Semiparametric mixture regression models have recently been proposed to model competing risks data in survival analysis. In particular, Ng and McLachlan (Stat Med 22:1097–1111, 2003) and Escarela and Bowater (Commun Stat Theory Methods 37:277–293, 2008) have investigated the computational issues associated with the nonparametric maximum likelihood estimation method in a multinomial logistic/proportional hazards mixture model. In this work, we rigorously establish the existence, consistency, and asymptotic normality of the resulting nonparametric maximum likelihood estimators. We also propose consistent variance estimators for both the finite and infinite dimensional parameters in this model.