Statistics and Its Interface Volume 12 (2019) 561–571 Destructive power series long-term survival model with complex activation schemes Diego I. Gallardo ∗,† , Heleno Bolfarine, Antonio C. Pedroso-de-Lima, and Jose S. Romeo A new destructive cure rate model is introduced based on a family of power series distribution for the number of con- current causes related to the event of interest. A mixture of first and last activation schemes is considered. For pa- rameter estimation a classical approach based on maximum likelihood methodology is implemented. The performance of estimation procedure is evaluated based on a small scale simulation study. The model is also considered on a real data example, involving congestive heart failure patients. AMS 2000 subject classifications: Primary 62N01, 62N01; secondary 62N02. Keywords and phrases: Cure rate models, Competing risks, Power series distribution. 1. INTRODUCTION Cure rate models have become the ad-hoc choice when the event of interest may not be attainable for a fraction of individuals in the population. A possible way to deal with this situation is to consider that there are a random num- ber M of possible concurrent causes of failure, with cor- responding latent times given by continuous non-negative random variables W 1 ,...,W M . Conditionally on M = m, these quantities are assumed to be independent and iden- tically distributed (iid ) so that the failure time T is given by T =  min(W 1 ,...,W M ), if M> 0; ∞, if M =0. Different distributions for M and W j have been ex- tensively considered by several authors. The seminal work by Berkson and Gage [1] assumed the combination Bernoulli/Exponential models. Several decades later, Chen et al. [2] considered a Poisson/Weibull structure for the problem. Based on the same Weibull distribution for the la- tent times, Rodrigues et al. [3], [4] proposed a more flexible framework assuming, respectively, Negative Binomial and ∗ Corresponding author. † The first author has been partially supported by FONDECYT (Fondo Nacional de Desarrollo Cient´ ıfico y Tecnol´ ogico) Grant 11160670. COM-Poisson distributions for M . Cancho et al. [5] con- sidered Geometric/Birnbaum-Saunders models and, later on, Cancho et al. [6] studied the combination Power se- ries/Weibull. Negative Binomial/Generalized Gamma and Power series/Beta-Weibull models were considered by Or- tega et al. [7], [8]. Cordeiro et al. [9] examined the Neg- ative Binomial/Birnbaum-Saunders combination and re- cently Gallardo et al. [10] developed the model based on the Yule-Simon/Weibull distributions. Rodrigues et al. [11] elaborated a more general model. Assuming the availability of some intervention, they con- sidered that out of M original risk factors, only a number D(≤ M ) remains in effect. For instance, in oncological stud- ies, M usually represents the number of carcinogenic cells for a patient that has some evidence of cancer. After an initial treatment, D of such cells would remain active. Therefore, considering the cure as M = 0 (i.e. the patient would not have any remaining carcinogenic cells) would be contradic- tory. In such a case, cure will be achieved when D = 0. In their initial proposal, Rodrigues et al. [11] imposed the weighted Poisson distribution for M , with Poisson and Neg- ative Binomial distributions as special cases. Conditionally on M = m, the random variable D is assumed to have a Binomial distribution with size m and success probability p, i.e., each initial concurrent causes can be independently activated with probability p. Under the constraint D ≤ M , one has the destructive structure. Other possibilities have been considered elsewhere (see, e.g., Yang and Chen [12]). Let W 1 ,...,W D be activation times related to non- destroyed causes, assumed to be conditionally independent (given D = d) and identically distributed. The correspond- ing failure time is then given by T = min(W 1 ,...,W D ) for D> 0 and T = ∞ for D = 0. This representation is known in the literature as first activation scheme (FA). In Cooner et al. [13] one can find a more general activation scheme. Specifically, they assume that it is necessary to have acti- vation of a random number of underlying causes, say R, to have the event of interest. Under their definition, R =1 would correspond to the FA scheme. It is conceivable then to consider situations where, instead of the minimum, it would be required to have the maximum among all concur- rent times (R = M or R = D in the non-destructive and destructive models, respectively). This scheme is known as the last activation scheme (LA). In addition, Cooner et al.