Wavelet Methods in Modern Statistics: Theory, Applications, and Recent Advances

Wavelet techniques have emerged as a fundamental instrument in contemporary statistical analysis due to their unique capacity to capture information simultaneously across time (or space) and scale. This dual localization property renders wavelets especially suitable for modeling non-stationary signals, heterogeneous structures, and data affected by noise or outliers. Over the past three decades, wavelet theory has progressed from a predominantly mathematical framework to a versatile statistical methodology with wide-ranging applications in nonparametric regression, time series modeling, survival analysis, statistical process control, multivariate analysis, and Bayesian inference. This review presents a comprehensive synthesis of wavelet-based statistical methodologies, with particular focus on applied and methodological developments in regression modeling, time series analysis, robustness, and quality control. Emphasis is placed on hybrid wavelet–statistical frameworks and recent advances that integrate wavelet shrinkage techniques with Bayesian approaches, robust estimation methods, and machine learning algorithms. By consolidating foundational theory alongside contemporary applications, this review highlights the adaptability, efficiency, and practical relevance of wavelet methods in addressing modern statistical challenges. Keywords:Wavelet methods; Discrete wavelet transform; Wavelet shrinkage; Nonparametric regression; Time series analysis; ARIMA and VAR models; Bayesian wavelet modeling; Robust statistics; Statistical process control; Multivariate wavelets; Outlier detection; Hybrid wavelet frameworks.