 Dataplot Vol 2 Vol 1

# PARPDF

Name:
PARPDF (LET)
Type:
Library Function
Purpose:
Compute the Pareto probability density function with shape parameters and a.
Description:
The standard form of the Pareto probability density function is: with and a denoting the tail length shape parameter and the lower bound parameter, respectively. The default value of a is 1.

Note that although the a parameter is typically called a location parameter (and it is in the sense that it defines the lower bound), it is not a location parameter in the technical sense that the following relation does not hold: For this reason, Dataplot treats a as a shape parameter. In Dataplot, the a shape parameter is optional with a default value of 1.

Syntax:
LET <y> = PARPDF(<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 = PARPDF(3,1.5)
LET A = PARPDF(3,1.5,2)
LET Y = PARPDF(X,GAMMA,A,LOC,SCALE)
Note:
The Pareto distribution can be extended with location and scale parameters using the relationship Most applications of the Pareto distribution use the standard form (i.e., location = 0 and scale = 1).

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

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

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

LET GAMMA1 = <value>
LET GAMMA2 = <value>
LET A = <value>
PARETO PPCC PLOT Y
PARETO KS PLOT Y

The default values for gamma1 and gamma2 are 0.2 and 10, respectively. Note that only the gamma parameter is estimated for these plots. The default value of A is 1. If the value of A is greater than the data minimum, then it is automatically set to the data minimum.

You can generate maximum likelihood estimates for the Pareto distribution with the command

PARETO MAXIMUM LIKELIHOOD Y

The maximum likelihood estimate of the lower bound parameter is: This estimate is used in the following equation to find the maximum likelihood estimate of the tail length parameter: Default:
None
Synonyms:
None
Related Commands:
 PARCDF = Compute the Pareto cumulative distribution function. PARCHAZ = Compute the Pareto cumulative hazard function. PARHAZ = Compute the Pareto hazard function. PARPPF = Compute the Pareto percent point 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. EXPPDF = Compute the exponential probability density function.
Reference:
"Continuous Univariate Distributions: Volume 1", Second Edition, Johnson, Kotz, and Balakrishnan, Wiley, 1994, chapter 19.
Applications:
Distributional Modeling
Implementation Date:
1994/4
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
.
TITLE Gamma = 1
PLOT PARPDF(X,1) FOR X = 1 0.1 10
TITLE Gamma = 2
PLOT PARPDF(X,2) FOR X = 1 0.1 10
TITLE Gamma = 5
PLOT PARPDF(X,5) FOR X = 1 0.1 10
TITLE Gamma = 0.5
PLOT PARPDF(X,0.5) FOR X = 1 0.1 10
END OF MULTIPLOT
.
MOVE 50 97
JUSTIFICATION CENTER
TEXT Pareto PDF Functions
``` Date created: 8/23/2006
Last updated: 8/23/2006