UTSPDF

Name:

UTSPDF (LET) Type:

Library Function Purpose:

Compute the uneven two-sided power probability density function. Description:

f(x,a,b,d,n1,n3,alpha) = [alpha*n1*n3/(alpha*(b-a)*n3+(d-b)*n1)]
*((x-a)/(b-a))**(n1-1) a <= x < b;
= [n1*n3/(alpha*(b-a)*n3+(d-b)*n1)]*((d-x)/(d-b))**(n3-1)
b <= x < d;
= 0 x < a, x >= d

where

₁

₃

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 alpha parameter is referred to as a jump paramter (It controls the size of the discontinuity at x = b. If alpha = 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

Kotz and Van Dorp show that the standard uneven two-sided power distribution can also be given as

f(x,theta,n1,n3,pi1) = pi1*(n1/theta)*(x/theta)^(n1-1)
0 <= x < theta;
= (1-pi1)*(n3/(1-theta))*((1-x)/(1-theta))^(n1-1)
theta <= x < 1

where

pi1 = alpha*theta*n3/(alpha*theta*n3 + (1-theta)*n1);
= theta*alpha*n3/(theta*(alpha*n3 -n1) + n1)

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 alpha = 1 is referred to as the generalized two-sided power distribution.

Syntax:

Examples:

Note:

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:

Beyond Beta: Other Continuous Families of Distributions with Bounded Support and Applications

Applications:

Distributional Modeling Implementation Date:

2007/10 Program:

 
LABEL CASE ASIS
TITLE CASE ASIS
TITLE OFFSET 2
.
MULTIPLOT CORNER COORDINATES 0 0 100 95
MULTIPLOT SCALE FACTOR 2
.
MULTIPLOT 2 2
LET A = 0
LET B = 0.4
LET D = 1.0
LET N1 = 2
LET N3 = 0.5
.
LET ALPHA = 0.5
TITLE N1 = ^n1, N3 = ^n3, Alpha = ^alpha
PLOT UTSPDF(X,A,B,D,N1,N3,ALPHA) FOR X = A 0.01 D
.
LET ALPHA = 1.0
TITLE N1 = ^n1, N3 = ^n3, Alpha = ^alpha
PLOT UTSPDF(X,A,B,D,N1,N3,ALPHA) FOR X = A 0.01 D
.
LET ALPHA = 2.0
TITLE N1 = ^n1, N3 = ^n3, Alpha = ^alpha
PLOT UTSPDF(X,A,B,D,N1,N3,ALPHA) FOR X = A 0.01 D
.
LET ALPHA = 5.0
TITLE N1 = ^n1, N3 = ^n3, Alpha = ^alpha
PLOT UTSPDF(X,A,B,D,N1,N3,ALPHA) FOR X = A 0.01 D
.
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Uneven Two-Sided Power Probability Density Functions
.
MULTIPLOT 2 2
LET N1 = 0.5
LET N3 = 2
.
LET ALPHA = 0.5
TITLE N1 = ^n1, N3 = ^n3, Alpha = ^alpha
PLOT UTSPDF(X,A,B,D,N1,N3,ALPHA) FOR X = A 0.01 D
.
LET ALPHA = 1.0
TITLE N1 = ^n1, N3 = ^n3, Alpha = ^alpha
PLOT UTSPDF(X,A,B,D,N1,N3,ALPHA) FOR X = A 0.01 D
.
LET ALPHA = 2.0
TITLE N1 = ^n1, N3 = ^n3, Alpha = ^alpha
PLOT UTSPDF(X,A,B,D,N1,N3,ALPHA) FOR X = A 0.01 D
.
LET ALPHA = 5.0
TITLE N1 = ^n1, N3 = ^n3, Alpha = ^alpha
PLOT UTSPDF(X,A,B,D,N1,N3,ALPHA) FOR X = A 0.01 D
.
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Uneven Two-Sided Power Probability Density Functions
.
MULTIPLOT 2 2
LET N1 = 2
LET N3 = 2
LET A2 = 0.01
.
LET ALPHA = 0.5
TITLE N1 = ^n1, N3 = ^n3, Alpha = ^alpha
PLOT UTSPDF(X,A,B,D,N1,N3,ALPHA) FOR X = A2 0.01 D
.
LET ALPHA = 1.0
TITLE N1 = ^n1, N3 = ^n3, Alpha = ^alpha
PLOT UTSPDF(X,A,B,D,N1,N3,ALPHA) FOR X = A2 0.01 D
.
LET ALPHA = 2.0
TITLE N1 = ^n1, N3 = ^n3, Alpha = ^alpha
PLOT UTSPDF(X,A,B,D,N1,N3,ALPHA) FOR X = A2 0.01 D
.
LET ALPHA = 5.0
TITLE N1 = ^n1, N3 = ^n3, Alpha = ^alpha
PLOT UTSPDF(X,A,B,D,N1,N3,ALPHA) FOR X = A2 0.01 D
.
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Uneven Two-Sided Power Probability Density Functions

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