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Dataplot Vol 2 Vol 1

ALPHAZ

Name:
    ALPHAZ (LET)
Type:
    Library Function
Purpose:
    Compute the alpha hazard function with shape parameter alpha.
Description:
    The alpha distribution has the following hazard function:

      h(x;alpha) = NORPDF(t)/(x**2*(NORCDF(alpha)-NORCDF(t))     
x > 0; alpha > 0

    with alpha denoting the shape parameter and where

      t = alpha - 1/x

      NORCDF = the standard normal cumulative distribution function

      NORPDF = the standard normal probability density function

    This distribution can be generalized with location and scale parameters using the relation

      h(x;alpha,loc,scale) = (1/scale)*h((x-loc)/scale;alpha,0,1)
    If Y has a normal distribution with location and scale parameters mu and sigma truncated to the left of 0, then X = 1/Y has an alpha distribution with shape parameter alpha = mu/sigma and scale parameter sigma.

    This distribution has application in reliability.

Syntax:
    LET <y> = ALPHAZ(<x>,<alpha>,<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 alpha hazard value is stored;
                <alpha> 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 = ALPHAZ(0.3,2.5)
    LET A = ALPHAZ(X1,2.5,0,10)
    PLOT ALPHAZ(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 = ALPHAZ(X,ALPHA,BETA,LOC,SCALE)

    to

      LET A = ALPHAZ(X,ALPHA,LOC,SCALE)

    This was done since BETA is in fact a scale parameter.

Default:
    None
Synonyms:
    None
Related Commands:
    ALPPDF = Compute the alpha probability density function.
    ALPCDF = Compute the alpha cumulative distribution function.
    ALPPPF = Compute the alpha percent point function.
    ALPCHAZ = Compute the alpha cumulative hazard function.
    PEXPDF = Compute the exponential power 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.
    NORCDF = Compute the normal cumulative distribution function.
Reference:
    Johnson, Kotz, and Balakrishnan (1994), "Continuous Univariate Distributions--Volume 1", Second Edition, John Wiley and Sons, p. 173.

    Salvia (1985), "Reliability applications of the Alpha Distribution", IEEE Transactions on Reliability, Vol. R-34, No. 3, pp. 251-252.

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 ALPHA  = 0.5
    TITLE ALPHA = ^alpha
    PLOT ALPHAZ(X,ALPHA) FOR X = 0.01  0.01  5
    .
    LET ALPHA  = 1
    TITLE ALPHA = ^alpha
    PLOT ALPHAZ(X,ALPHA) FOR X = 0.01  0.01  5
    .
    LET ALPHA  = 2
    TITLE ALPHA = ^alpha
    PLOT ALPHAZ(X,ALPHA) FOR X = 0.01  0.01  5
    .
    LET ALPHA  = 5
    TITLE ALPHA = ^alpha
    PLOT ALPHAZ(X,ALPHA) FOR X = 0.01  0.01  5
    .
    END OF MULTIPLOT
    .
    JUSTIFICATION CENTER
    MOVE 50 97
    TEXT Alpha 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.