Unconditional Distributions Obtained from Conditional Speciﬁcation Models with Applications in Risk Theory E. G´ omez-D´ eniz a and E. Calder´ ın–Ojeda b Abstract Bivariate distributions, speciﬁed in terms of their conditional distribu- tions, provide a powerful tool to obtain ﬂexible distributions. These distributions play an important role in specifying the conjugate prior in certain multiparameter Bayesian settings. In this paper, the condi- tional speciﬁcation technique is applied to look for more ﬂexible dis- tributions than the traditional ones used in the actuarial literature, as the Poisson, negative binomial and others. The new speciﬁcation draws inferences about parameters of interest in problems appearing in actuarial statistics. Two unconditional (discrete) distributions ob- tained are studied and used in the collective risk model to compute the right-tail probability of the aggregate claim size distribution. Com- parisons with the compound Poisson and compound negative binomial are made. Keywords: Actuarial, Collective Risk Model, Conditional Distributions; Bivariate Distributions 1 Introduction Many statisticians applying their studies in actuarial framework have exam- ined the distribution of annual claim numbers (Willmot (1987), Kokonendji and Khoudar (2004), Meng et al. (1999), Sarabia et al. (2004), Denuit et al. (2007), Klugman et al. (2008) and G´ omez-D´ eniz et al. (2008a, 2008b); among others). One of the most popular models in this context is the Poisson- gamma, which is based on the assumptions that the portfolio is heterogeneous and that all policyholders have constant but unequal underlying risks of hav- ing an accident. Under the above hypotheses, it is straightforward to prove that the unconditional distribution of the number of claims follows a negative