## Using Tweedie Parameter to Identify Distributions

In the development of operational loss models, it is important to identify which distribution should be used to model operational risk measures, e.g. frequency and severity. For instance, why should we use the Gamma distribution instead of the Inverse Gaussian distribution to model the severity?

In my previous post https://statcompute.wordpress.com/2016/11/20/modified-park-test-in-sas, it is shown how to use the Modified Park test to identify the mean-variance relationship and then decide the corresponding distribution of operational risk measures. Following the similar logic, we can also leverage the flexibility of the Tweedie distribution to accomplish the same goal. Based upon the parameterization of a Tweedie distribution, the variance = Phi * (Mu ** P), where Mu is the mean and P is the power parameter. Depending on the specific value of P, the Tweedie distribution can accommodate several important distributions commonly used in the operational risk modeling, including Poisson, Gamma, Inverse Gaussian. For instance,

- With P = 0, the variance would be independent of the mean, indicating a Normal distribution.
- With P = 1, the variance would be in a linear form of the mean, indicating a Poisson-like distribution
- With P = 2, the variance would be in a quadratic form of the mean, indicating a Gamma distribution.
- With P = 3, the variance would be in a cubic form of the mean, indicating an Inverse Gaussian distribution.

In the example below, it is shown that the value of P is in the neighborhood of 1 for the frequency measure and is near 3 for the severity measure and that, given P closer to 3, the Inverse Gaussian regression would fit the severity better than the Gamma regression.

library(statmod) library(tweedie) profile1 <- tweedie.profile(Claim_Count ~ Age + Vehicle_Use, data = AutoCollision, p.vec = seq(1.1, 3.0, 0.1), fit.glm = TRUE) print(profile1$p.max) # [1] 1.216327 # The P parameter close to 1 indicates that the claim_count might follow a Poisson-like distribution profile2 <- tweedie.profile(Severity ~ Age + Vehicle_Use, data = AutoCollision, p.vec = seq(1.1, 3.0, 0.1), fit.glm = TRUE) print(profile2$p.max) # [1] 2.844898 # The P parameter close to 3 indicates that the severity might follow an Inverse Gaussian distribution BIC(glm(Severity ~ Age + Vehicle_Use, data = AutoCollision, family = Gamma(link = log))) # [1] 360.8064 BIC(glm(Severity ~ Age + Vehicle_Use, data = AutoCollision, family = inverse.gaussian(link = log))) # [1] 350.2504

Together with the Modified Park test, the estimation of P in a Tweedie distribution is able to help us identify the correct distribution employed in operational loss models in the context of GLM.