The rapid emergence of massive datasets in various fields poses a serious challenge to traditional statistical methods. Meanwhile, it provides opportunities for researchers to develop novel algorithms. Inspired by the idea of divide-and-conquer, various distributed frameworks for statistical estimation and inference have been proposed. They were developed to deal with large-scale statistical optimization problems. This paper aims to provide a comprehensive review for related literature. It includes parametric models, nonparametric models, and other frequently used models. Their key ideas and theoretical properties are summarized. The trade-off between communication cost and estimate precision together with other concerns is discussed.

The semiparametric partially linear varying coefficient models (SPLVCM) are frequently used in statistical modelling, but most existing methods were built on either the least-square or likelihood-based methods, which are very sensitive to the outliers and their efficiency may be significantly reduced for heavy tail error distribution. This paper proposes a new efficient and robust estimation procedure for the SPLVCM based on modal regression. We establish the asymptotic normality of proposed estimators for both the parametric and nonparametric parts, and show that the estimators achieve the best convergence rate. Moreover, we develop a variable selection procedure to select significant parametric components for the SPLVCM and prove the method possessing the oracle property. We also discuss the bandwidth selection and propose an expectation–maximisation-type algorithm for the proposed estimation procedure. Some simulation results and real data analysis confirm that the newly proposed method works very competitively compared to other existing methods.

The beta regression models are commonly used by practitioners to model variables that assume values in the standard unit interval (0, 1). In this paper, we consider the issue of variable selection for beta regression models with varying dispersion (VBRM), in which both the mean and the dispersion depend upon predictor variables. Based on a penalized likelihood method, the consistency and the oracle property of the penalized estimators are established. Following the coordinate descent algorithm idea of generalized linear models, we develop new variable selection procedure for the VBRM, which can efficiently simultaneously estimate and select important variables in both mean model and dispersion model. Simulation studies and body fat data analysis are presented to illustrate the proposed methods.