The main objective of this article is to investigate the empirical performances of the Bayesian approach in analyzing structural equation models with small sample sizes. The traditional maximum likelihood (ML) is also included for comparison. In the context of a confirmatory factor analysis model and a structural equation model, simulation studies are conducted with the different magnitudes of parameters and sample sizes n = da, where d = 2, 3, 4 and 5, and a is the number of unknown parameters. The performances are evaluated in terms of the goodness-of-fit statistics, and various measures on the accuracy of the estimates. The conclusion is: for data that are normally distributed, the Bayesian approach can be used with small sample sizes, whilst ML cannot.

This paper develops a computationally efficient procedure for analysis of structural equation models with continuous and polytomous variables. A partition maximum likelihood approach is used to obtain the first stage estimates of the thresholds and the polyserial and polychoric correlations in the underlying correlation matrix. Then, based on the joint asymptotic distribution of the first stage estimator and an appropriate weight matrix, a generalized least squares approach is employed to estimate the structural parameters in the correlation structure. Asymptotic properties of the estimators are derived. Some simulation studies are conducted to study the empirical behaviours and robustness of the procedure, and compare it with some existing methods.

Varying coefficient models are useful extensions of the classical linear models. Under the condition that the coefficient functions possess about the same degrees of smoothness, the model can easily be estimated via simple local regression. This leads to the one-step estimation procedure. In this paper, we consider a semivarying coefficient model which is an extension of the varying coefficient model, which is called the semivarying-coefficient model. Procedures for estimation of the linear part and the nonparametric part are developed and their associated statistical properties are studied. The proposed methods are illustrated by some simulation studies and a real example. © 2001 Elsevier Science (USA) AMS 2000 subject classifications: 62G05; 65G08; 62G20.

Cook's [J. Roy. Statist. Soc. Ser. B 48 (1986) 133-169] local influence approach based on normal curvature is an important diagnostic tool for assessing local influence of minor perturbations to a statistical model. However, no rigorous approach has been developed to address two fundamental issues: the selection of an appropriate perturbation and the development of influence measures for objective functions at a point with a nonzero first derivative. The aim of this paper is to develop a differential-geometrical framework of a perturbation model (called the perturbation manifold) and utilize associated metric tensor and affine curvatures to resolve these issues. We will show that the metric tensor of the perturbation manifold provides important information about selecting an appropriate perturbation of a model. Moreover, we will introduce new influence measures that are applicable to objective functions at any point. Examples including linear regression models and linear mixed models are examined to demonstrate the effectiveness of using new influence measures for the identification of influential observations.

perturbation, influence graph, case weight perturbation, Cook's distance, eigen-values and eigenvectors,