Abstract

This paper formulates the nonparametric maximum likelihood es-timation of probability measures and generalizes the consistency result on the maximum likelihood estimator (MLE). We drop the indepen-dence assumption on the underlying stochastic process and replace it with the assumption that the stochastic process is stationary and ergodic. The present proof employs Birkhoff’s ergodic theorem and the martingale convergence theorem. The main result is applied to the parametric and nonparametric maximum likelihood estimation of density functions.