Penalized Quantile Regression Methods For High Dimensional Models And Their Applications

Explosion of data is a burning issue in the modern era due to more variables as compared to observations therefore penalization has been employed to resolve that kind of issues. The least shrinkage selection operator (Lasso) penalty and its hybrids have become increasingly useful in this perspective. These methods are usually based on ordinary least squares (OLS), which provides an incomplete picture of response relationships in case of outliers, particularly if the primary interest resides in the tail ends of the distribution of the outcome. The quantile regression (QR) offers an alternative approach in which the influence of independent covariates on the outcome can be specified at any location along the distribution of the outcome.This thesis presents the penalized variable selection through QR with diverging number of parameters. The convergence rate of proposed estimator with smoothly clipped absolute deviation (SCAD) penalty is studied. Moreover, the oracle property with a proper selection of tuning parameter for QR under certain regularity conditions is also established. In addition, rank correlation screening (RCS) method is used to accommodate ultra-high dimensional data settings.Besides this, the Ridge and SCAD penalty functions which are combined to cater the problem of collinearity in high dimensions because such structures can contribute well in prediction accuracy. The consistency, asymptotic normality, and oracle property are established for sparse QR with a diverging number of parameters. The rate of convergence of the combined penalized estimator is also established.Moreover, one step estimator for the ultra-high dimensional quantile linear model with SCAD is also developed in this study. The model selection consistency is verified under certain conditions.Through numerical investigations, it is demonstrated that the proposed methods lead to estimations with competitive or higher efficiency than the standard QR estimation methods. For practical applications, the ordinary least square and Lasso techniques were used for the selection of most significant traits contributing towards seed yield in Mungbean plants with morphological and yield related traits and to develop the prediction model.