A minimum discrepancyestimator in parameter estimation

In statistical estimation theory, a satisfactory esti-mator should be able to embody a large portion of the availableinformation, which may be known a priori or provided by thedata. Hence, the loss of information is minimum when this esti-mator is employed. In recent work, an estimator criterion basedon the discrepancy between the estimator's error covarianceand its information lower bound was proposed. Conceptually,this criterion is a measure of the loss of information carriedby a parameter estimator based on the Bayesian approach.A minimum discrepancy estimator (MDE) was derived underthe linearity assumption. It was, however, pointed out that theminimal information loss could not be guaranteed by the linearversion. Moreover, some good asymptotic properties were notobtainable. Therefore, in this paper, the existence and unique-ness conditions of the general MDE are studied under certainregularity conditions. The MDE can be obtained by solving aFredholm equation of the second kind. Furthermore, it is shownto be consistent and asymptotically efficient. As a result, the MDEis ensured to have the minimum loss of information in linitcsamples and no loss of information when sample size tends toinfinity. 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 suitablychosen, the MDE is superior to the maximum-likelihood estimator(MLE) on the basis of deficiency.