The SIMSYSTEM Procedure
Pearson System of Distributions
The Pearson system of continuous distributions was introduced by Karl Pearson beginning in the 1890s (Pearson 1895, 1901, 1916). For an explanation of Pearson’s derivation, see Chapter 4 of Elderton and Johnson (1969) and Chapter 12 of Johnson, Kotz, and Balakrishnan (1994). Rodriguez (2006) and Ord (2006) provide overviews of the Pearson system.
Each probability density function 
in the Pearson system is a solution of the differential equation
The location, scale, and shape parameters of the density function depend on the constants a, 
, 
, and 
. Solving the equation for various combinations of these constants gives rise to different families of distributions, which Pearson designated by using roman numerals. As pointed out by Johnson, Kotz, and Balakrishnan (1994), this numbering scheme does not have a clear basis, but it is well established in the statistical literature.
Subsequent to Pearson’s work, most families in the Pearson system were re-derived in other contexts, where they received other names that are familiar in modern statistical work. For instance, the distributions in the Pearson Type III family are now commonly known as the gamma distributions. The parameters table that is produced by the SIMSYSTEM procedure identifies each Pearson distribution used for simulation by its Pearson type and its common name (if there is one). This table also provides the location, scale, and shape parameter(s) for each distribution.
There are three main types of distributions in the Pearson system:
the Type I family, known as beta distributions
the Type IV family, which has no familiar counterpart
the Type VI family, known as F distributions
Each of these three families has two shape parameters. Their regions in the 
plane complement each other and completely cover the feasible region.
The Pearson system also includes families of distributions that are limiting forms of the main types and are referred to as transition types:
the normal distribution (no associated type number)
the Type II family, known as symmetric beta distributions
the Type III family, known as gamma distributions
the Type V family, known as inverse gamma distributions
the Type VII family, known as Student’s t distributions
Finally, the Pearson system also includes families of distributions that are special cases of the more important types:
Type VIII, known as power distributions
Type IX, known as power distributions
Type X, known as exponential distributions
Type XI, known as power distributions
Type XII, a special case of Type I
These additional types are not required for defining distributions that cover the full skewness/kurtosis range, so they are not included in the PROC SIMSYSTEM results.
The following subsections provide the probability density function for each Pearson family. The parameter notation matches that of the parameters table produced by the procedure.
Pearson Normal Family
The family of normal distributions belongs to the Pearson system in the sense that it is a limiting form of each of the three main types of Pearson distributions. Pearson did not assign a roman numeral to the normal family, but it could be referred to as the "Type 0 family."
The probability density function for the normal family is
For normal distributions, 
and 
.
Pearson Type I Family
Pearson Type I probability density functions are obtained as solutions to Pearson’s differential equation when the roots of 
are real and have opposite signs.
The Type I family is composed of the beta distributions, whose probability density functions are of the form
where a is the location parameter, 
is the scale parameter, and 
and 
are shape parameters (
and 
). The range of the distribution is bounded below by a and above by b.
The standard form of the beta distribution is obtained by setting 
and 
, and the standard beta density function is
Beta density functions come in a variety of shapes:
unimodal (bell-shaped) when
and 
unimodal and symmetric when
, , and 
rectangular when
(uniform distributions) bimodal (U-shaped) when
and 
U-shaped and symmetric when
, , and J-shaped or reverse J-shaped when
is not positive
For more information about beta distributions, see Chapter 25 of Johnson, Kotz, and Balakrishnan (1995).
Pearson Type II Family
The Pearson Type II family is a limiting form of the Type I family. It is composed of the symmetric beta distributions, which have identical shape parameters 
, and its probability density functions are of the general form
where a is the location parameter, is the scale parameter, and . The range of the distribution is bounded below by a and above by b.
Setting 
and gives the particular form
which can be rewritten as
Some references present this as the standard form of the Type II density function.
For distributions in the Type II family, and 
. The uniform distribution is a special case of the Type II family.
Pearson Type III Family
The Pearson Type III family is composed of the gamma distributions, whose probability density functions are of the form
where 
is the location parameter, 
is the scale parameter, and is the shape parameter (). The range of the distribution is bounded below by .
For distributions in the Type III family,
The standard form of the gamma distribution is obtained by setting 
and 
:
Gamma density functions come in a variety of shapes:
unimodal (bell-shaped) when
reverse J-shaped and finite at
when 
(exponential distributions) reverse J-shaped and tending to infinity as y tends to 0
Within the gamma family, three subfamilies are particularly well-known by other names:
Gamma distributions where
are referred to as exponential distributions and form the Pearson Type X family. Gamma distributions for which
is an integer are referred to as Erlang distributions. Standard gamma distributions where
and 
for 
are referred to as chi-square distributions with 
degrees of freedom.
For more information about gamma distributions, see Chapter 17 of Johnson, Kotz, and Balakrishnan (1994).
Pearson Type IV Family
Pearson Type IV probability density functions are obtained as solutions to Pearson’s differential equation when the roots of are complex.
The Pearson Type IV family is composed of distributions whose probability density functions are of the form
where is the location parameter, is the scale parameter, m and are shape parameters (
), and K is the constant of integration.
Elderton and Johnson (1969, pp. 58–66) explain how to compute K and the parameters given 
, 
, 
, and 
. Johnson, Kotz, and Balakrishnan (1994, pp. 18–19) note that integrating the expression for to obtain the cumulative distribution function leads to intractable mathematics. The SIMSYSTEM procedure uses specialized numerical techniques to compute K, the distribution function, and the corresponding quantile function.
Type IV distributions are unimodal and bell-shaped. No common statistical distributions are of the Type IV form.
Pearson Type V Family
Pearson Type V probability density functions are obtained as solutions to Pearson’s differential equation when 
.
The Pearson Type V family includes the inverse gamma distributions, whose probability density functions are of the form
where is the location parameter, is the scale parameter (
), and is the shape parameter (). Inverse gamma densities are unimodal and either bell-shaped or skewed.
The Type V family is a transition family; its values of skewness and kurtosis lie on a curve in the 
plane that separates the regions of the Type IV and Type VI families. The equation for this curve is
Pearson Type VI Family
Pearson Type VI probability density functions are obtained as solutions to Pearson’s differential equation when the roots of are real and have the same sign. If the roots 
and 
are both negative and 
, then the densities have the general form
where K is a constant of integration, 
, and 
. For more information, see Johnson, Kotz, and Balakrishnan (1994, chap. 12).
Pearson Type VI probability density functions can be expressed as
where is the location parameter, is the scale parameter (), and 
and 
are shape parameters (
, 
). For and , this reduces to the standard form
which is the density function of 
, the ratio of two independent variables 
and 
with chi-square distributions whose degrees of freedom are and , respectively (Johnson, Kotz, and Balakrishnan 1995, chap. 27). Pearson Type VI distributions are also known as beta distributions of the second kind (Johnson, Kotz, and Balakrishnan 1995, chap. 25).
Pearson Type VI distributions are closely related to F distributions (Johnson, Kotz, and Balakrishnan 1995, chap. 27). The ratio
has the F distribution with degrees of freedom and . The density function of F is
Pearson Type VII Family
Pearson Type VII probability density functions are obtained as solutions to Pearson’s differential equation when 
, 
, and 
.
Pearson Type VII densities have the general form
where K is a constant of integration.
For distributions in the Type VII family, and 
.
Pearson Type VII distributions are closely related to Student’s t distributions, which have the form
Note that must be positive but does not need to be an integer. The t distributions have skewness and kurtosis 
, which exist if 
. For more information about the t family, see Johnson, Kotz, and Balakrishnan (1995, chap. 28) .
Parameter Values for Pearson Distributions
The purpose of this subsection is to explain how parameter values that the SIMSYSTEM procedure computes for the well-known distributions that underlie most of the Pearson system are related to the parameters used by the other functions in SAS to work with these distributions. These distributions are special cases of the Pearson system, and they are identified in the parameters table when the combination of skewness and kurtosis specified matches the theoretical skewness and kurtosis for these cases. The tables that are provided are helpful for understanding what the procedure is computing in each particular case, and they are also helpful if you want to reuse parameter values from the procedure to compute your own simulations in DATA step code by using the RAND function.
Table 2 summarizes the Pearson types, their corresponding standard distributions, and the mathematical form of the density for these standard distributions in terms of the parameters displayed in the "Parameters for Pearson Distributions" table. In Table 2, 
denotes the gamma function, and 
denotes the beta function. The one exception is Type IV, which is essentially a skewed version of Student’s t distribution. There is no closed form for the constant C in the Type IV density; internally, it is computed by using specialized methods.
Table 2: Pearson Distributions for 
| Type | Standard Distribution | Probability Density Function | Constant C |
|---|---|---|---|
| Type I | Beta(,) |  |  |
| Type II | Beta(,) |  |  |
| Type III | Gamma() |  |  |
| Type IV |  | No closed form | |
| Type V | Inverse gamma() |  | |
| Type VI |  |  |  |
| Type VII | Student’s  |  |  |
The precise numeric values of these parameters are available as hidden columns named Shap1Val and Shap2Val in the underlying ODS object for this table. For illustration, Table 3 shows DATA step code that is equivalent to how PROC SIMSYSTEM generates random variates for these distributions, after it determines the appropriate values of these shape parameters.
Table 3: Pearson Sampling for
| Type | Standard Distribution | Standard Random Variate |
|---|---|---|
| Type I | Beta(,) | rand(’beta’,Shap1Val,Shap2Val) |
| Type II | Beta(,) | rand(’beta’,Shap1Val,Shap1Val) |
| Type III | Gamma() | rand(’gamma’,Shap1Val) |
| Type IV | ||
| Type V | Inverse gamma() | 1/rand(’gamma’,Shap1Val) |
| Type VI | | rand(’f’,Shap1Val,Shap2Val) |
| Type VII | Student’s | rand(’t’,Shap1Val) |
Again, the standard random variate for the Type IV distribution is blank because there is no standard distribution that corresponds to this type.
In order to achieve the specified mean and standard deviation, the argument for these standard distributions is further shifted by a value and scaled by a value ; these values are displayed as the Shift and Scale columns in the "Parameters for Pearson Distributions" table. If the specified value of skewness is negative, then internally, the procedure solves for a distribution that has a mean and skewness of the opposite sign, and it adjusts the sign of and values to produce a distribution that has the specified moments.
The manner in which the distributions in the Pearson system partition the space of possible skewness/kurtosis values is shown in Figure 10.
Figure 10: Pearson Distributions
