We study the sufficient conditions for the consistency of the maximum likelihood estimator (MLE) under a certain parametric distribution families where E(log f(X)) may not exist or may not be finite, and f is the density of the random variable $X$. The existing sufficient conditions in the classical textbooks (e.g., Ferguson (1996), Casella and Berger (2001), among others) for the consistency of MLEs are not applicable to these families. We establish another set of sufficient conditions and apply them to these class of parametric families.