HPCDM Procedure

Descriptive Statistics

This section provides computational details for the descriptive statistics that are computed for each aggregate loss sample. You can also save these statistics in an OUTSUM= data set by specifying appropriate keywords in the OUTSUM statement.

This section gives specific details about the moment statistics. For more information about the methods of computing percentile statistics, see the description of the PCTLDEF= option in the UNIVARIATE procedure in the Base SAS Procedures Guide: Statistical Procedures.

Standard algorithms (Fisher 1973) are used to compute the moment statistics. The computational methods that the HPCDM procedure uses are consistent with those that other SAS procedures use for calculating descriptive statistics.

Mean

The sample mean is calculated as

y overbar equals StartFraction sigma-summation Underscript i equals 1 Overscript n Endscripts y Subscript i Baseline Over n EndFraction

where n is the size of the generated aggregate loss sample and is the ith value of the aggregate loss.

Standard Deviation

The standard deviation is calculated as

s equals StartRoot StartFraction 1 Over d EndFraction sigma-summation Underscript i equals 1 Overscript n Endscripts left-parenthesis y Subscript i Baseline minus y overbar right-parenthesis squared EndRoot

where n is the size of the generated aggregate loss sample, is the ith value of the aggregate loss, is the sample mean, and d is the divisor controlled by the VARDEF= option in the PROC HPCDM statement:

d equals StartLayout Enlarged left-brace 1st Row 1st Column n minus 1 2nd Column if VARDEF equals DF left-parenthesis default right-parenthesis 2nd Row 1st Column n 2nd Column if VARDEF equals upper N EndLayout

Skewness

The sample skewness, which measures the tendency of the deviations to be larger in one direction than in the other, is calculated as

StartFraction 1 Over d Subscript s Baseline EndFraction sigma-summation Underscript i equals 1 Overscript n Endscripts left-parenthesis StartFraction y Subscript i Baseline minus y overbar Over s EndFraction right-parenthesis cubed

where n is the size of the generated aggregate loss sample, is the ith value of the aggregate loss, is the sample mean, s is the sample standard deviation, and is the divisor controlled by the VARDEF= option in the PROC HPCDM statement:

d Subscript s Baseline equals StartLayout Enlarged left-brace 1st Row 1st Column StartFraction left-parenthesis n minus 1 right-parenthesis left-parenthesis n minus 2 right-parenthesis Over n EndFraction 2nd Column if VARDEF equals DF left-parenthesis default right-parenthesis 2nd Row 1st Column n 2nd Column if VARDEF equals upper N EndLayout

If VARDEF=DF, then n must be greater than 2.

The sample skewness can be positive or negative; it measures the asymmetry of the data distribution and estimates the theoretical skewness , where and are the second and third central moments. Observations that are normally distributed should have a skewness near zero.

Kurtosis

The sample kurtosis, which measures the heaviness of tails, is calculated as in Table 2 depending on the value that you specify in the VARDEF= option.

Table 2: Formulas for Kurtosis

VARDEF= Value	Formula
DF (default)
N

In these formulas, n is the size of the generated aggregate loss sample, is the ith value of the aggregate loss, is the sample mean, and s is the sample standard deviation. If VARDEF=DF, then n must be greater than 3.

The sample kurtosis measures the heaviness of the tails of the data distribution. It estimates the adjusted theoretical kurtosis denoted as , where and is the fourth central moment. Observations that are normally distributed should have a kurtosis near zero.

Last updated: June 19, 2025