In practice, practitioners usually use a data set with commonly available exogenous covariates in order to explain the variable $Y_i$, known in this case as the endogenous variate. That is, suppose that for the $i$th observation the covariates $x$ and $z$ are available. In order to adapt the model to this framework, we need to relate these covariates with endogenous variables via the parameters $\theta$ and $\omega$. This can be done through the following links:

with ${\overrightarrow{\beta }}^{\mathrm{T}}=({\beta }_1,\mathrm{\dots },{\beta }_k)$ and ${\overrightarrow{\gamma }}^{\mathrm{T}}=({\gamma }_1,\mathrm{\dots },{\gamma }_k)$ vectors of unknown regression parameters associated with covariates. Of course, in practice it is common to suppose that the design matrix $X$ and $Z$ are the same.

A nice reformulation of the zero-inflated model above was proposed recently by Ghosh et al (2006), who considered that the zero-inflated model can be represented as $Y=V(1-B)$, where $B$ is a Bernoulli, Bernoulli($p$), random variable, and $V$ independently to $B$ has a discrete distribution on the power series, $\mathrm{PS}(\theta )$. Under this representation, the mean $E(Y)$ and variance $var(Y)$ can be rewritten as

where $\delta =var(V)/E(V)$ denotes the coefficient of dispersion of the latent random variable, $V$. If $V$ does not have an underdispersed distribution (ie, $\delta \ge 1$), then the distribution of $Y$ is overdispersed. On the other hand, if $V$ does have an underdispersed distribution (ie, ), then $Y$ has an underdispersed distribution if and only if . In their paper, Ghosh et al (2006) suppose that $V$ follows the power series distribution.

In our paper, two discrete distributions belonging to the power series distributions will be considered. They are the Poisson distribution with parameter $\theta >0$ and the geometric distribution with parameter $1/(1+\theta )$, $\theta >0$. These models will be denoted as the ZIP model and the zero-inflated geometric (ZIG) model.

In this section, a Bayesian methodology is carried out; this allows us to estimate the model above in a simple way, facilitating the process of incorporating covariates and providing exact posterior inference up to a Monte Carlo error. This model can easily accommodate multiple continuous and categorical predictors.

From a Bayesian point of view, prior distributions for $\omega$ and $\theta$ will be required. In this sense, and looking to the loglikelihood in (2.11), it is adequate to assume a Beta prior distribution for $\omega$ and the natural conjugate prior (Ghosh et al 2006) for the power series distribution in the following way:

Both assumptions establish a congruent model, present important computational advantages and, in addition, have a tradition in Bayesian statistical literature. However, in this paper, the covariates that affect $\omega$ and $\mu$ are fixed. So, we specify independent prior distributions for the parameters of the regression models, ie, $\overrightarrow{\beta }$ and $\overrightarrow{\gamma }$, as follows:

where the constants $a_{\beta }$, $b_{\beta }$, $a_{\gamma }$ and $b_{\gamma }$ are assumed to be known. In particular, $a_{\beta }=a_{\gamma }=-{10}^5$ and $b_{\beta }=b_{\gamma }={10}^5$, which expresses our lack of knowledge about the regression parameters. These noninformative uniform distributions are appropriate if no prior knowledge is available about the likely range of values of the parameters (see Lempers (1971) and Mitchell and Beauchamp (1988), among others).

Given that the prior distributions for parameters have been assessed, the next procedure is to combine the likelihood function in (2.11) with priors in order to make a Bayesian inference. Since no closed forms are available for marginal posterior distributions, numerical approaches have to be used to generate them. The numerical approaches used are based on simulations from the posterior distributions, which are proper, since we consider proper priors, although with somewhat large variances. The simulation approach used is MCMC, which can be conducted using the WinBUGS software. MCMC is in this setting a powerful tool that allows us to get estimates of the parameters involved. As is well known, MCMC is a method of posterior simulation and leads us to compute the posterior density function for arbitrary points in the parameter space. With MCMC, it is possible to generate samples from an arbitrary posterior density and use these samples to approximate expectations of quantities of interest, such as the mean or second-order moment. Several other aspects of the MCMC also contribute to its success. The methodology is very simple and consists of generating simulated samples from that posterior density, even though the density corresponds to unknown distributions. In this context, Gibbs sampling is a natural estimation method. Reasonable choices for starting values of $\overrightarrow{\beta }$ and $\overrightarrow{\gamma }$ for the MCMC simulation can be obtained by standard Poisson and negative binomial regression models using any statistical software package, such as Stata. In this work, all simulations were done using WinBUGS (Spiegelhalter et al 1999). We run three parallel chains and a single long chain for diagnostic assessment (checked using Coda software). A total of 100 000 iterations were carried out (after another 100 000 iterations of burn-in). A complete Gibbs sampling algorithm is outlined in Ghosh et al (2006). If Bayesian estimation in considered for the ZIP and ZIG models, these models will be denoted by BZIP and BZIG, respectively.

In particular, the data studied contains information from the policyholders of an Australian insurance company between 2004 and 2005. It describes certain characteristics related to the vehicles and the policyholders. The database contains 67 856 policies, of which 63 232 (93.18$\%$) have no claims. This is high-frequency data, although the methodology proposed here is also valid for models with low-frequency data. Table 1 shows a descriptive summary of the dependent and independent variables.

Table 2 shows some measures that lead us to consider departures from the Poisson distribution. These measures are the proportion of zeros, $p_0$, the cumulant, ${\kappa }_3$, the zero-inflation index, $z_i=1+\log (p_0)/\mu$, and the third central moment inflation index, $\kappa ={\kappa }_3/\mu -1$ (see Puig and Valero (2006) for details). In this table, we can see the sample values of these measures and their corresponding estimated values; these are obtained using the Poisson, geometric, ZIP and ZIG distributions after estimating the corresponding parameters using the maximum-likelihood method. We can see that the geometric distribution (in its inflated and non-inflated at zero versions) outperforms the Poisson one. In order to test whether there are too many zeros for the Poisson distribution, a Van den Broek (1995) score test has been conducted. This measure tests the null hypothesis ${\mathrm{H}}_0:\omega =0$ and is defined by

where ${\widehat{p}}_{0i}=\widehat{\text{Prob}}(Y=0)$, the estimate probability of zero in the Poisson regression, and $\overline{y}$ is the average of the count observations. This statistic has a chi-squared distribution with 1 degree of freedom. The score statistic is given by 121 392.61 ($p\text{-value}\ll 0.0001$), which provides evidence that the observed zeros exceed the zeros limit of the Poisson distribution.

For each policy, the initial information for the period considered and the existence (or otherwise) of at least a claim are reported within this yearly period. In total, four explanatory variables are considered, together with a dependent variable representing the number of claims. Vehicle value is represented in 10 000 Australian dollars. Vehicle age is equal to 1 if the vehicle is relatively young (seven years old or less). Gender is equal to 1 if the policyholder is a man. This variable is included in the model for didactic purposes, but, as expected, it is not relevant in any of the models considered. Finally, a categorical variable is considered to represent the age of the policyholder by dividing this feature into three dummies: young, medium and old ages. In this sense, we try to identify if there is/are age sets with a higher propensity to make claims. Several authors have previously used the age variable in a dichotomous way (see Boucher et al (2007), Bermúdez (2009) and Pérez et al (2014), among others).

Table 3 shows results under the Bayesian estimation of the ZIP model. As we can see, vehicle value and older policyholders (in relation to medium-aged policyholders) are relevant for the chance of being in zero-state. The positive value of the first coefficient indicates that this chance increases with the value of the vehicle (with significance at $1\%$). The negative scope of old age indicates that the chance of being in zero-state decreases for older policyholders in relation to medium-aged policyholders, at $5\%$ significance. However, an intercept of $-1.883$ (with significance at 1$\%$) indicates that the average number of claims is lower for medium-aged policyholders. Further, the higher the vehicle value, the lower the average number of claims expected, again at $1\%$ significance. These results are consistent with the ${\widehat{\gamma }}_i$ zero-state coefficients.

Table 4 shows results under the Bayesian estimation of the ZIG model. Now, being in the medium-aged class increases the chance of zero-state (the $\alpha$-intercept is relevant at $10\%$). The average number of claims increases if the vehicle value is high (at $1\%$ significance) and decreases in the medium- and old-aged classes (more for older policyholders). For the youngest people, the average number of claims is expected to be greater than for the other two groups of policyholders (with $1\%$ significance). Finally, the average number of claims decreases if the age of the vehicle increases.

Finally, it is interesting to compare the results from a Bayesian perspective with those obtained using standard methodology. Table 5 shows the results under two standard Poisson and negative binomial distributions, along with their zero-inflated versions. We observe that they detect the same relevant factors as the BZIG model, but these models do not distinguish between the zero-state and average number of claims factors.

Although there are a variety of methodologies to validate several models for a given data set, the DIC, which is a generalization of the AIC and BIC, is useful in Bayesian model selection problems where the posterior distributions of the models have been obtained by MCMC simulation. Recall that the DIC is only valid when the posterior distribution is approximately multivariate normal, which is the case considered here. Out of all the criteria, the model that best fits a data set is the model with the smallest value. As we can see in Table 6, frequentist models are a worse fit than the BZIP and BZIG models. The deviance of BZIP is the smallest, but the confidence interval with $5\%$ significance is $[9336,9889]$; this includes the deviance of the BZIG model, so there is no significative difference in terms of fitting between these two models. By comparing Tables 4 and 5, we can observe that the estimated coefficients differ considerably, although the signs and the relevant factors remain the same. So, we can say that the results from the BZIG model are more consistent (than those from the BZIP model) with respect to the results from the classical models in Table 5, providing a much better fit.