with denoting the shape parameter.
This distribution can be generalized with location and scale parameters using the relation
This distribution was proposed by Dhillon as useful distribution for reliability applications since it can have increasing, decreasing, or bathtub shaped hazard functions.
where <x> is a number, parameter, or variable containing positive values;
<y> is a variable or a parameter (depending on what <x> is) where the computed exponential power cdf value is stored;
<beta> is a positive number, parameter, or variable that specifies the shape parameter;
<loc> is a number, parameter, or variable that specifies the location parameter;
<scale> is a positive number, parameter, or variable that specifies the scale parameter;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
If <loc> and <scale> are omitted, they default to 0 and 1, respectively.
LET A = PEXCDF(X1,2.5,0,10)
PLOT PEXCDF(X,2.5,0,3) FOR X = 0.1 0.1 10
This was done since ALPHA is in fact a scale parameter (in the articles listed in the References section, ALPHA is actually the reciprocal of the scale parameter).
Dhillon (1981), "Life Distributions", IEEE Transactions on Reliability, Vol. R-30, No. 5, pp. 457-459.
2007/11: Corrected the second shape parameter to be the scale parameter
LABEL CASE ASIS TITLE CASE ASIS TITLE OFFSET 2 . MULTIPLOT 2 2 MULTIPLOT CORNER COORDINATES 0 0 100 95 MULTIPLOT SCALE FACTOR 2 . LET BETA = 0.5 TITLE BETA = ^beta PLOT PEXCDF(X,BETA) FOR X = 0.01 0.01 2 . LET BETA = 1 TITLE BETA = ^beta PLOT PEXCDF(X,BETA) FOR X = 0.01 0.01 2 . LET BETA = 2 TITLE BETA = ^beta PLOT PEXCDF(X,BETA) FOR X = 0.01 0.01 2 . LET BETA = 5 TITLE BETA = ^beta PLOT PEXCDF(X,BETA) FOR X = 0.01 0.01 2 . END OF MULTIPLOT . JUSTIFICATION CENTER MOVE 50 97 TEXT Exponential Power Cumulative Distribution Functions
Date created: 11/27/2007