The complementary Weibull geometric distribution

In this paper, we proposed a new three-parameters lifetime distribution with unimodal, increasing and decreasing hazard rate. The new distribution, the complementary Weibull geometric (CWG), is complementary to the Weibull-geometric (WG) model proposed by Barreto-Souza et al. (The Weibull-Geometric distribution, J. Statist. Comput. Simul. 1 (2010), pp. 1-13). The CWG distribution arises on a latent complementary risks scenarios, where the lifetime associated with a particular risk is not observable, rather we observe only the maximum lifetime value among all risks. The properties of the proposed distribution are discussed, including a formal proof of its probability density function and explicit algebraic formulas for its reliability and hazard rate functions, moments, density of order statistics and their moments. We provide expressions for the Renyi and Shannon entropies. The parameter estimation is based on the usual maximum likelihood approach. We obtain the observed information matrix and discuss inferences issues. We report a hazard function comparison study between the WG distribution and our complementary one. The flexibility and potentiality of the new distribution is illustrated by means of three real dataset, where we also made a comparison between Weibull, WG and CWG modelling approach.