Dataplot Vol 2 Vol 1

# GL2PDF

Name:
GL2PDF (LET)
Type:
Library Function
Purpose:
Compute the type 2 generalized logistic probability density function with shape parameter .
Description:
The standard form of the type 2 generalized logistic distribution has the probability density function:

The general form of the type 2 generalized logistic probability density function can be obtained by replacing x in the above formula with (x-loc)/scale.

Syntax:
LET <y> = GL2PDF(<x>,<alpha>,<loc>,<scale>)
<SUBSET/EXCEPT/FOR qualification>
where <x> is a variable, number or parameter;
<alpha> is a number or parameter that specifies the value of the shape parameter;
<loc> is a number or parameter that specifies the value of the location parameter;
<scale> is a number or parameter that specifies the value of the scale parameter;
<y> is a variable or a parameter (depending on what <x> is) where the computed generalized logistic type 2 pdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

The location and scale parameters are optional.

Examples:
LET A = GL2PDF(3,2)
LET X2 = GL2PDF(X1,ALPHA)
PLOT GL2PDF(X,ALPHA) FOR X = -5 0.01 5
Note:
For the type 2 parameterization, a value of = 1 results in the logistic distribution. Values of < 1 result in left-skewed distributions and values of > 1 result in right-skewed distributions.
Note:
Generalized logistic type 2 random numbers, probability plots, and goodness of fit tests can be generated with the commands:

LET ALPHA = <value>
LET Y = GENERALIZED LOGISTIC TYPE 2 RANDOM NUMBERS ...
FOR I = 1 1 N
GENERALIZED LOGISTIC TYPE 2 PROBABILITY PLOT Y
GENERALIZED LOGISTIC TYPE 2 KOLMOGOROV SMIRNOV ...
GOODNESS OF FIT Y
GENERALIZED LOGISTIC TYPE 2 CHI-SQUARE ...
GOODNESS OF FIT Y

The following commands can be used to estimate the shape parameter for the generalized logistic type 2 distribution:

LET ALPHA1 = <value>
LET ALPHA2 = <value>
GENERALIZED LOGISTIC TYPE 2 PPCC PLOT Y
GENERALIZED LOGISTIC TYPE 2 KS PLOT Y

The default values for ALPHA1 and ALPHA2 are 0.1 and 10, respectively.

Bootstrap samples for these plots can be obtained with the commands

LET ALPHA1 = <value>
LET ALPHA2 = <value>
BOOTSTRAP GENERALIZED LOGISTIC TYPE 2 PLOT Y
BOOTSTRAP GENERALIZED LOGISTIC TYPE 2 KS PLOT Y
Note:
Johnson, Kotz, and Balakrishnan (see Reference section below) also define type 1, type 3, type 4, and a parameterization due to Hoskings generalized logistic distributions. These are also supported by Dataplot (see the Related Commands section below).

If a random variable, X, follows a type 2 generalized logistic distribution, then -X follows a type 1 generalized logistic distribution.

Default:
None
Synonyms:
None
Related Commands:
 GL2CDF = Compute the generalized logistic type 2 cumulative distribution function. GL2PPF = Compute the generalized logistic type 2 percent point function. GLOPDF = Compute the generalized logistic type 1 probability density function. GL3PDF = Compute the generalized logistic type 3 probability density function. GL4PDF = Compute the generalized logistic type 4 probability density function. GL5PDF = Compute the generalized logistic (Hosking parameterization) probability density function. LOGPDF = Compute the logistic probability density function. NORPDF = Compute the normal probability density function. LGNPDF = Compute the logmormal probability density function.
Reference:
"Continuous Univariate Distributions - 2", 2nd. Ed., Johnson, Kotz, and Balakrishnan, John Wiley, 1994 (pp. 140-147).
Applications:
Distributional Modeling
Implementation Date:
2006/3
Program:
```
LET A = DATA 0.5  1  2  5
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 2
LABEL CASE ASIS
TITLE CASE ASIS
TITLE DISPLACEMENT 2
X1LABEL X
Y1LABEL Probability Density
X1LABEL DISPLACEMENT 12
Y1LABEL DISPLACEMENT 15
.
LOOP FOR K = 1 1 4
LET ALPHA = A(K)
TITLE Alpha = ^ALPHA
PLOT GL2PDF(X,ALPHA) FOR X = -5  0.01  5
END OF LOOP
END OF MULTIPLOT
CASE ASIS
MOVE 50 97
JUSTIFICATION CENTER
TEXT Generalized Logistic Type 2 PDF's
```

Date created: 3/27/2006
Last updated: 3/27/2006