The Tweedie distribution is often used in the insurance industry to explain the effect of independent variables (regressors) on the distribution of losses. PROC SEVERITY provides a predefined scaled Tweedie distribution (STWEEDIE) that enables you to model the regression effects on the scale parameter. The scale regression model has its own advantages such as the ability to easily account for inflation effects. This example illustrates how that model can be used to evaluate the effect of regressors on the mean of the Tweedie distribution, which is useful in problems such ratemaking and pure premium modeling.
Assume a Tweedie process, whose mean is affected by regressors , as follows:

where represents the base value of the mean (you can think of as , where is the intercept). This model for the mean is identical to the popular generalized linear model for the mean with a logarithmic link function. More interestingly, it parallels the model used by PROC SEVERITY for the scale parameter , which is as follows

where represents the base value of the scale parameter. As described in the section Tweedie Distributions, for the parameter range , the mean of the Tweedie distribution is given by

where is the Poisson mean parameter of the scaled Tweedie distribution. This relationship enables you to use the scale regression model to infer the effect of regressors on the mean of the distribution.
Let the data set Work.Test_Sevtw
contain a sample generated from a Tweedie distribution with dispersion parameter , index parameter , and the mean parameter that is affected by three regression variables x1
, x2
, and x3
as follows:

Thus, the population values of regression parameters are , , , and . You can find the code used to generate the sample in the PROC SEVERITY sample program sevex05.sas
.
The following PROC SEVERITY step uses the sample in Work.Test_Sevtw
data set to estimate the parameters of the scale regression model for the predefined scaled Tweedie distribution (STWEEDIE)
with the dual quasiNewton (QUANEW) optimization technique:
proc severity data=test_sevtw outest=estw covout print=all plots=none; loss y; scalemodel x1x3; dist stweedie; nloptions tech=quanew; run;
The dual quasiNewton technique is used because it requires only the firstorder derivatives of the objective function, and it is harder to compute reasonably accurate estimates of the secondorder derivatives of Tweedie distribution’s PDF with respect to the parameters.
Some of the key results prepared by PROC SEVERITY are shown in Output 23.5.1 and Output 23.5.2. The distribution information and the convergence results are shown in Output 23.5.1.
Output 23.5.1: Convergence Results for the STWEEDIE Model with Regressors
Distribution Information  

Name  stweedie 
Description  Tweedie Distribution with Scale Parameter 
Number of Distribution Parameters  3 
Number of Regression Parameters  3 
Convergence Status for stweedie Distribution 

Convergence criterion (FCONV=2.220446E16) satisfied. 
Optimization Summary for stweedie Distribution  

Optimization Technique  Dual QuasiNewton 
Number of Iterations  41 
Number of Function Evaluations  156 
Log Likelihood  1044.3 
The final parameter estimates of the STWEEDIE regression model are shown in Output 23.5.2. The estimate that is reported for the parameter Theta
is the estimate of the base value . The estimates of regression coefficients , , and are indicated by the rows of x1
, x2
, and x3
, respectively.
Output 23.5.2: Parameter Estimates for the STWEEDIE Model with Regressors
Parameter Estimates for stweedie Distribution  

Parameter  Estimate  Standard Error 
t Value  Approx Pr > t 
Theta  0.82532  0.42135  1.96  0.0511 
Lambda  16.40072  21.40657  0.77  0.4442 
P  1.75168  0.33675  5.20  <.0001 
x1  0.27991  0.09906  2.83  0.0050 
x2  0.76666  0.10338  7.42  <.0001 
x3  3.03252  0.10169  29.82  <.0001 
If your goal is to explain the effect of regressors on the scale parameter, then the output displayed in Output 23.5.2 is sufficient. But, if you want to compute the effect of regressors on the mean of the distribution, then some postprocessing needs to be done. Using the relationship between and , can be written in terms of the parameters of the STWEEDIE model as

This shows that the parameters are identical for the mean and the scale model, and the base value of the mean model is

The estimate of and the standard error associated with it can be computed by using the property of the functions of maximum likelihood estimators (MLE). If represents a totally differentiable function of parameters , then the MLE of has an asymptotic normal distribution with mean and covariance , where is the MLE of , is the estimate of covariance matrix of , and is the gradient vector of with respect to evaluated at . For , the function is . The gradient vector is




You can write a DATA step that implements these computations by using the parameter and covariance estimates prepared by PROC
SEVERITY step. The DATA step program is available in the sample program sevex05.sas
. The estimates of prepared by that program are shown in Output 23.5.3. These estimates and the estimates of as shown in Output 23.5.2 are reasonably close (that is, within one or two standard errors) to the parameters of the population from which the sample
in Work.Test_Sevtw
data set was drawn.
Output 23.5.3: Estimate of the Base Value Mu0 of the Mean Parameter
Parameter  Estimate  Standard Error 
t Value  Approx Pr > t 

Mu0  4.47156  0.42283  10.5752  0 
Another effect of using the scaled Tweedie distribution to model the regression effects is that the regressors also affect the variance of the Tweedie distribution. The variance is related to the mean as , where is the dispersion parameter. Using the relationship between the parameters TWEEDIE and STWEEDIE distributions as described in the section Tweedie Distributions, the regression model for the dispersion parameter is




Subsequently, the regression model for the variance is




In summary, PROC SEVERITY enables you to estimate regression effects on various parameters and statistics of the Tweedie model.