Estimation Given Conditionals from an Exponential Family

Modeling the Conditional Dependence between Discrete and Continuous Random Variables with Applications in Insurance

“…Certainly, the major limitation of this type of models, built by conditional specification, lies in the difficulty that both the marginal distributions and the joint distribution depend on a normalization constant that in most cases is very difficult or even impossible to calculate in closed form (see for example [14]). Obviously, the difficulty grows when the two variables on which the model depends have different support or it is even more complicated when one variable is discrete and the other one is continuous, as the case considered here.…”

“…Therefore, the model provided in ( 14) is suitable for including covariates since the marginal means are given by µ 1 and µ 2 . The expression (X, N) ∼ BGN B(γ, µ 1 , µ 2 ) is used to denote a bivariate random variable (X, N) that follows the distribution given in (14), with marginals gamma and negative binomial. Graphs of the density function for different parameter values and their corresponding contour plots are shown in Figure 1, revealing that the density seems to be unimodal for different values of its parameters.…”

“…In addition the fitted correlation obtained from (15) by using the maximum likelihood estimates of the parameters provided in Table 3 is 0.592. For the bivariate model without covariates parameters estimates together with their corresponding standard errors are provided in Table 3 by using the parametrization of the bivariate model given in (14). In addition, the minimum value of the negative of the log-likelihood function (NLL) and the total sample size is displayed.…”