A parameter value (or rule producing such a value from data) that maximizes the likelihood function of observed data under a specified parametric model; operationally obtained by maximizing the log-likelihood.

Principle

Choose parameters that make the observed data most probable under the model; optimization exploits gradient or second-order information and uses regularity conditions to ensure well-behaved estimators.

Demonstration

For independent Bernoulli trials with unknown success probability θ, the likelihood is θ^{k}(1−θ)^{n−k}; the maximum-likelihood estimator is θ̂ = k/n, the sample proportion of successes.

Misapplication

Using the estimator without checking model identifiability or regularity (e.g., boundary parameters, infinite Fisher information) or applying it blindly when the model is severely misspecified, yielding biased or inconsistent estimates.

Consequence

Under standard regularity, the estimator is consistent and concentrates around the true parameter as data accumulate; it yields point estimates that are invariant to monotone reparameterizations of the likelihood and often underpins asymptotic inference.

Reversal

A Bayesian estimator conditions on a prior and minimizes posterior expected loss; it incorporates prior information and yields a different object (a posterior distribution or posterior-optimal point estimate) rather than pure likelihood maximization.

Boundary

Applies within parametric models where the likelihood is well-defined and optimizable; excludes nonparametric procedures that do not specify a finite-dimensional likelihood family and settings lacking identifiability or sufficient regularity.

Semantic Tension

Method-of-moments estimators match population moments to sample moments and can be simpler to compute, while maximum-likelihood targets likelihood maximization and often attains efficiency under correct specification.

Synthesis

The maximum-likelihood estimator selects parameter values that maximize the probability of the observed data under a parametric model, providing a principled, optimization-based point estimate whose large-sample properties follow from regularity conditions.