The SEVERITY Procedure

Example 23.6 Fitting Distributions to Interval-Censored Data

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 @@;
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

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;

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 Output 23.6.1. The histogram, kernel density, and EDF plots all indicate that the data is 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.

Output 23.6.1: 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 23.6.2. 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.

Output 23.6.2: Turnbull EDF Estimation Summary for Interval-Censored Data

Turnbull EDF Estimation Summary
Technique EM with Maximum Likelihood Check
Convergence Status Converged
Number of Iterations 2
Maximum Absolute Relative Error 1.8406E-16
Maximum Absolute Reduced Gradient 1.7764E-15
Estimates Maximum Likelihood

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

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

Some of the key results prepared by PROC SEVERITY are shown in Output 23.6.3 through Output 23.6.4. According to the Model Selection Table in Output 23.6.3, all distribution models have converged. The All Fit Statistics Table in Output 23.6.3 indicates that the generalized Pareto distribution (GPD) 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.

Output 23.6.3: Statistics of Fit for Interval-Censored Data

The SEVERITY Procedure

Input Data Set

Model Selection Table
Distribution Converged Anderson-Darling
Burr Yes 0.00103 Yes
Exp Yes 0.09936 No
Gamma Yes 0.04608 No
Igauss Yes 0.12301 No
Logn Yes 0.01884 No
Pareto Yes 0.00739 No
Gpd Yes 0.00739 No
Weibull Yes 0.03293 No

All Fit Statistics Table
Distribution -2 Log
Burr 41.41112 * 47.41112   51.41112   48.31888   0.08974 * 0.00103 * 0.0000816 *
Exp 42.14768   44.14768 * 44.64768 * 44.45026 * 0.26412   0.09936   0.01866  
Gamma 41.92541   45.92541   47.63969   46.53058   0.19569   0.04608   0.00759  
Igauss 42.34445   46.34445   48.05874   46.94962   0.34514   0.12301   0.02562  
Logn 41.62598   45.62598   47.34027   46.23115   0.16853   0.01884   0.00333  
Pareto 41.45480   45.45480   47.16908   46.05997   0.11423   0.00739   0.0009084  
Gpd 41.45480   45.45480   47.16908   46.05997   0.11423   0.00739   0.0009084  
Weibull 41.76272   45.76272   47.47700   46.36789   0.17238   0.03293   0.00472  

The P-P plots of Output 23.6.4 show that both GPD and BURR have a close fit between EDF and CDF estimates, although BURR has slightly better fit, which is also indicated by the EDF-based statistics. Given that BURR is a generalization of the GPD and that the plots do not offer strong evidence in support of the more complex distribution, GPD seems like a good choice for this data.

Output 23.6.4: P-P Plots of Burr and GPD for Interval-Censored Data