Estimation of the Geometric Distribution in the Light of Future Data

Abstract

The maximum likelihood method in view of future data (i.e., the maximization of expected loglikelihood) enables estimates of geometric distribution parameter. This estimator is de ned as an estimator in which n (number of data) in the maximum likelihood estimator is replaced with (n + a0); a0 takes a value such as -1 or -0:5. The value of a0 re ects knowledge about the range where the parameter is to be found. Therefore, when we know that the true parameter of a population lie in a particular range, this method gives a larger expected log-likelihood than the maximum likelihood estimator. Simple simulations show that this new estimator gives anticipated results. The characteristic of the estimator with (n + a0) is similar to that for the mean squared error (MSE), that is, the expectation of the sum of the squared di erence between the true parameter and its estimate. This new methodology in which estimators are modi ed using some constants for yielding better estimators in terms of prediction will contribute to various elds where the number of data is not very large.