SEVERITY Procedure

defines a function that returns the value of the cumulative distribution function (CDF) of the distribution at the specified values of the random variable and distribution parameters.

If you want to consider this distribution as a candidate distribution when you estimate a response variable model with regression effects, then the first parameter of this distribution must be a scale parameter or log-transformed scale parameter. In other words, if the distribution has a scale parameter, then the following equation must be satisfied:

If the distribution has a log-transformed scale parameter, then the following equation must be satisfied:

Here is a sample structure of the function for a distribution named 'FOO':

 function FOO_CDF(x, P1, P2);
     /* Code to compute CDF by using x, P1, and P2 */

     F = <computed CDF>;
     return (F);
 endsub;

defines a subroutine that specifies constant parameters. A parameter is constant if it is required for defining a distribution but is not subject to optimization in PROC SEVERITY. Constant parameters are required to be part of the model in order to compute the PDF or the CDF of the distribution. Typically, values of these parameters are known a priori or estimated using some means other than the maximum likelihood method used by PROC SEVERITY. You can estimate them inside the dist_PARMINIT subroutine. Once initialized, the parameters remain constant in the context of PROC SEVERITY; that is, they retain their initial value. PROC SEVERITY estimates only the nonconstant parameters.

Here is a sample structure of the subroutine for a distribution named 'FOO' that has P3 and P5 as its constant parameters, assuming that distribution has at least three parameters:

 subroutine FOO_CONSTANTPARM(p5, p3);
 endsub;

Note the following points when you specify the constant parameters:

defines a function that returns a description of the distribution.

Here is a sample structure of the function for a distribution named 'FOO':

 function FOO_DESCRIPTION() $48;
     length desc $48;
     desc = "A model for a continuous distribution named foo";
     return (desc);
 endsub;

There is no restriction on the length of the description (the length of 48 used in the previous example is for illustration purposes only). However, if the length is greater than 256, then only the first 256 characters appear in the displayed output and in the _DESCRIPTION_ variable of the OUTMODELINFO= data set. Hence, the recommended length of the description is less than or equal to 256.

defines a function that returns the natural logarithm of the specified core function of the distribution at the specified values of the random variable and distribution parameters. The core keyword can be PDF, CDF, or SDF.

Here is a sample structure of the function for the core function PDF of a distribution named 'FOO':

 function FOO_LOGPDF(x, P1, P2);
     /* Code to compute LOGPDF by using x, P1, and P2 */

     l = <computed LOGPDF>;
     return (l);
 endsub;

defines a subroutine that returns lower bounds for the parameters of the distribution. If this subroutine is not defined for a given distribution, then the SEVERITY procedure assumes a lower bound of 0 for each parameter. If a lower bound of is returned for a parameter , then the SEVERITY procedure assumes that (strict inequality). If a missing value is returned for some parameter, then the SEVERITY procedure assumes that there is no lower bound for that parameter (equivalent to a lower bound of ).

Here is a sample structure of the subroutine for a distribution named 'FOO':

 subroutine FOO_LOWERBOUNDS(p1, p2);
     outargs p1, p2;

     p1 = <lower bound for P1>;
     p2 = <lower bound for P2>;
 endsub;

defines a subroutine that returns the initial values for the distribution’s parameters given an empirical distribution function (EDF) estimate.

Here is a sample structure of the subroutine for a distribution named 'FOO':

 subroutine FOO_PARMINIT(dim, x{*}, nx{*}, F{*}, Ftype, p1, p2);
     outargs p1, p2;

     /* Code to initialize values of P1 and P2 by using
        dim, x, nx, and F */

     p1 = <initial value for p1>;
     p2 = <initial value for p2>;
 endsub;

defines a function that returns the value of the probability density function (PDF) of the distribution at the specified values of the random variable and distribution parameters.

If you want to consider this distribution as a candidate distribution when you estimate a response variable model with regression effects, then the first parameter of this distribution must be a scale parameter or log-transformed scale parameter. In other words, if the distribution has a scale parameter, then the following equation must be satisfied:

If the distribution has a log-transformed scale parameter, then the following equation must be satisfied:

Here is a sample structure of the function for a distribution named 'FOO':

 function FOO_PDF(x, P1, P2);
     /* Code to compute PDF by using x, P1, and P2 */

     f = <computed PDF>;
     return (f);
 endsub;

defines a function that returns the quantile of the distribution at the specified value of the CDF for the specified values of distribution parameters.

Here is a sample structure of the function for a distribution named 'FOO':

 function FOO_QUANTILE(c, P1, P2);
     /* Code to compute quantile by using c, P1, and P2 */

     Q = <computed quantile>;
     return (Q);
 endsub;

defines a function that returns a keyword to identify the transform that needs to be applied to the scale parameter to convert it to the first parameter of the distribution.

If you want to use this distribution for modeling regression effects, then the first parameter of this distribution must be a scale parameter. However, for some distributions, a typical or convenient parameterization might not have a scale parameter, but one of the parameters can be a simple transform of the scale parameter. As an example, consider a typical parameterization of the lognormal distribution with two parameters, location and shape , for which the PDF is defined as follows:

You can reparameterize this distribution to contain a parameter instead of the parameter such that . The parameter would then be a scale parameter. However, if you want to specify the distribution in terms of and (which is a more recognized form of the lognormal distribution) and still allow it as a candidate distribution for estimating regression effects, then instead of writing another distribution with parameters and , you can simply define the distribution with as the first parameter and specify that it is the logarithm of the scale parameter.

If you do not specify this function, then the IDENTITY transform is assumed.

Here is a sample structure of the function for a distribution named 'FOO':

 function FOO_SCALETRANSFORM() $8;
     length xform $8;
     xform = "IDENTITY";
     return (xform);
 endsub;

defines a function that returns the value of the survival distribution function (SDF) of the distribution at the specified values of the random variable and distribution parameters.

If you want to consider this distribution as a candidate distribution when estimating a response variable model with regression effects, then the first parameter of this distribution must be a scale parameter or log-transformed scale parameter. In other words, if the distribution has a scale parameter, then the following equation must be satisfied:

If the distribution has a log-transformed scale parameter, then the following equation must be satisfied:

Here is a sample structure of the function for a distribution named 'FOO':

 function FOO_SDF(x, P1, P2);
     /* Code to compute SDF by using x, P1, and P2 */

     S = <computed SDF>;
     return (S);
 endsub;

defines a subroutine that returns upper bounds for the parameters of the distribution. If this subroutine is not defined for a given distribution, then the SEVERITY procedure assumes that there is no upper bound for any of the parameters. If an upper bound of is returned for a parameter , then the SEVERITY procedure assumes that (strict inequality). If a missing value is returned for some parameter, then the SEVERITY procedure assumes that there is no upper bound for that parameter (equivalent to an upper bound of ).

Here is a sample structure of the subroutine for a distribution named 'FOO':

 subroutine FOO_UPPERBOUNDS(p1, p2);
     outargs p1, p2;

     p1 = <upper bound for P1>;
     p2 = <upper bound for P2>;
 endsub;

defines a subroutine that returns the gradient vector of the specified core function of the distribution at the specified values of the random variable and distribution parameters. The core keyword can be PDF, CDF, SDF, LOGPDF, LOGCDF, or LOGSDF.

Here is a sample structure of the function for the core function CDF of a distribution named 'FOO':

 subroutine FOO_CDFGRADIENT(x, P1, P2, grad{*});
     outargs grad;

     /* Code to compute gradient by using x, P1, and P2 */
     grad[1] = <partial derivative of CDF w.r.t. P1
                evaluated at x, P1, P2>;
     grad[2] = <partial derivative of CDF w.r.t. P2
                evaluated at x, P1, P2>;
 endsub;

defines a subroutine that returns the Hessian matrix of the specified core function of the distribution at the specified values of the random variable and distribution parameters. The core keyword can be PDF, CDF, SDF, LOGPDF, LOGCDF, or LOGSDF.

Here is a sample structure of the subroutine for the core function LOGSDF of a distribution named 'FOO':

 subroutine FOO_LOGSDFHESSIAN(x, P1, P2, hess{*});
     outargs hess;

     /* Code to compute Hessian by using x, P1, and P2 */
     hess[1] = <second order partial derivative of LOGSDF
                w.r.t. P1 evaluated at x, P1, P2>;
     hess[2] = <second order partial derivative of LOGSDF
                w.r.t. P1 and P2 evaluated at x, P1, P2>;
     hess[3] = <second order partial derivative of LOGSDF
                w.r.t. P2 evaluated at x, P1, P2>;
 endsub;