Parameter Initialization for Predefined Distribution Models

The definition of each distribution model also contains a name_PARMINIT subroutine to initialize the parameters. The parameters are initialized by using the method of moments for all the distributions, except for the gamma and the Weibull distributions. For the gamma distribution, approximate maximum likelihood estimates are used. For the Weibull distribution, the method of percentile matching is used.

Given observations of the severity value (), the estimate of th raw moment is denoted by and computed as

The 100th percentile is denoted by (). By definition, satisfies

where . PROC SEVERITY uses the following practical method of computing . Let denote the empirical distribution function (EDF) estimate at a severity value . This estimate is computed by PROC SEVERITY and supplied to the name_PARMINIT subroutine. Let and denote two consecutive values in the array of values such that and . Then, the estimate is computed as

where and .

Let denote the smallest double-precision floating-point number such that . This machine precision constant can be obtained by using the CONSTANT function in Base SAS software.

The details of how parameters are initialized for each predefined distribution model are as follows:

BURR

The parameters are initialized by using the method of moments. The th raw moment of the Burr distribution is:

Three moment equations () need to be solved for initializing the three parameters of the distribution. In order to get an approximate closed form solution, the second shape parameter is initialized to a value of . If , then simplifying and solving the moment equations yields the following feasible set of initial values:

If , then the parameters are initialized as follows:

EXP

The parameters are initialized by using the method of moments. The th raw moment of the exponential distribution is:

Solving yields the initial value of .

GAMMA

The parameter is initialized by using its approximate maximum likelihood (ML) estimate. For a set of iid observations (), drawn from a gamma distribution, the log likelihood, , is defined as follows:

Using a shorter notation of to denote and solving the equation yields the following ML estimate of :

Substituting this estimate in the expression of and simplifying gives

Let be defined as follows:

Solving the equation yields the following expression in terms of the digamma function, :

The digamma function can be approximated as follows:

This approximation is within 1.4% of the true value for all the values of except when is arbitrarily close to the positive root of the digamma function (which is approximately 1.461632). Even for the values of that are close to the positive root, the absolute error between true and approximate values is still acceptable ( for ). Solving the equation that arises from this approximation yields the following estimate of :

If this approximate ML estimate is infeasible, then the method of moments is used. The th raw moment of the gamma distribution is:

Solving and yields the following initial value for :

If (almost zero sample variance), then is initialized as follows:

After computing the estimate of , the estimate of is computed as follows:

Both the maximum likelihood method and the method of moments arrive at the same relationship between and .

GPD

The parameters are initialized by using the method of moments. Notice that for , the CDF of the generalized Pareto distribution (GPD) is:

This is equivalent to a Pareto distribution with scale parameter and shape parameter . Using this relationship, the parameter initialization method used for the PARETO distribution model is used to get the following initial values for the parameters of the GPD distribution model:

If (almost zero sample variance) or , then the parameters are initialized as follows:

IGAUSS

The parameters are initialized by using the method of moments. Note that the standard parameterization of the inverse Gaussian distribution (also known as the Wald distribution), in terms of the location parameter and shape parameter , is as follows (Klugman, Panjer, and Willmot 1998, p. 583):

For this parameterization, it is known that the mean is and the variance is , which yields the second raw moment as (computed by using ).

The predefined IGAUSS distribution model in PROC SEVERITY uses the following alternate parameterization to allow the distribution to have a scale parameter, :

The parameters (scale) and (shape) of this alternate form are related to the parameters and of the preceding form such that and . Using this relationship, the first and second raw moments of the IGAUSS distribution are:

Solving and yields the following initial values:

If (almost zero sample variance), then the parameters are initialized as follows:

LOGN

The parameters are initialized by using the method of moments. The th raw moment of the lognormal distribution is:

Solving and yields the following initial values:

PARETO

The parameters are initialized by using the method of moments. The th raw moment of the Pareto distribution is:

Solving and yields the following initial values:

If (almost zero sample variance) or , then the parameters are initialized as follows:

WEIBULL

The parameters are initialized by using the percentile matching method. Let and denote the estimates of the th and th percentiles, respectively. Using the formula for the CDF of Weibull distribution, they can be written as

Simplifying and solving these two equations yields the following initial values:

where . These initial values agree with those suggested in Klugman, Panjer, and Willmot (1998).

A summary of the initial values of all the parameters for all the predefined distributions is given in Table 22.4. The table also provides the names of the parameters to use in the INIT= option in the DIST statement if you want to provide a different initial value.