The Research Of Profile Likelihood In Semiparametric Models

Semiparametric model combines the flexibility of nonparametric regression andparsimony of linear regression. Semiparametric models are important and appliedwidely in economic, biological and medical studies. The profle likelihood methodis attractive and has been used extensively in various semiparametric estimationproblems. Severini and Wong showed that the asymptotic variance of the proflelikelihood estimator attains the semiparametric efciency bound. The basic idea isto replace the unknown function by its nonparametric (kernel) estimate. We applythe profle likelihood in three aspects in this paper.First, we propose a penalized profle likelihood for simultaneously selectingsignifcant variables and estimating unknown parameters in the multiple linearregression models. Abrevaya, Hausman and Khan tells that the multiple linearregression model is essentially a semiparametric model, since the density functionof its error term is typically unknown. Our proposed penalized profle likelihoodmethod not only enjoys the oracle properties but also performs extraordinarily welleven when the variance of the error is infnite. Furthermore, our proposed approachperforms better than the adaptive Lasso and the adaptively penalized compositequantile regression approach. Our simulation studies show that the proposed penal-ized profle likelihood method possesses higher probability of correctly selecting theexact model than the adaptively penalized weighted composite quantile regression.Moreover, exact model selection via our proposed approach is robust regardless ofthe error distribution.Second, motivated by the modifed Cholesky decomposition, we propose a pro-fle likelihood approach for estimating the mean and the covariance structures ef-ciently in the longitudinal semiparametric partially linear models. Both theoreticaland empirical results indicate that properly taking into account the within-subject correlation among the responses using our method can substantially improve ef-ciency. Moreover, our proposed estimation procedure is computationally efcient.Third, based on kernel smoothing techniques, we propose two simple esti-mators of the log odds-ratio function for sparse data. Regression analysis of theodds ratios for sparse data has received considerable attention. However, exist-ing works are restricted to the parametric case, and a parametric model may bea misspecifcation, which may lead to biased and inefcient estimators. So, weproposed nonparametric models for the odds ratio for sparse data. In the future,based on the current work, we will model the odds ratio semiparametrically andpropose profle likelihood or spline framework approach for estimating the oddsratio semiparametric models.