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Dataplot Vol 2 Vol 1

RIGPDF

Name:
    RIGPDF (LET)
Type:
    Library Function
Purpose:
    Compute the reciprocal inverse Gaussian probability density function with shape parameters gamma and mu.
Description:
    The reciprocal inverse Gaussian distribution is the distribution of (1/X) when X has an inverse Gaussian distribution. It has the following probability density function:

      f(x;gamma,mu) = SQRT(gamma/(2*PI*x)) *
 EXP(-gamma*(1-mu*x)**2/(2*mu**2*x))
 x >= 0; gamma, mu > 0

    with gamma and mu denoting the shape parameters.

    The reciprocal inverse Gaussian distribution can be computed in terms of the inverse Gaussian distribution by

      RIGPDF(x;gamma,mu) = IGPDF((1/x);gamma,mu)/(x**2)

    with IGPDF denoting the probability density function of the inverse Gaussian distribution. Dataplot uses this relationship to compute the probability density function.

    The reciprocal inverse Gaussian distribution has mean (gamma + mu)/(gamma*mu) and standard deviation SQRT((gamma + 2*mu)/(gamma**2*mu)).

    The reciprocal inverse Gaussian distribution can be generalized with location and scale parameters in the usual way.

Syntax:
    LET <y> = RIGPDF(<x>,<gamma>,<mu>,<loc>,<scale>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a variable or a parameter;
                <gamma> is number or parameter that specifies the first shape parameter;
                <mu> is number or parameter that specifies the second shape parameter;
                <loc> is number or parameter that specifies the location parameter;
                <scale> is number or parameter that specifies the scale parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed reciprocal inverse Gaussian pdf value is stored;
    and where the <SUBSET/EXCEPT/FOR qualification> is optional.

    Note that the location and scale parameters are optional.

Examples:
    LET A = RIGPDF(3,2,1)
    LET A = RIGPDF(A1,2,1)
    LET X2 = RIGPDF(X1,2,3)
    PLOT RIGPDF(X,2,1.5) FOR X = 0.1 0.1 5
Note:
    Random numbers, probability plots, and Kolmogorov-Smirnov and chi-square goodness of fit tests can be generated with the commands:

      LET GAMMA = <value>
      LET MU = <value>
      LET Y = RECIPROCAL INVERSE GAUSSIAN RANDOM NUMBERS ...
        FOR I = 1 1 N
      RECIPROCAL INVERSE GAUSSIAN PROBABILITY PLOT Y
      RECIPROCAL INVERSE GAUSSIAN KOLMOGOROV-SMIRNOV ...
        GOODNESS OF FIT Y
      RECIPROCAL INVERSE GAUSSIAN CHI-SQUARE FIT Y

    The following commands can be used to generate estimates for the shape parameters of the reciprocal inverse Gaussian distribution:

      LET GAMMA1 = <value>
      LET GAMMA2 = <value>
      LET MU1 = <value>
      LET MU2 = <value>
      RECIPROCAL INVERSE GAUSSIAN PPCC PLOT Y
      RECIPROCAL INVERSE GAUSSIAN KS PLOT Y

    The default values for GAMMA1 and GAMMA2 are 0.5 and 25. The default values for MU1 and MU2 are 0.5 and 25.

Default:
    None
Synonyms:
    None
Related Commands:
    RIGCDF = Compute the reciprocal inverse Gaussian cumulative distribution function.
    RIGCHAZ = Compute the reciprocal inverse Gaussian cumulative hazard function.
    RIGHAZ = Compute the reciprocal inverse Gaussian hazard function.
    RIGPDF = Compute the reciprocal inverse Gaussian probability density function.
    RIGPPF = Compute the reciprocal inverse Gaussian percent point function.
    IGPDF = Compute the reciprocal inverse Gaussian probability density function.
    CHSPDF = Compute the chi-square probability density function.
    FPDF = Compute the F probability density function.
    NORPDF = Compute the normal probability density function.
    TPDF = Compute the t probability density function.
    WEIPDF = Compute the Weibull probability density function.
    WALPDF = Compute the Wald probability density function.
    FLPDF = Compute the fatigue life probability density function.
Reference:
    "Continuous Univariate Distributions--Volume 1", Second Edition, Johnson, Kotz, and Balakrishnan, Wiley, 1994, chapter 15.

    "Statistical Distributions", Third Edition, Evans, Hastings, and Peacock, Wiley, 2000, pp. 114-116.

Applications:
    Distributional Modeling
Implementation Date:
    1990/5: Original implementation
    2003/12: Modified to treat mu as a shape parameter instead of a location parameter.
Program:
     
    Y1LABEL Probability
    X1LABEL X
    LABEL CASE ASIS
    X1LABEL DISPLACEMENT 12
    Y1LABEL DISPLACEMENT 12
    MULTIPLOT SCALE FACTOR 2
    MULTIPLOT 2 2
    MULTIPLOT CORNER COORDINATES 0 0 100 95
    TITLE GAMMA = 2, MU = 1
    PLOT RIGPDF(X,2,1) FOR X = 0.01  0.01  5
    TITLE GAMMA = 5, MU = 1
    PLOT RIGPDF(X,5,1) FOR X = 0.01  0.01  5
    TITLE GAMMA = 2, MU = 2
    PLOT RIGPDF(X,2,2) FOR X = 0.01  0.01  5
    TITLE GAMMA = 5, MU = 2
    PLOT RIGPDF(X,5,2) FOR X = 0.01  0.01  5
    END OF MULTIPLOT
    JUSTIFICATION CENTER
    MOVE 50 97
    CASE ASIS
    TEXT Reciprocal Inverse Gaussian PDF
        
    plot generated by sample program

Date created: 7/7/2004
Last updated: 7/7/2004
Please email comments on this WWW page to alan.heckert@nist.gov.