Glossary

Maximum Likelihood Estimation (MLE)

Definition

Maximum likelihood estimation is an estimation method that selects parameter values maximising the likelihood function — the joint probability of the observed data given the model. MLE is the dominant estimation paradigm in modern statistics: it underlies logistic regression, GARCH models, structural equation models, and many others. Under regularity conditions, MLEs are consistent, asymptotically efficient, and asymptotically normal.

Why It Matters

MLE provides a unified, principled approach to parameter estimation across an enormous range of models. Rather than deriving ad hoc estimators for each application, researchers specify a likelihood function and let optimisation find the best-fitting parameters. The theoretical guarantees of MLE — consistency, efficiency, and asymptotic normality — mean that with sufficient data, MLE produces the best possible estimates. However, MLE can be sensitive to model misspecification and may produce biased estimates in small samples, making model diagnostics essential.

Example

In a logistic regression predicting loan default, the likelihood function represents the probability of observing the pattern of defaults and non-defaults in the data for any given set of coefficients. The MLE algorithm searches for the coefficient values that maximise this likelihood. If the model includes credit score and debt-to-income ratio, the resulting MLE coefficients tell us how each predictor shifts the log-odds of default.

Related Terms

Software Notes

SPSS: MLE is used automatically in Analyze > Regression > Binary Logistic, Analyze > Generalized Linear Models, and other maximum-likelihood-based procedures
R: glm() for MLE of GLMs; optim() for custom likelihood functions; mle2() from the bbmle package for flexible MLE
Stata: Most estimation commands use MLE by default (e.g., logit, probit, regress with MLE option); custom likelihoods via ml commands