SKEWNESS FOR PARAMETERS IN GENERALIZED LINEAR MODELS

In this paper we give an asymptotic formula of order n ^?1/2, where n is the sample size, for the skewness of the distribution of the maximum likelihood estimates of the linear parameters in generalized linear models. The formula is given in matrix notation and is very suitable for computer implementation. Several special cases are discussed. We also give asymptotic formulae for the skewness of the distribution of the maximum likelihood estimates of the dispersion and precision parameters.     

Reduced-rank estimation of the difference between two covariance matrices

We consider m×m$m\times m$ covariance matrices, _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$, which satisfy _Σ2-_Σ1=Δ${\Sigma }_2-{\Sigma }_1=\Delta$, where Δ$\Delta$ has a specified rank. Maximum likelihood estimators of _Σ1${\Sigma }_1$ and _Σ2${\Sigma }_2$ are obtained when sample covariance matrices having Wishart distributions are available and rank(Δ)$\text{rank}(\Delta )$ is known. The likelihood ratio statistic for a test about the value of rank(Δ)$\text{rank}(\Delta )$ is also given and some properties of its null distribution are obtained. The methods developed in this paper are illustrated through an example.     

Testing the Existence of Change-Point in NHPP Software Reliability Models

In the software testing process, the nature of the failure data is affected by many factors, such as the testing environment, testing strategy, and resource allocation. These factors are unlikely to all be kept stable during the entire process of software testing. As a result, the statistical structure of the failure data is likely to experience major changes. Recently, some useful non homogeneous Poisson process (NHPP) models with change-point are proposed. However, in many realistic situations, whether a change-point exists is unknown. Furthermore, some real data seem to have two or more change-points. In this article we propose test statistics to test the existence of change-point(s). The experimental results of real data show that our tests perform well.     

BARTLETT CORRECTED LIKELIHOOD RATIO TESTS IN LOCATION-SCALE NONLINEAR MODELS

The paper derives Bartlett corrections for improving the chisquare approximation to the likelihood ratio statistics in a class of location-scale family of distributions, which encompasses the elliptical family of distributions and also asymmetric distributions such as the extreme value distributions. We present, in matrix notation, a Bartlett corrected likelihood ratio statistic for testing that a subset of the nonlinear regression coefficients in this class of models equals a given vector of constants. The formulae derived are simple enough to be used analytically to obtain several Bartlett corrections in a variety of important models. We show that these formulae generalize a number of previously published results. We also present simulation results comparing the sizes and powers of the usual likelihood ratio tests and their Bartlett corrected versions when the scale parameter is considered known and when this parameter is uncorrectly specified.     

Tests Concerning Equicorrelation Matrices with Grouped Normal Data

ABSTRACT

This article considers three practical hypotheses involving the equicorrelation matrix for grouped normal data. We obtain statistics and computing formulae for common test procedures such as the score test and the likelihood ratio test. In addition, statistics and computing formulae are obtained for various small sample procedures as proposed in Skovgaard (2001 Skovgaard , I. M. . ( 2001 ). Likelihood asymptotics . Scand. J. Statist. . 28 : 3 – 32 . [CROSSREF] [Crossref], [Web of Science ®] , [Google Scholar]). The properties of the tests for each of the three hypotheses are compared using Monte Carlo simulations.     

Accent on Teaching Materials

Estimation of covariance components in the multivariate random-effect model with nested covariance structure is discussed. There are two covariance matrices to be estimated, namely, the between-group and the within-group covariance matrices. These two covariance matrices are most often estimated by forming a multivariate analysis of variance and equating mean square matrices to their expectations. Such a procedure involves taking the difference between the between-group mean square and the within-group mean square matrices, and often produces an estimated between-group covariance matrix that is not nonnegative definite. We present estimators of the two covariance matrices that are always proper covariance matrices. The estimators are the restricted maximum likelihood estimators if the random effects are normally distributed. The estimation procedure is extended to more complicated models, including the twofold nested and the mixed-effect models. A numerical example is presented to illustrate the use of the estimation procedure.     

Separation and Lack of Separation of Subpopulation in the Mixed Distributions

This is a comparative study between the estimates of parameters of mixed distributions in the case of the possibility of separating the units of subpopulation or the absence of that possibility under the progressive type I censored test data. An iterative procedure is developed and tested numerically to obtain new estimators and their variance–covariance matrix. Finally, we will use the exact distribution of the maximum likelihood estimators as well as its asymptotic distribution and the parametric bootstrap method; then, we will discuss the construction of confidence intervals for the mean parameter and their performance is assessed through Monte Carlo simulations.     

On the wald,lagrangian multiplier and likelihood ratio tests when the information matrix is singular

Summary Modified formulas for the Wald and Lagrangian multiplier statistics are introduced and considered together with the likelihood ratio statistics for testing a typical null hypothesisH ₀ stated in terms of equality constraints. It is demonstrated, subject to known standard regularity conditions, that each of these statistics and the known Wald statistic has the asymptotic chi-square distribution with degrees of freedom equal to the number of equality constraints specified byH ₀ whether the information matrix is singular or nonsingular. The results of this paper include a generalization of the results of Sively (1959) concerning the equivalence of the Wald, Lagrange multiplier and likelihood ratio tests to the case of singular information matrices.     

A note on sequential ML estimates and their asymptotic covariances

A marginal and sequential maximum likelihood estimation method is described which can be used instead of full information maximum likelihood estimation if the latter method is unfeasible. It is shown that the sequential procedure yields strongly consistent and asymptotically normal estimates under relatively general regularity conditions. It is shown that the covariance matrix of the sequential ML estimator does not coincide with the inverse of the Fisher information matrix. Hence, the corrected covariance matrix is derived. The application of the sequential procedure to the multivariate probit model with dichotomous, ordered categorical, single-sided censored and double-sided censored endogenous variables is included. This research was partially supported by a dissertation grant of theStudienstiftung des Deutschen Volkes. Comments and suggestions on earlier drafts by Gerhard Arminger, Giorgio Calzolari, Bernd Kortzen and an anonymous referee are gratefully acknowledged.     

Empirical likelihood method for the multivariate accelerated failure time models

In applications, multivariate failure time data appears when each study subject may potentially experience several types of failures or recurrences of a certain phenomenon, or failure times may be clustered. Three types of marginal accelerated failure time models dealing with multiple events data, recurrent events data and clustered events data are considered. We propose a unified empirical likelihood inferential procedure for the three types of models based on rank estimation method. The resulting log-empirical likelihood ratios are shown to possess chi-squared limiting distributions. The properties can be applied to do tests and construct confidence regions without the need to solve the rank estimating equations nor to estimate the limiting variance-covariance matrices. The related computation is easy to implement. The proposed method is illustrated by extensive simulation studies and a real example.     

Penalized and Shrinkage Estimation in the Cox Proportional Hazards Model

This article considers the shrinkage estimation procedure in the Cox's proportional hazards regression model when it is suspected that some of the parameters may be restricted to a subspace. We have developed the statistical properties of the shrinkage estimators including asymptotic distributional biases and risks. The shrinkage estimators have much higher relative efficiency than the classical estimator, furthermore, we consider two penalty estimators—the LASSO and adaptive LASSO—and compare their relative performance with that of the shrinkage estimators numerically. A Monte Carlo simulation experiment is conducted for different combinations of irrelevant predictors and the performance of each estimator is evaluated in terms of simulated mean squared error. Simulation study shows that the shrinkage estimators are comparable to the penalty estimators when the number of irrelevant predictors in the model is relatively large. The shrinkage and penalty methods are applied to two real data sets to illustrate the usefulness of the procedures in practice.     

Statistical Analysis for the Inverse Trinomial Distribution

This article considers parameter estimation, goodness of fit, likelihood ratio and score tests, and model selection by Akaike information criterion for the inverse trinomial (IT) distribution, a classical one-dimensional random walk distribution. The IT distribution has a cubic variance function of the mean and is a generalization of the negative binomial distribution. Basic distributional properties and expressions for the probability mass function, recurrence formula, moments, and score functions are also presented.     

Improved likelihood-based inference for the stationary AR(2) model

An improved likelihood-based method based on Fraser et al. (1999) is proposed in this paper to test the significance of the second lag of the stationary AR(2) model. Compared with the test proposed by Fan and Yao (2003) and the signed log-likelihood ratio test, the proposed method has remarkable accuracy. Simulation studies are performed to illustrate the accuracy of the proposed method. Application of the proposed method on historical data is presented to demonstrate the implementation of this method. Furthermore, the method can be extended to the general AR(p) model.     

Simulations on the Jelinski-Moranda model of software reliability; application of some parametric bootstrap methods

In software reliability theory many different models have been proposed and investigated. some of these models intuitively match reality better than others. The properties of certain statistical estimation procedures in connection with these models are also model-dependent. In this paper we investigate how well the maximum likelihood estimation procedure and the parametric bootstrap behave in the case of the very well-known software reliability model suggested by Jelinski and Moranda (1972). For this study we will make use of simulated data.