The HPSEVERITY 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 HPSEVERITY 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;

The following PROC HPSEVERITY step fits all the predefined distributions to the data in the Work.Gdental data set:

/* Fit all predefined distributions */
proc hpseverity data=gdental edf=turnbull print=all criterion=aicc;
   loss / rc=lowerbd lc=upperbd;
   weight count;
   dist _predef_;
   performance nthreads=1;
run;

The EDF= option in the PROC HPSEVERITY statement specifies that the Turnbull’s method be used for EDF estimation. The LOSS statement specifies the left and right boundaries 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 no response variable is specified in the LOSS statement, which is allowed as long as each observation in the input data set is censored. The PERFORMANCE statement specifies that just one thread of execution be used, to minimize the overhead associated with multithreading, because the input data set is very small.

Some of the key results prepared by PROC HPSEVERITY are shown in Output 21.5.1. According to the "Model Selection" table in Output 21.5.1, all distribution models have converged. The "All Fit Statistics" table in Output 21.5.1 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.

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.