This article presents a comprehensive overview of Maximum Likelihood Estimation (MLE), tracing its origins from Ronald A. Fisher’s pioneering work to its modern-day applications across scientific disciplines. MLE is introduced as a foundational statistical method for parameter estimation, valued for its efficiency, consistency, and asymptotic normality under regular conditions.
The article outlines the core principles of MLE, beginning with the formulation of the likelihood and log-likelihood functions. It explains how maximizing these functions yields estimators that best align with observed data. Detailed derivations are provided for normal and binomial distributions, and a full table summarizes MLE formulas across numerous statistical distributions, showcasing MLE’s flexibility.
Practical aspects are emphasized, including analytical and numerical optimization techniques (e.g., Newton-Raphson, EM algorithm), computational challenges, and software tools in R, Python, and MATLAB. MLE’s role in model selection through Likelihood Ratio Tests, AIC, and BIC is explained, with examples from regression, PCA, clustering, and time series analysis.
Applications of MLE are illustrated in economics, biology, engineering, machine learning, survival analysis, and natural language processing. It is shown to be essential for tasks ranging from phylogenetics to text classification.
The article also discusses limitations—such as sensitivity to model specification and outliers—and introduces Stein’s phenomenon, raising questions about MLE’s optimality in high-dimensional settings. It concludes with emerging directions in robust MLE, big data analytics, and AI integration.
For derivations, use-case tables, and detailed applications, see the full encyclopedia entry in the International Encyclopedia of Statistical Science.