Shrinkage and penalized estimators in weighted least absolute deviations regression models

Variable Selection and Regularization in Quantile Regression via Minimum Covariance Determinant Based Weights

The importance of variable selection and regularization procedures in multiple regression analysis cannot be overemphasized. These procedures are adversely affected by predictor space data aberrations as well as outliers in the response space. To counter the latter, robust statistical procedures such as quantile regression which generalizes the well-known least absolute deviation procedure to all quantile levels have been proposed in the literature. Quantile regression is robust to response variable outliers but very susceptible to outliers in the predictor space (high leverage points) which may alter the eigen-structure of the predictor matrix. High leverage points that alter the eigen-structure of the predictor matrix by creating or hiding collinearity are referred to as collinearity influential points. In this paper, we suggest generalizing the penalized weighted least absolute deviation to all quantile levels, i.e., to penalized weighted quantile regression using the RIDGE, LASSO, and elastic net penalties as a remedy against collinearity influential points and high leverage points in general. To maintain robustness, we make use of very robust weights based on the computationally intensive high breakdown minimum covariance determinant. Simulations and applications to well-known data sets from the literature show an improvement in variable selection and regularization due to the robust weighting formulation.

Stein-type shrinkage estimators in gamma regression model with application to prostate cancer data

Gamma regression is applied in several areas such as life testing, forecasting cancer incidences, genomics, rainfall prediction, experimental designs, and quality control. Gamma regression models allow for a monotone and no constant hazard in survival models. Owing to the broad applicability of gamma regression, we propose some novel and improved methods to estimate the coefficients of gamma regression model. We combine the unrestricted maximum likelihood (ML) estimators and the estimators that are restricted by linear hypothesis, and we present Stein-type shrinkage estimators (SEs). We then develop an asymptotic theory for SEs and obtain their asymptotic quadratic risks. In addition, we conduct Monte Carlo simulations to study the performance of the estimators in terms of their simulated relative efficiencies. It is evident from our studies that the proposed SEs outperform the usual ML estimators. Furthermore, some tabular and graphical representations are given as proofs of our assertions. This study is finally ended by appraising the performance of our estimators for a real prostate cancer data.

A robust quantitative structure-activity relationship modelling of influenza neuraminidase a/PR/8/34 (H1N1) inhibitors based on the rank-bridge estimator

Linear regression model is frequently encountered in quantitative structure-activity relationship (QSAR) modelling. The traditional estimation of regression model parameters is based on the normal assumption of the response variable (biological activity) and therefore, it is sensitive to outliers or heavy-tailed distributions. Robust penalized regression methods have been given considerable attention because they combine the robust estimation method with penalty terms to perform QSAR parameter estimation and variable selection (descriptor selection) simultaneously. In this paper, based on bridge penalty, a robust QSAR model of the influenza neuraminidase a/PR/8/34 (H1N1) inhibitors is proposed as a resistant method to the existence of outliers or heavy-tailed errors. The basic idea is to combine the rank regression and the bridge penalty together to produce the rank-bridge method. The rank-bridge model is internally and externally validated based on Qint2 , QLGO2 , QBoot2 , MSEtrain , Y-randomization test, Qext2 , MSEtest and the applicability domain (AD). The validation results indicate that the rank-bridge model is robust and not due to chance correlation. In addition, the results indicate that the descriptor selection and prediction performance of the rank-bridge model for training dataset outperforms the other two used modelling methods. Rank-bridge model shows the highest Qint2 , QLGO2 and QBoot2 , and the lowest MSEtrain . For the test dataset, rank-bridge model shows higher external validation value ( Qext2 = 0.824), and lower value of MSEtest compared with the other methods, indicating its higher predictive ability.