Minimax estimation with divergence loss function

Abstract

We propose a class of loss functions for parameter estimation obtained by considering the discrepancy between the probability distribution p.d. with the unknown parameter and the p.d. with the parameter replaced by the estimate, the discrepancy being measured by the divergence functional. The minimax estimate of the binomial distribution is determined using the divergence loss function. The minimax estimate is Bayes with respect to a CDF F ∗ which is obtained by maximizing the conditional mutual information function. The CDF F ∗ is a staircased function. The minimax estimate is compared with other estimates, such as the maximum likelihood estimate, the minimax estimate with the quadratic loss function, etc. It is found that the estimate derived here is satisfactory both for large and small samples, unlike the maximum likelihood estimate which is satisfactory only with large samples and the minimax estimate with the quadratic loss function which is satisfactory only with small samples. The theory given here can be easily extended for estimation in multinomial distributions.