The analysis of high-dimensional data—characterized by a significantly greater number of variables than observations—poses essential statistical difficulties, such as multicollinearity, overfitting, sensitivity to outliers, and challenges in model interpretability. This research presents a collection of robust and adaptable statistical techniques aimed at improving inference, prediction, and feature selection within high-dimensional contexts. We formulate hybrid approaches that integrate penalized regression methods (e.g., adaptive Lasso, elastic net) with robust estimation strategies (e.g., M-estimators, Huber loss, and rank-based techniques) to reduce the impact of heavy-tailed distributions and contaminated data points. Additionally, we implement dimension reduction through sparse principal component analysis and robust factor models to maintain signal integrity while eliminating noise. Nonparametric machine learning techniques—such as robust random forests and kernel-based approaches—are modified for the high-dimensional context using stability selection and subsample aggregation. Simulation experiments across various levels of contamination and correlation structures reveal that the proposed methodologies deliver enhanced performance regarding bias, variance, and robustness against outliers. Applications in genomics, image recognition, and financial risk assessment demonstrate the practical applicability and generalizability of the proposed framework. This study adds to the expanding body of literature on high-dimensional statistics by providing computationally efficient, interpretable, and resilient tools that are appropriate for real-world, large-scale data analysis under less-than-ideal conditions.

Keywords: High-Dimensional data, Robust statistics, Penalized regression, Outlier resistance, Regularization methods