The SEVERITY Procedure

A Simple Example of Fitting Predefined Distributions

The simplest way to use PROC SEVERITY is to fit all the predefined distributions to a set of values and let the procedure identify the best fitting distribution.

Consider a lognormal distribution, whose probability density function (PDF) $f$ and cumulative distribution function (CDF) $F$ are as follows, respectively, where $\Phi $ denotes the CDF of the standard normal distribution:

\[  f(x; \mu , \sigma ) = \frac{1}{x \sigma \sqrt {2 \pi }} e^{-\frac{1}{2}\left(\frac{\log (x) - \mu }{\sigma }\right)^2} \quad \text {and} \quad F(x; \mu , \sigma ) = \Phi \left(\frac{\log (x) - \mu }{\sigma }\right)  \]

The following DATA step statements simulate a sample from a lognormal distribution with population parameters $\mu = 1.5$ and $\sigma =0.25$, and store the sample in the variable Y of a data set Work.Test_sev1:

/*------------- Simple Lognormal Example -------------*/
data test_sev1(keep=y label='Simple Lognormal Sample');
   call streaminit(45678);
   label y='Response Variable';
   Mu = 1.5;
   Sigma = 0.25;
   do n = 1 to 100;
      y = exp(Mu) * rand('LOGNORMAL')**Sigma;

The following statements fit all the predefined distribution models to the values of Y and identify the best distribution according to the corrected Akaike’s information criterion (AICC):

proc severity data=test_sev1 crit=aicc;
   loss y;
   dist _predefined_;

The PROC SEVERITY statement specifies the input data set along with the model selection criterion, the LOSS statement specifies the variable to be modeled, and the DIST statement with the _PREDEFINED_ keyword specifies that all the predefined distribution models be fitted.

Some of the default output displayed by this step is shown in Figure 23.1 through Figure 23.5. First, information about the input data set is displayed followed by the Model Selection Table, as shown in Figure 23.1. The model selection table displays the convergence status, the value of the selection criterion, and the selection status for each of the candidate models. The Converged column indicates whether the estimation process for a given distribution model has converged, might have converged, or failed. The Selected column indicates whether a given distribution has the best fit for the data according to the selection criterion. For this example, the lognormal distribution model is selected, because it has the lowest value for the selection criterion.

Figure 23.1: Data Set Information and Model Selection Table

The SEVERITY Procedure

Input Data Set
Label Simple Lognormal Sample

Model Selection Table
Distribution Converged Corrected Akaike's
Information Criterion
Burr Yes 322.50845 No
Exp Yes 508.12287 No
Gamma Yes 320.50264 No
Igauss Yes 319.61652 No
Logn Yes 319.56579 Yes
Pareto Yes 510.28172 No
Gpd Yes 510.20576 No
Weibull Yes 334.82373 No

Next, two comparative plots are prepared. These plots enable you to visually verify how the models differ from each other and from the nonparametric estimates. The plot in Figure 23.2 displays the cumulative distribution function (CDF) estimates of all the models and the estimates of the empirical distribution function (EDF). The CDF plot indicates that the Exp (exponential), Pareto, and Gpd (generalized Pareto) distributions are a poor fit as compared to the EDF estimate. The Weibull distribution is also a poor fit, although not as poor as exponential, Pareto, and Gpd. The other four distributions seem to be quite close to each other and to the EDF estimate.

Figure 23.2: Comparison of EDF and CDF Estimates of the Fitted Models

Comparison of EDF and CDF Estimates of the Fitted Models

The plot in Figure 23.3 displays the probability density function (PDF) estimates of all the models and the nonparametric kernel and histogram estimates. The PDF plot enables better visual comparison between the Burr, Gamma, Igauss (inverse Gaussian), and Logn (lognormal) models. The Burr and Gamma differ significantly from the Igauss and Logn distributions in the central portion of the range of Y values, while the latter two fit the data almost identically. This provides a visual confirmation of the information in the Model Selection Table of Figure 23.1, which indicates that the AICC values of Igauss and Logn distributions are very close.

Figure 23.3: Comparison of PDF Estimates of the Fitted Models

Comparison of PDF Estimates of the Fitted Models

The comparative plots are followed by the estimation information for each of the candidate models. The information for the lognormal model, which is the best fitting model, is shown in Figure 23.4. The first table displays a summary of the distribution. The second table displays the convergence status. This is followed by a summary of the optimization process which indicates the technique used, the number of iterations, the number of times the objective function was evaluated, and the log likelihood attained at the end of the optimization. Since the model with lognormal distribution has converged, PROC SEVERITY displays its statistics of fit and parameter estimates. The estimates of Mu=1.49605 and Sigma=0.26243 are quite close to the population parameters of Mu=1.5 and Sigma=0.25 from which the sample was generated. The $p$-value for each estimate indicates the rejection of the null hypothesis that the estimate is 0, implying that both the estimates are significantly different from 0.

Figure 23.4: Estimation Details for the Lognormal Model

The SEVERITY Procedure

Distribution Information
Name Logn
Description Lognormal Distribution
Number of Distribution Parameters 2

Convergence Status for Logn Distribution
Convergence criterion (GCONV=1E-8) satisfied.

Optimization Summary for Logn Distribution
Optimization Technique Trust Region
Number of Iterations 2
Number of Function Evaluations 8
Log Likelihood -157.72104

Fit Statistics for Logn Distribution
-2 Log Likelihood 315.44208
Akaike's Information Criterion 319.44208
Corrected Akaike's Information Criterion 319.56579
Schwarz's Bayesian Information Criterion 324.65242
Kolmogorov-Smirnov Statistic 0.50641
Anderson-Darling Statistic 0.31240
Cramer-von Mises Statistic 0.04353

Parameter Estimates for Logn Distribution
Parameter Estimate Standard
t Value Approx
Pr > |t|
Mu 1.49605 0.02651 56.43 <.0001
Sigma 0.26243 0.01874 14.00 <.0001

The parameter estimates of the Burr distribution are shown in Figure 23.5. These estimates are used in the next example.

Figure 23.5: Parameter Estimates for the Burr Model

Parameter Estimates for Burr Distribution
Parameter Estimate Standard
t Value Approx
Pr > |t|
Theta 4.62348 0.46181 10.01 <.0001
Alpha 1.15706 0.47493 2.44 0.0167
Gamma 6.41227 0.99039 6.47 <.0001