is determined and compared for the situation when no prior information is available. The results of the comparison advocate the use of the Polya (beta-binomial) prior distribution in hypergeometric sampling.     

Change-point detection in a shape-restricted regression model

A regression model with a possible structural change and with a small number of measurements is considered. A priori information about the shape of the regression function is used to formulate the model as a linear regression model with inequality constraints and a likelihood ratio test for the presence of a change-point is constructed. The exact null distribution of the test statistic is given. Consistency of the test is proved when the noise level goes to zero. Numerical approximations to the powers against various alternatives are given and compared with the powers of the k-linear-r-ahead recursive residuals tests and CUSUM tests. Performance of four different estimators of the change-point is studied in a Monte Carlo experiment. An application of the procedures to some real data is also presented.     

On the distribution of a second degree polynomial statistic in singular normal var1aies

This note provides an alternative proof of a basic theorem on the distribution of a second degree polynomial statistic in noncentral singular normal variates originally proven by Rayner and Livingstone (1965)     

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.     

Objective Bayesian analysis for a spatial model with nugget effects

The focus of this paper is objective priors for spatially correlated data with nugget effects. In addition to the Jeffreys priors and commonly used reference priors, two types of “exact” reference priors are derived based on improper marginal likelihoods. An “equivalence” theorem is developed in the sense that the expectation of any function of the score functions of the marginal likelihood function can be taken under marginal likelihoods. Interestingly, these two types of reference priors are identical.     

Testing for two states in a hidden Markov model

The authors consider hidden Markov models (HMMs) whose latent process has m ≥ 2 states and whose state‐dependent distributions arise from a general one‐parameter family. They propose a test of the hypothesis m = 2. Their procedure is an extension to HMMs of the modified likelihood ratio statistic proposed by Chen, Chen & Kalbfleisch (2004) for testing two states in a finite mixture. The authors determine the asymptotic distribution of their test under the hypothesis m = 2 and investigate its finite‐sample properties in a simulation study. Their test is based on inference for the marginal mixture distribution of the HMM. In order to illustrate the additional difficulties due to the dependence structure of the HMM, they show how to test general regular hypotheses on the marginal mixture of HMMs via a quasi‐modified likelihood ratio. They also discuss two applications.     

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.     

Testing for normality in linear regression models

The importance of the normal distribution for fitting continuous data is well known. However, in many practical situations data distribution departs from normality. For example, the sample skewness and the sample kurtosis are far away from 0 and 3, respectively, which are nice properties of normal distributions. So, it is important to have formal tests of normality against any alternative. D'Agostino et al. [A suggestion for using powerful and informative tests of normality, Am. Statist. 44 (1990), pp. 316–321] review four procedures Z ²(g ₁), Z ²(g ₂), D and K ² for testing departure from normality. The first two of these procedures are tests of normality against departure due to skewness and kurtosis, respectively. The other two tests are omnibus tests. An alternative to the normal distribution is a class of skew-normal distributions (see [A. Azzalini, A class of distributions which includes the normal ones, Scand. J. Statist. 12 (1985), pp. 171–178]). In this paper, we obtain a score test (W) and a likelihood ratio test (LR) of goodness of fit of the normal regression model against the skew-normal family of regression models. It turns out that the score test is based on the sample skewness and is of very simple form. The performance of these six procedures, in terms of size and power, are compared using simulations. The level properties of the three statistics LR, W and Z ²(g ₁) are similar and close to the nominal level for moderate to large sample sizes. Also, their power properties are similar for small departure from normality due to skewness (γ₁≤0.4). Of these, the score test statistic has a very simple form and computationally much simpler than the other two statistics. The LR statistic, in general, has highest power, although it is computationally much complex as it requires estimates of the parameters under the normal model as well as those under the skew-normal model. So, the score test may be used to test for normality against small departure from normality due to skewness. Otherwise, the likelihood ratio statistic LR should be used as it detects general departure from normality (due to both skewness and kurtosis) with, in general, largest power.