Dataplot Vol 2 Vol 1

# PA2PDF

Name:
PA2PDF (LET)
Type:
Library Function
Purpose:
Compute the Pareto probability density function of the second kind with shape parameters and a.
Description:
The standard form of the Pareto probability density function of the second kind is:

with and a denoting the shape parameters. The a parameter is optional and has a default value of 1.

The Pareto distribution of the second kind is sometimes referred to as the Lomax distribution.

Syntax:
LET <y> = PA2PDF(<x>,<gamma>,<a>,<loc>,<scale>)
<SUBSET/EXCEPT/FOR qualification>
where <x> is a variable, a number, or a parameter;
<gamma> is a number or parameter that specifies the tail length shape parameter;
<a> is a number or parameter that specifies the optional lower bound shape parameter;
<loc> is a number or parameter that specifies the optional location parameter;
<scale> is a number or parameter that specifies the optional scale parameter;
<y> is a variable or a parameter (depending on what <x> is) where the computed Pareto pdf value is saved;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

The a, loc, and scale parameters are all optional.

Examples:
LET A = PA2PDF(3,1.5)
LET A = PA2PDF(3,1.5,0.5)
LET X2 = PA2PDF(X1,GAMMA,A)
PLOT PA2PDF(X,GAMMA,A) FOR X = 0 0.01 10
Note:
The Pareto distribution of the second kind can be extended with location and scale parameters by using the relationship

Note:
Pareto second kind random numbers, probability plots, and goodness of fit tests can be generated with the commands:

LET GAMMA = <value>
LET A = <value>
LET Y = PARETO SECOND KIND RANDOM NUMBERS FOR I = 1 1 N
PARETO SECOND KIND PROBABILITY PLOT Y
PARETO SECOND KIND PROBABILITY PLOT Y X
PARETO SECOND KIND PROBABILITY PLOT Y XLOW XHIGH
PARETO SECOND KIND KOLMOGOROV SMIRNOV GOODNESS OF FIT Y
PARETO SECOND KIND CHI-SQUARE GOODNESS OF FIT Y X
PARETO SECOND KIND CHI-SQUARE GOODNESS OF FIT Y XLOW XHIGH

The following commands can be used to estimate the shape parameters for the Pareto distribution of the second kind:

LET A = <value>
LET GAMMA1 = <value>
LET GAMMA2 = <value>
PARETO SECOND KIND PPCC PLOT Y
PARETO SECOND KIND PPCC PLOT Y X
PARETO SECOND KIND PPCC PLOT Y XLOW XHIGH
PARETO SECOND KIND KS PLOT Y
PARETO SECOND KIND KS PLOT Y X
PARETO SECOND KIND KS PLOT Y XLOW XHIGH

The default values for gamma1 and gamma2 are 0.2 and 10, respectively. Note that only the parameter is estimated for these plots. The default value of A is 1.

Note:
Johnson, Kotz, and Balakrishnan (see Reference section below) define Pareto distributions of the first, second, third, and fourth kinds. Dataplot supports Pareto distributions of the first and second kinds.
Default:
None
Synonyms:
None
Related Commands:
 PA2CDF = Compute the Pareto second kind cumulative distribution function. PA2PPF = Compute the Pareto second kind percent point function. PARPDF = Compute the Pareto probability density function. GEPPDF = Compute the generalized Pareto probability density function. EV1PDF = Compute the extreme value type I probability density function. WEIPDF = Compute the Weibull probability density function. NORPDF = Compute the normal probability density function. LGNPDF = Compute the lognormal probability density function. EXPPDF = Compute the exponential probability density function.
Reference:
"Continuous Univariate Distributions: Volume 1", 2nd. Ed., Johnson, Kotz, and Balakrishnan, John Wiley, 1994, (chapter 20).
Applications:
Distributional Modeling, Income Distributions
Implementation Date:
1995/10
Program:
```
MULTIPLOT 2 2
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 2
.
CASE ASIS
TITLE CASE ASIS
LABEL CASE ASIS
TITLE DISPLACEMENT 2
Y1LABEL DISPLACEMENT 15
X1LABEL DISPLACEMENT 12
Y1LABEL Probability Density
X1LABEL X
.
LET A = 1
TITLE Gamma = 0.1
PLOT PA2PDF(X,0.1,A) FOR X = 0.01  0.01  5
TITLE Gamma = 1
PLOT PA2PDF(X,1,A) FOR X = 0.01  0.01  5
TITLE Gamma = 2
PLOT PA2PDF(X,2,A) FOR X = 0.01  0.01  5
TITLE Gamma = 5
PLOT PA2PDF(X,5,A) FOR X = 0.01  0.01  5
END OF MULTIPLOT
.
MOVE 50 97
JUSTIFICATION CENTER
TEXT Pareto Second Kind PDF Functions
```

Date created: 8/23/2006
Last updated: 8/23/2006