On information-Estimation relationships over binomial and negative binomial models

In recent years, a number of results have been developed which connect information measures and estimation measures under various models, including, predominantly, Gaussian and Poisson models. More recent results due to Gil Taborda and Pérez-Cruz relate the relative entropy to certain mismatched estimation errors in the context of binomial and negative binomial models, where, unlike in the case of Gaussian and Poisson models, the conditional mean estimates concern models of different orders than those of the original model. In this paper, a different set of results in simple forms are developed for binomial and negative binomial models, where the conditional mean estimates are produced through the original models. The new results are consistent with previous results for Gaussian and Poisson models.