This paper explores the properties of estimators obtained by maximizing a likelihood function associated with a family of probability distributions that may not include the true distribution. The authors focus on pseudo maximum likelihood methods, determining the families of pseudo-likelihood functions that provide consistent and asymptotically normal estimators of parameters in the true distribution. Specifically, they show that exponential families of type I provide such estimators for first-order moments, and that any family with this property is necessarily an exponential family of type I. They also propose a generalization of these methods for estimating second-order moments and define exponential families of type II, which are the only ones providing consistent estimators for both first and second moments. The paper includes proofs and examples to support these findings, and discusses the application of the scoring method to exponential families of type I.This paper explores the properties of estimators obtained by maximizing a likelihood function associated with a family of probability distributions that may not include the true distribution. The authors focus on pseudo maximum likelihood methods, determining the families of pseudo-likelihood functions that provide consistent and asymptotically normal estimators of parameters in the true distribution. Specifically, they show that exponential families of type I provide such estimators for first-order moments, and that any family with this property is necessarily an exponential family of type I. They also propose a generalization of these methods for estimating second-order moments and define exponential families of type II, which are the only ones providing consistent estimators for both first and second moments. The paper includes proofs and examples to support these findings, and discusses the application of the scoring method to exponential families of type I.