Applications of Fixed Point Theory in Statistical Estimation and Probabilistic Analysis

Abstract

This paper explores the applications of fixed point theory in statistical estimation and probabilistic analysis, highlighting its role in iterative methods and equilibrium modeling. Fixed points are crucial in understanding the convergence of statistical algorithms like the Expectation-Maximization (EM) method, iterative least squares, and optimization techniques. Additionally, their relevance in probabilistic contexts, such as steady- state distributions in Markov chains and convergence in probabilistic metric spaces, is examined.

Real-world examples, including rainfall data, financial time series, and epidemiological trends, illustrate

the practical utility of fixed point methods in parameter estimation and model fitting. By integrating fixed point theory with statistical and probabilistic approaches, this work provides a unified framework to address complex inference problems, offering insights into algorithmic convergence and robust analysis. Potential avenues for further research in this interdisciplinary domain are also discussed.