Consistency of maximum likelihood estimators in finite mixture models of the union of W-type families

In finite mixture models, maximum likelihood estimators have good properties, such as efficiency, consistency, and asymptotic normality under some uniform integrability assumptions on the mixture and its derivatives up to the third order. These conditions are frequently not easy to check because complex computations on bounding a lot of derivatives are involved. We give results implying these conditions for a new class of families of distributions, W-type families, which make it easier to check the conditions in many cases. Many useful and known families of distributions such as Weibull, Generalized Gamma, Log-gamma, inverse Log-gamma, inverse Gaussian, and all of the exponential families are W-type families. Hence, these results have broad applications.