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Generalizing lifetime distributions is always precious for applied statisticians. In this paper, we introduce a new family of distributions by compounding the Burr XII and power series distributions. The compounding procedure follows the key idea by Adamidis and Loukas ( Statist. Probab. Lett. 39 (1998) 35–42) or, more generally, by Chahkandi and Ganjali ( Comput. Statist. Data Anal. 53 (2009) 4433–4440) and Morais and Barreto-Souza ( Comput. Statist. Data Anal. 55 (2011) 1410–1425). The proposed family includes as a basic exemplar the Burr XII distribution. We provide some mathematical properties including moments, quantile and generating functions, order statistics and their moments, Kullback–Leibler divergence and Shannon entropy. The estimation of the model parameters is performed by maximum likelihood and the inference under large sample. Two special models of the new family are investigated in details. We illustrate the potential of the new family by means of two applications to real data. It provides better fits to these data than other important lifetime models available in the literature.
Silva et al. (Thu,) studied this question.