Poisson Models With Dynamic Random Effects and Nonnegative Credibilities Per Period

Abstract

This paper provides a toolbox for the credibility analysis of frequency risks, with allowance for the seniority of claims and of risk exposure. We use Poisson models with dynamic and second-order stationary random effects that ensure nonnegative credibilities per period. We specify classes of autocovariance functions that are compatible with positive random effects and that entail nonnegative credibilities regardless of the risk exposure. Random effects with nonnegative generalized partial autocorrelations are shown to imply nonnegative credibilities. This holds for ARFIMA(0, d, 0) models. The AR(p) time series that ensure nonnegative credibilities are specified from their precision matrices. The compatibility of these semiparametric models with log-Gaussian random effects is verified. Gaussian sequences with ARFIMA(0, d, 0) specifications, which are then exponentiated entrywise, provide positive random effects that also imply nonnegative credibilities. Dynamic random effects applied to Poisson distributions are retained as products of two uncorrelated and positive components: the first is time-invariant, whereas the autocovariance function of the second vanishes at infinity and ensures nonnegative credibilities. The limit credibility is related to the three levels for the length of the memory in the random effects. The limit credibility is less than one in the short memory case, and a formula is provided.

The Measures of Accuracy of Claim Frequency Credibility Predictor

“…The GLMM is also considered in [16], where its application is presented for an analysis of longitudinal claims data by generalizing the normal assumption in Bühlmann-Straub model to Poisson and negative binomial distributions. The Poisson claim frequency, but in the form of Poisson mixtures (not as the GLMM), is also discussed in [17] assuming time-varying random effects and the frequency risks dependent on the latent individual factor corresponding to, e.g., the behavioral risk factors. These behavioral factors in credibility prediction can be analyzed in the case of the access to telematic data too.…”

“…The empirical distribution presented in Table 1 clearly demonstrates a strong positive asymmetry. The aim is to compute the values of predictors for the year 2011, based on (8), and assess the prediction accuracy, based on (15) and our proposals: (16) and (17), mainly for HYBRIDtrans f er, for which no past data are available but also for other risk factors. What is important is that the presentation of the results will not be limited to the values of the predictors and estimates of accuracy measures.…”

“…Hence, the distribution can be interpreted as the distribution of a mixture of the bootstrap absolute prediction errors for all risk factors with equal weights. Then, quantiles of these values are computed, as defined in (17), denoted by green lines in Figure 3. For example, the estimate of QMAPE for order p = 0.99 equals 3.92 (see also Table 2).…”

Wishart‐gamma random effects models with applications to nonlife insurance

“…Indeed, while the approximate linear credibility approach, worked out in this setting by Pinquet (1998) and Englund, Guillen, Gustafsson, Nielsen, and Nielsen (2008), is easy to implement and more robust to model misspecification (see Hong & Martin, 2020), it may fail to capture the nonlinearity of the pricing formula as demonstrated by Lu (2018). The linear credibility premium may even become negative, as documented by Pinquet (2020) who finds that the conditions for the credibility coefficients to be positive are quite complicated and unless they are imposed ex ante, the credibility premium can potentially be negative, rendering this approach problematic, especially from a regulation point of view. The only tractable multivariate random effects count model we are aware of is proposed by Badescu, Lin, Tang, and Valdez (2015), who assume a mixture of Erlang distributions for the random effects (i.e., a mixture of Gamma distributions with integer degrees of freedom).…”

On the ordering of credibility factors

“…One popular extension is the dynamic random effects (or state-space) model. However, while the latter allows for time-varying heterogeneity, its application to the credibility analysis should be conducted with care due to the possibility of negative credibilities per period [see Pinquet (2020a)]. Another important but underexplored topic is the ordering of the credibility factors in a monotonous manner-recent claims ought to have larger weights than the old ones.…”