Maximum Likelihood Estimation of Parameters of a Random Variable Using Monte Carlo Methods

In a parametric estimation framework, this paper proposes different properties for the maximum likelihood estimators of unknown parameters of a given random variable having a known distribution, where different parameter estimation cases are studied. The Refined Descriptive Sampling (RDS) method is chosen to generate samples used for the estimation purpose. Then, we compare the RDS maximum likelihood estimators to their competitors provided by simple random samples with the same size and issued from the same distribution, through their Fisher information. Furthermore, the Maximum likelihood RDS mean is written as a function of its corresponding empirical estimator where the expression can be used to determine the estimator value when a refined descriptive sample is provided. All these results allow us to conclude that the proposed Maximum Likelihood Estimation (MLE) using refined descriptive samples is more efficient than that already obtained from simple random samples, which means that MLE using RDS has advantage in estimating parameters when the samples are not independent and identically distributed. Some Monte Carlo simulations are provided to validate the obtained theoretical results.