Wavelet Methods in Statistics: A Comprehensive Review

Wavelet methods have become a central tool in modern statistical analysis due to their ability to represent data simultaneously in time (or space) and scale. This property makes wavelets particularly effective for analyzing non-stationary signals, heterogeneous data, and processes contaminated by noise or outliers. Over the last three decades, wavelet theory has evolved from a primarily mathematical construction into a powerful statistical framework applied to nonparametric regression, time series analysis, survival analysis, quality control, multivariate modeling, and Bayesian inference. This review provides a comprehensive overview of wavelet-based statistical methods with particular emphasis on applied and methodological contributions in regression modeling, time series analysis, robustness, and quality control. Special attention is given to hybrid wavelet–statistical models and recent developments integrating wavelet shrinkage with Bayesian, robust, and machine-learning-based approaches. The review synthesizes both classical foundations and a wide body of recent applied research, highlighting the flexibility and effectiveness of wavelets in addressing contemporary statistical challenges.