On Some Properties of $$\alpha $$ α -Mixtures

The $$\alpha $$ α -mixtures is a new, flexible family of distributions that includes many mixture models as special cases. This paper is mainly focused on relevant stochastic comparisons and ageing properties of $$\alpha $$ α -mixtures of survival functions. In particular, we prove that ageing properties of $$\alpha $$ α -mixtures for additive and multiplicative hazards models depend on the properties of the baseline failure rate functions and the corresponding conditional moments of mixing distributions. Partial orderings of the finite $$\alpha $$ α -mixtures in the sense of the usual stochastic order and the hazard rate order are discussed. Finally, we extend some results on the shape of the mixture failure rate obtained in the literature for usual mixtures to the case of $$\alpha $$ α -mixtures.