The Complementary Exponentiated Exponential Geometric Lifetime Distribution

We proposed a new family of lifetime distributions, namely, complementary exponentiated exponential geometric distribution. This new family arises on a latent competing risk scenario, where the lifetime associated with a particular risk is not observable but 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 survival and hazard functions, moments, rth moment of the ith order statistic, mean residual lifetime, and modal value. Inference is implemented via a straightforwardly maximum likelihood procedure. The practical importance of the new distribution was demonstrated in three applications where our distribution outperforms several former lifetime distributions, such as the exponential, the exponential-geometric, the Weibull, the modified Weibull, and the generalized exponential-Poisson distribution. 1. Introduction Several new classes of models have been introduced in recent years grounded in the simple exponential distribution. The main idea is to propose lifetime distributions which can accommodate practical applications where the underlying hazard functions are nonconstant, presenting monotone shapes, since the exponential distribution does not provide a reasonable fit in such situations. For instance, we can cite [1], which proposed a variation of the exponential distribution, the exponential geometric (EG) distribution, with decreasing hazard function, [2], which introduced the exponentiated exponential distribution as a generalization of the usual exponential distribution, which can accommodate data with increasing and decreasing hazard functions, [3], which proposed a generalized exponential distribution, which can accommodate data with increasing and decreasing hazard functions, [4], which proposed the exponentiated type distributions extending the Fréchet, gamma, Gumbel, and Weibull distributions, [5], which proposed another modification of the exponential distribution with decreasing hazard function, [6], which generalizes the distribution proposed by [5] by including a power parameter in this distribution, which can accommodate increasing, decreasing, and unimodal hazard functions, [7], which proposed the Poisson-exponential distribution, and [8], which proposed the complementary exponential geometric distribution, which is complementary to the exponential geometric distribution proposed by [1]. The last two proposed distributions accommodate increasing hazard functions. In this paper,