MaximumLikelihoodFrameworks

Maximum likelihood estimation is a method for estimating the parameters of a statistical model by choosing the values that maximize the likelihood of the observed data under the model. The goal is to select parameter values that make the observed data most probable, given the assumed distribution.

Consider data x1, ..., xn drawn independently from a distribution with density f(x; theta). The likelihood is L(theta) = ∏ f(xi; theta) and the log-likelihood is l(theta) = ∑ log f(xi; theta). The maximum likelihood estimator (MLE) is theta_hat = argmax_theta L(theta) = argmax_theta l(theta). In many cases, the MLE is found by solving score equations, while in others numerical optimization is required.

Maximization is often performed on the log-likelihood rather than the likelihood because the log function is monotone, turning products into sums and simplifying calculations. When closed-form solutions do not exist, algorithms such as gradient ascent, Newton-Raphson, or specialized methods like the expectation-maximization (EM) algorithm are employed, especially in models with latent variables.

Key properties arise under regularity conditions: MLEs are consistent (converge to the true parameter as sample size grows), asymptotically normal, and efficient in large samples (attaining the Cramér–Rao lower bound). They are invariant under smooth reparametrizations: if theta_hat maximizes l, then g(theta_hat) maximizes the log-likelihood for the transformed parameter g(theta).

Example: estimating the mean mu of a normal distribution with known variance sigma^2 yields mu_hat = x_bar. In practice, MLEs may require numerical methods, and the presence of nuisance parameters can lead to using profile likelihoods.

Limitations include sensitivity to model misspecification, potential small-sample bias, and non-uniqueness or boundary issues in some problems; prior information is not incorporated unless a Bayesian framework is used.