Bayesian Empirical Likelihood Estimation of Quantile Structural Equation Models

Structural equation model(SEM) is a multivariate analysis tool that has been widely applied to many fields such as biomedical and social sciences. In the traditional SEM, it is often assumed that random errors and explanatory latent variables follow the normal distribution, and the effect of explanatory latent variables on outcomes can be formulated by a mean regression-type structural equation. But this SEM may be inappropriate in some cases where random errors or latent variables are highly nonnormal. The authors develop a new SEM, called as quantile SEM(QSEM), by allowing for a quantile regression-type structural equation and without distribution assumption of random errors and latent variables. A Bayesian empirical likelihood(BEL) method is developed to simultaneously estimate parameters and latent variables based on the estimating equation method. A hybrid algorithm combining the Gibbs sampler and Metropolis-Hastings algorithm is presented to sample observations required for statistical inference. Latent variables are imputed by the estimated density function and the linear interpolation method. A simulation study and an example are presented to investigate the performance of the proposed methodologies.

Adjusted Empirical Likelihood Estimation of Distribution Function and Quantile with Nonignorable Missing Data

This paper considers the estimation problem of distribution functions and quantiles with nonignorable missing response data. Three approaches are developed to estimate distribution functions and quantiles, i.e., the Horvtiz-Thompson-type method, regression imputation method and augmented inverse probability weighted approach. The propensity score is specified by a semiparametric expo- nential tilting model. To estimate the tilting parameter in the propensity score, the authors propose an adjusted empirical likelihood method to deal with the over-identified system. Under some regular conditions, the authors investigate the asymptotic properties of the proposed three estimators for distri- bution functions and quantiles, and find that these estimators have the same asymptotic variance. The jackknife method is employed to consistently estimate the asymptotic variances. Simulation studies are conducted to investigate the finite sample performance of the proposed methodologies.

Consistency and asymptotic normality of profilekernel and backfitting estimators in semiparametric reproductive dispersion nonlinear models

Semiparametric reproductive dispersion nonlinear model (SRDNM) is an extension of nonlinear reproductive dispersion models and semiparametric nonlinear regression models, and includes semiparametric nonlinear model and semiparametric generalized linear model as its special cases. Based on the local kernel estimate of nonparametric component, profile-kernel and backfitting estimators of parameters of interest are proposed in SRDNM, and theoretical comparison of both estimators is also investigated in this paper. Under some regularity conditions, strong consistency and asymptotic normality of two estimators are proved. It is shown that the backfitting method produces a larger asymptotic variance than that for the profile-kernel method. A simulation study and a real example are used to illustrate the proposed methodologies.