A minimum discrepancy estimator in parameter estimation

In statistical estimation theory, a satisfactory estimator should be able to embody a large portion of the available information, which may be known a priori or provided by the data. Hence, the loss of information is minimum when this estimator is employed. In previous work, an estimator criterion based on the discrepancy between the estimator's error covariance and its information lower bound was proposed. Conceptually, this criterion is a measure of the loss of information carried by a parameter estimator based on the Bayesian approach. A minimum discrepancy estimator (MDE) was derived under the linearity assumption. It was, however, pointed out that the minimal information loss could not be guaranteed by the linear version. Moreover, some good asymptotic properties were not obtainable. Therefore, in this paper, the existence and uniqueness conditions of the general MDE are studied under certain regularity conditions. The MDE can be obtained by solving a Fredholm equation of the second kind. Furthermore, it is shown to be consistent and asymptotically efficient. As a result, the MDE is ensured to have the minimum loss of information in finite samples and no loss of information when sample size tends to infinity. Examples indicate that if the prior information is vague, the MDE is superior to the minimum variance estimator (MVE) in terms of information loss. If the prior distribution is suitably chosen, the MDE is superior to the maximum-likelihood estimator (MLE) on the basis of deficiency.