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Dataplot Vol 2 Vol 1

PEXHAZ

Name:
    PEXHAZ (LET)
Type:
    Library Function
Purpose:
    Compute the exponential power hazard function with shape parameter beta.
Description:
    The exponential power distribution is typically defined in terms of its hazard function:

      h(x;beta,mu,alpha) = (beta/alpha)*((x-mu)/alpha)^(beta-1)* EXP(((x-mu)/scale)^beta) x > mu; beta, alpha > 0

    with beta, alpha, and mu denoting the shape, scale, and location parameters, respectively.

    The case where mu = 0 and alpha = 1 is referred to as the standard exponential power distribution.

    This distribution was proposed by Dhillon as useful distribution for reliability applications since it can have increasing, decreasing, or bathtub shaped hazard functions.

Syntax:
    LET <y> = PEXHAZ(<x>,<beta>,<loc>,<scale>)
                            <SUBSET/EXCEPT/FOR qualification>
    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 hazard 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.

Examples:
    LET A = PEXHAZ(0.3,2.5)
    LET A = PEXHAZ(X1,2.5,0,10)
    PLOT PEXHAZ(X,2.5,0,3) FOR X = 0.1 0.1 10
Note:
    The 11/2007 version changed the syntax for this function from

      LET A = PEXHAZ(X,ALPHA,BETA,LOC,SCALE)

    to

      LET A = PEXHAZ(X,BETA,LOC,SCALE)

    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).

Default:
    None
Synonyms:
    None
Related Commands:
    PEXPDF = Compute the exponential power probability density function.
    PEXCDF = Compute the exponential power cumulative distribution function.
    PEXPPF = Compute the exponential power percent point function.
    PEXCHAZ = Compute the exponential power cumulative hazard function.
    ALPPDF = Compute the alpha probability density function.
    WEIPDF = Compute the Weibull probability density function.
    LGNPDF = Compute the log-normal probability density function.
    NORPDF = Compute the normal probability density function.
References:
    Johnson, Kotz, and Balakrishnan (1994), "Continuous Univariate Distributions--Volume 2", Second Edition, John Wiley and Sons, pp. 643-644.

    Dhillon (1981), "Life Distributions", IEEE Transactions on Reliability, Vol. R-30, No. 5, pp. 457-459.

Applications:
    Reliability, accelerated life testing
Implementation Date:
    1998/4
    2007/11: Corrected the second shape parameter to be the scale parameter
Program: 
     
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 PEXHAZ(X,BETA) FOR X = 0.01  0.01  2
.
LET BETA  = 1
TITLE BETA = ^beta
PLOT PEXHAZ(X,BETA) FOR X = 0.01  0.01  2
.
LET BETA  = 2
TITLE BETA = ^beta
PLOT PEXHAZ(X,BETA) FOR X = 0.01  0.01  2
.
LET BETA  = 5
TITLE BETA = ^beta
PLOT PEXHAZ(X,BETA) FOR X = 0.01  0.01  2
.
END OF MULTIPLOT
.
JUSTIFICATION CENTER
MOVE 50 97
TEXT Exponential Power Hazard Functions
    plot generated by sample program

Date created: 11/27/2007
Last updated: 11/27/2007
Please email comments on this WWW page to alan.heckert@nist.gov.