 Dataplot Vol 2 Vol 1

# UTSPDF

Name:
UTSPDF (LET)
Type:
Library Function
Purpose:
Compute the uneven two-sided power probability density function.
Description:
The uneven two-sided power distribution has the following probability density function: where

a <= b <= d, n1, n3, > 0

The parameters a and d are lower and upper limit parameters. The b parameter is a threshold parameter (the distribution has a discontinuity at this point). The parameter is referred to as a jump paramter (It controls the size of the discontinuity at x = b. If = 1, there is no discontinuity at x = b). The n1 and n3 parameters are shape parameters.

The case where a = 0 and d = 1 is referred to as the standard uneven two-sided power distribution. The a and d parameters are lower and upper limit parameters. These are related to location and scale parameters as follows

loc = a
scale = d - a

Kotz and Van Dorp show that the standard uneven two-sided power distribution can also be given as where = b 0 ≤ ≤ 1

Kotz and Van Dorp use this form to derive some of the properties of this distribution.

The unveven two-sided power distribution is a generalization of the two-sided power distribution. It is also related to the (the center part of the generalized trapezoid distribution shrinks to a single point). See Van Dorp and Kotz for details.

The special case where = 1 is referred to as the generalized two-sided power distribution.

Syntax:
LET <y> = UTSPDF(<x>,<a>,<b>,<d>,<n1>,<n3>,<alpha>)
<SUBSET/EXCEPT/FOR qualification>
where <x> is a variable, number, or parameter containing values in the interval (a,d);
<a> is a number, parameter, or variable that specifies the first shape parameter;
<b> is a number, parameter, or variable that specifies the second shape parameter;
<d> is a number, parameter, or variable that specifies the third shape parameter;
<n1> is a number, parameter, or variable that specifies the fourth shape parameter;
<n3> is a number, parameter, or variable that specifies the fifth shape parameter;
<alpha> is a number, parameter, or variable that specifies the sixth shape parameter;
<y> is a variable or a parameter (depending on what <x> is) where the computed pdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
LET A = UTSPDF(0.65,0,0.2,1,2,2,0.5)
LET Y = UTSPDF(X,0,0.8,1,2,2,0.5)
LET Y = UTSPDF(X,A,B,D,N1,N3,ALPHA)
Note:
Uneven two-sided power random numbers, probability plots, and goodness of fit tests can be generated with the commands:

LET A = <value>
LET B = <value>
LET D = <value>
LET N1 = <value>
LET N3 = <value>
LET ALPHA = <value>
LET Y = UNEVEN TWO-SIDED POWER RANDOM NUMBERS ...
FOR I = 1 1 N
UNEVEN TWO-SIDED POWER PROBABILITY PLOT Y
UNEVEN TWO-SIDED POWER PROBABILITY PLOT Y2 X2
UNEVEN TWO-SIDED POWER PROBABILITY PLOT Y3 XLOW XHIGH
UNEVEN TWO-SIDED POWER KOLMOGOROV SMIRNOV ...
GOODNESS OF FIT Y
UNEVEN TWO-SIDED POWER CHI-SQUARE ...
GOODNESS OF FIT Y2 X2
UNEVEN TWO-SIDED POWER CHI-SQUARE ...
GOODNESS OF FIT Y3 XLOW XHIGH

Note that

A ≤ data minimum < B < data maximum ≤ D
Default:
None
Synonyms:
None
Related Commands:
 UTSCDF = Compute the uneven two-sided power cumulative distribution function. UTSPPF = Compute the uneven two-sided power percent point function. TSPPDF = Compute the two-sided power probability density function. POWPDF = Compute the power probability density function. GTRPDF = Compute the generalized trapezoid probability density function. TSOPDF = Compute the two-sided ogive probability density function. OGIPDF = Compute the ogive probability density function. TSSPDF = Compute the two-sided slope probability density function. SLOPDF = Compute the slope probability density function. BETPDF = Compute the Beta probability density function. JSBPDF = Compute the Johnson SB probability density function.
Reference:
Kotz and Van Dorp (2004), "Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications", World Scientific Publishing Company, Chapter 6.
Applications:
Distributional Modeling
Implementation Date:
2007/10
Program:

Date created: 12/17/2007
Last updated: 12/17/2007