Buhlmann credibility model in predicting claim frequency that follows heterogeneous Weibull count distribution

In insurance, policyholder’s claim experience is usually too limited to be given full credibility in predicting claim frequency, but policyholder’s risk is usually a part of a large risk class which collective claim experience can provide information for credible statistical prediction. Credibility could be used to consider both information. By assuming that the policyholder’s number of claims, given the policyholder’s risk parameter, follows Weibull count distribution over its risk class, this paper explains Buhlmann credibility in predicting claim frequency. Weibull count distribution relaxes the equidispersion assumption of Poisson distribution. Thus, Weibull count distribution can handle non-equidispersed count data. This paper also shows that for a certain type of past claim frequency data, the use of Poisson assumption in Buhlmann credibility model could result in very low credibility factor which underrates the policyholder’s experience.


Introduction
One approach to predict a policyholder's claim frequency in a certain period is by calculating the mean of that policyholder's claim frequency in previous policy periods. However, in most cases, policyholder's claim experience is too limited to be given full credibility in predicting claim frequency. Moreover, policyholder's risk is usually a part of a large risk class which collective claim experience can provide information for credible statistical prediction. One of the methods to take both informations into account is linear credibility theory first explained by Buhlmann in 1920 [2]. Credibility, in this paper, assigns weight to policyholder's experience estimate and the collective experience estimate to predict policyholder's claim frequency.
One of the distributions often used to model policyholder's claim frequency is Poisson count model which has exponential interarrival times. However, Poisson count model is only valid if the data satisfiy equidispersion assumption (the variance of the data equals its mean). Applying Poisson count model to the significantly overdispersed (the variance is more than its mean) or underdispersed (the variance is less than its mean) data could lead to misspesification of the distribution of the data [1]. Weibull distribution which is a generalization of exponential, is considered in this paper to develop a count model that can handle both overdispersion and underdispersion. Weibull interarrival times could handle overdispersed data when its shape parameter (c) is 0 1, c  and underdispersed data when 1, c  and is reduced as exponential when 1 c  [4]. The count model is called Weibull count model. This paper explains Weibull count and heterogeneous Weibull count distribution. Heterogeneous Weibull count distribution could be used to fit the data distribution of a policyholder's past claim frequency in order to predict the next period claim frequency. By assuming that the policyholder's number of claims, given the policyholder's risk parameter, follows Weibull count distribution over its 2 risk class, this paper then explains an approach by Buhlmann credibility model in predicting claim frequency. Finally, we show that for a certain type of past claim frequency data, the use of Poisson assumption in Buhlmann credibility model could result in very low credibility factor which underrates the policyholder's experience.

Weibull count model
Weibull count model is obtained by Taylor expansion and convolution method. The model is in recursive form, and by mathematical induction we obtain that the general form for where The expectation of this model is given by

Heterogeneous Weibull count model
The unconditional probability that the number of events from unit i up until time t is n is obtained by mixing.
Finally, the heterogeneous Weibull count model could be written as follows  The expectation of this model is given by and its variance is given by

Buhlmann credibility in predicting claim frequency that follows heterogeneous Weibull count distribution
In this section we will derive a Buhlmann credibility model with assumption that the policyholder's number of events with unknown risk parameter follows Weibull count distribution. We derive this model the same way Buhlmann derived his credibility model in 1920 which is stated as follows.
   is the hypothetical means or mean of individual risks in a certain risk class.

 
v  is process variance or could also be described as the variability measure in policyholder's risk experience.
Before deriving the Buhlmann credibility model using Weibull count assumption, we first recall the Buhlmann credibility model if the assumption used is that the individual number of events, conditional on the policyholder's risk parameter, follows Poisson distribution. Suppose the number of claims of a policyholder in year j with unknown risk parameter ,  | j X  are assumed to be independently and identically Poisson distributed for 1,..., . jn  We also assume that the risk parameter follows gamma distribution with shape parameter r and rate .
a Then, Finally, we derive the Buhlmann credibility model assuming that the number of events of a policyholder, conditional on the policyholder's risk parameter, follows Weibull count distribution. The policyholder's risk parameter in this paper is denoted by . i  We assume that the value i  varies and represents heterogeneity among the individuals. We also assume that its random variable i  follows gamma distribution. The gamma assumption as the distribution of i  will result in heterogeneous Weibull count distribution as the unconditional distribution of the number of events, as shown in section 3. By this, now we can say that we will derive a Buhlmann credibility model assuming that the individual number of events follows heterogeneous Weibull count distribution.
It is assumed that the estimator function for ( 1) in N  is linear with It is also assumed that ( | ) ij i N  are independent for 1,..., , 1 j w w  [2]. Thus, the linear estimator for predictive random variable ( 1) iw N  is [2]   10 1 where the value of 01 , ,..., Thus, (3) and (4) becomes Note that we will predict the number of claims of policyholder i in period 1 w  given the information of that policyholder in periods 1,2,..., jw  which means 1 t  period or 1 year. Thus, Finally, with an assumption that the number of claims of policyholder in year , j given that the risk profile of policyholder is gamma distributed with shape r and rate , a follows Weibull count distribution, the Buhlmann credibility model to predict the number of claims of a policyholder i in a certain policy period could be stated as (21).