The SEVERITY Procedure

(View the complete code for this example.)

In some applications, the data available for modeling might not be exact. A commonly encountered scenario is the use of grouped data from an external agency, which for several reasons, including privacy, does not provide information about individual loss events. The losses are grouped into disjoint bins, and you know only the range and number of values in each bin. Each group is essentially interval-censored, because you know that a loss magnitude is in certain interval, but you do not know the exact magnitude. This example illustrates how you can use PROC SEVERITY to model such data.

The following DATA step generates sample grouped data for dental insurance claims, which is taken from Klugman, Panjer, and Willmot (1998):

/* Grouped dental insurance claims data
    (Klugman, Panjer, and Willmot, 1998) */
data gdental;
   input lowerbd upperbd count @@;
   datalines;
0 25 30  25 50 31  50 100 57  100 150 42  150 250 65  250 500 84
500 1000 45  1000 1500 10  1500 2500 11  2500 4000 3
;
run;

Often, when you do not know the nature of the data, it is recommended that you first explore the nature of the sample distribution by examining the nonparametric estimates of PDF and CDF. The following PROC SEVERITY step prepares the nonparametric estimates, but it does not fit any distribution because there is no DIST statement specified:

/* Prepare nonparametric estimates */
proc severity data=gdental print=all plots(histogram kernel)=all;
   loss / rc=lowerbd lc=upperbd;
   weight count;
run;

The LOSS statement specifies the left and right boundary of each group as the right-censoring and left-censoring limits, respectively. The variable count records the number of losses in each group and is specified in the WEIGHT statement. Note that there is no response or loss variable specified in the LOSS statement, which is allowed as long as each observation in the input data set is censored. The nonparametric estimates prepared by this step are shown in Figure 26. The histogram, kernel density, and EDF plots all indicate that the data are heavy-tailed. For interval-censored data, PROC SEVERITY uses Turnbull’s algorithm to compute the EDF estimates. The plot of Turnbull’s EDF estimates is shown to be linear between the endpoints of a censored group. The linear relationship is chosen for convenient visualization and ease of computation of EDF-based statistics, but you should note that theoretically the behavior of Turnbull’s EDF estimates is undefined within a group.

Figure 26: Nonparametric Distribution Estimates for Interval-Censored Data

With the PRINT=ALL option, PROC SEVERITY prints the summary of the Turnbull EDF estimation process as shown in Output 28.6.1. It indicates that the final EDF estimates have converged and are in fact maximum likelihood (ML) estimates. If they were not ML estimates, then you could have used the ENSUREMLE option to force the algorithm to search for ML estimates.

After exploring the nature of the data, you can now fit a set of heavy-tailed distributions to these data. The following PROC SEVERITY step fits all the predefined distributions to the data in the Work.Gdental data set:

/* Fit all predefined distributions */
proc severity data=gdental print=all plots(histogram kernel)=all
              criterion=ad;
   loss / rc=lowerbd lc=upperbd;
   weight count;
   dist _predef_;
run;

Some of the key results prepared by PROC SEVERITY are shown in Output 28.6.2 through Figure 27. According to the "Model Selection" table in Output 28.6.2, all distribution models have converged. The "All Fit Statistics" table in Output 28.6.2 indicates that the exponential distribution (EXP) has the best fit for data according to a majority of the likelihood-based statistics and that the Burr distribution (BURR) has the best fit according to all the EDF-based statistics.

The P-P plots in Figure 27 show that the Burr distribution clearly has a better fit between EDF and CDF estimates, confirming the information that is reported by the EDF-based statistics. When the best distributions that are chosen by the likelihood-based and EDF-based statistics are different, you need to decide which fit statistic best represents your objective. In this example, if your objective is to minimize the distance between EDF and CDF values, then you should choose the Burr distribution. On the other hand, if your objective is to maximize the likelihood of the observed data while minimizing the model complexity, then you should choose the exponential distribution. Note that the exponential distribution has worse (lower) raw likelihood than the Burr distribution, but it has better AIC, AICC, and BIC statistics than the Burr distribution because the exponential distribution has only one parameter compared to the three parameters of the Burr distribution. Further, the small sample size of 10 helps accentuate the role of model complexity in the AIC, AICC, and BIC statistics. If the sample size would have been larger, the exponential distribution might not have won according to the likelihood-based statistics.

Figure 27: P-P Plots of Burr and Exponential Distributions for Interval-Censored Data