# Overview: SEVERITY Procedure

The SEVERITY procedure estimates parameters of any arbitrary continuous probability distribution that is used to model the magnitude (severity) of a continuous-valued event of interest. Some examples of such events are loss amounts paid by an insurance company and demand of a product as depicted by its sales. PROC SEVERITY is especially useful when the severity of an event does not follow typical distributions, such as the normal distribution, that are often assumed by standard statistical methods.

PROC SEVERITY provides a default set of probability distribution models that includes the Burr, exponential, gamma, generalized Pareto, inverse Gaussian (Wald), lognormal, Pareto, Tweedie, and Weibull distributions. In the simplest form, you can estimate the parameters of any of these distributions by using a list of severity values that are recorded in a SAS data set. The values can optionally be grouped by a set of BY variables. PROC SEVERITY computes the estimates of the model parameters, their standard errors, and their covariance structure by using the maximum likelihood method for each of the BY groups.

PROC SEVERITY can fit multiple distributions at the same time and choose the best distribution according to a specified selection criterion. Seven different statistics of fit can be used as selection criteria. They are log likelihood, Akaike’s information criterion (AIC), corrected Akaike’s information criterion (AICC), Schwarz Bayesian information criterion (BIC), Kolmogorov-Smirnov statistic (KS), Anderson-Darling statistic (AD), and Cramér-von Mises statistic (CvM).

You can request the procedure to output the status of the estimation process, the parameter estimates and their standard errors, the estimated covariance structure of the parameters, the statistics of fit, estimated cumulative distribution function (CDF) for each of the specified distributions, and the empirical distribution function (EDF) estimate (which is used to compute the KS, AD, and CvM statistics of fit).

The following key features of PROC SEVERITY make it unique among SAS procedures that can estimate continuous probability distributions:

• PROC SEVERITY enables you to fit a distribution model when the severity values are truncated or censored or both. You can specify any combination of the following types of censoring and truncation effects: left-censoring, right-censoring, left-truncation, or right-truncation. This is especially useful in applications with an insurance-type model where a severity (loss) is reported and recorded only if it is greater than the deductible amount (left-truncation) and where a severity value greater than or equal to the policy limit is recorded at the limit (right-censoring). Another useful application is that of interval-censored data, where you know both the lower limit (right-censoring) and upper limit (left-censoring) on the severity, but you do not know the exact value. The procedure also enables you to specify a probability of observability for the left-truncated data, which is a probability of observing values greater than the left-truncation threshold. This additional information can be useful in certain applications to more correctly model the distribution of the severity of events.

When truncation or censoring is specified, PROC SEVERITY can compute the empirical distribution function (EDF) estimate by using either Kaplan-Meier’s product-limit estimator or Turnbull’s estimator. The former is used by default when only one form of censoring effect (right-censoring or left-censoring) is specified, whereas the latter is used by default when both left-censoring and right-censoring effects are specified.

• PROC SEVERITY enables you to define any arbitrary continuous parametric distribution model and to estimate its parameters. You just need to define the key components of the distribution, such as its probability density function (PDF) and cumulative distribution function (CDF), as a set of functions and subroutines written with the FCMP procedure, which is part of Base SAS software. As long as the functions and subroutines follow certain rules, PROC SEVERITY can fit the distribution model defined by them.

• PROC SEVERITY can model the effect of exogenous or regressor variables on a probability distribution, as long as the distribution has a scale parameter. A linear combination of the regressor variables is assumed to affect the scale parameter via an exponential link function.

If a distribution does not have a scale parameter, then either it needs to have another parameter that can be derived from a scale parameter by using a supported transformation or it needs to be reparameterized to have a scale parameter. If neither of these is possible, then regression effects cannot be modeled.

• PROC SEVERITY enables you to specify your own objective function to be optimized for estimating the parameters of a model. You can write SAS programming statements to specify the contribution of each observation to the objective function. You can use keyword functions such as _PDF_ and _CDF_ in order to generalize the objective function to any distribution. If you do not specify your own objective function, then PROC SEVERITY estimates the parameters of a model by maximizing the likelihood function of the data.

These features and the core functionality are described in detail in the following sections.