A Note on Maximum-Likelihood in the Case of Dependent Random Variables

SUMMARY The validity of certain statistical procedures depends on the “classical” properties of the method of maximum-likelihood, viz., its consistency and the fact that it leads to an estimator whose asymptotic variance can be derived easily from the likelihood function. These properties are well established for the case where successive observations are independent. Here it is shown that their extension to the dependent case involves suitable behaviour of two semi-martingales and a martingale, and general conditions are given which ensure this behaviour. While interest is centred on the theoretical rather than on the practical side of the problem, these conditions may have some practical value.