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Dataplot Vol 2 Vol 1

ALPPDF

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

      f(x;alpha) = NORPDF(alpha - (1/x))/{x**2*NORCDF(alpha)}     
x > 0; alpha > 0

    with alpha denoting the shape parameter and where

      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

      f(x;alpha,loc,scale) = (1/scale)*f((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> = ALPPDF(<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 pdf 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 = ALPPDF(0.3,2.5)
    LET A = ALPPDF(X1,2.5,0,10)
    PLOT ALPPDF(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 = ALPPDF(X,ALPHA,BETA,LOC,SCALE)

    to

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

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

Note:
    Alpha random numbers, probability plots, and goodness of fit tests can be generated with the commands:

      LET ALPHA = <value>
      LET Y = ALPHA RANDOM NUMBERS FOR I = 1 1 N
      ALPHA PROBABILITY PLOT Y
      ALPHA PROBABILITY PLOT Y2 X2
      ALPHA PROBABILITY PLOT Y3 XLOW XHIGH
      ALPHA KOLMOGOROV SMIRNOV GOODNESS OF FIT Y
      ALPHA CHI-SQUARE GOODNESS OF FIT Y2 X2
      ALPHA CHI-SQUARE GOODNESS OF FIT Y3 XLOW XHIGH

    The following commands can be used to estimate the alpha shape parameter for the alpha distribution:

      LET ALPHA1 = <value>
      LET ALPHA2 = <value>
      ALPHA PPCC PLOT Y
      ALPHA PPCC PLOT Y2 X2
      ALPHA PPCC PLOT Y3 XLOW XHIGH
      ALPHA KS PLOT Y
      ALPHA KS PLOT Y2 X2
      ALPHA KS PLOT Y3 XLOW XHIGH

    The default values for ALPHA1 and ALPHA2 are 0.5 and 10.

    The probability plot can then be used to estimate the location and scale (location = PPA0, scale = PPA1).

    The BOOTSTRAP DISTRIBUTION command can be used to find uncertainty intervals for the parameter estimates obtained from the ppcc plot and the ks plot methods.

Default:
    None
Synonyms:
    None
Related Commands:
    ALPCDF = Compute the alpha cumulative distribution function.
    ALPPPF = Compute the alpha percent point function.
    ALPHAZ = Compute the alpha hazard 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 1:
     
    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 ALPPDF(X,ALPHA) FOR X = 0.01  0.01  5
    .
    LET ALPHA  = 1
    TITLE ALPHA = ^alpha
    PLOT ALPPDF(X,ALPHA) FOR X = 0.01  0.01  5
    .
    LET ALPHA  = 2
    TITLE ALPHA = ^alpha
    PLOT ALPPDF(X,ALPHA) FOR X = 0.01  0.01  5
    .
    LET ALPHA  = 5
    TITLE ALPHA = ^alpha
    PLOT ALPPDF(X,ALPHA) FOR X = 0.01  0.01  5
    .
    END OF MULTIPLOT
    .
    JUSTIFICATION CENTER
    MOVE 50 97
    TEXT Alpha Probability Density Functions
        
    plot generated by sample program

Program 2:
     
    let alpha = 2.4
    let y = alpha random numbers for i = 1 1 200
    let y = 10*y
    let alphasav = alpha
    let amax = maximum y
    .
    alpha ppcc plot y
    let alpha1 = alpha - 1
    let alpha1 = max(alpha1,0.1)
    let alpha2 = alpha + 1
    y1label Correlation Coefficient
    x1label Alpha
    alpha ppcc plot y
    justification center
    move 50 6
    let alpha = shape
    text Alphahat = ^alpha (True Value: ^alphasav)
    .
    char x
    line bl
    y1label Data
    x1label Theoretical
    alpha prob plot y
    move 50 6
    text Location = ^ppa0, Scale = ^ppa1
    move 50 2
    text PPCC = ^ppcc
    char bl
    line so
    label
    .
    relative histogram y
    limits freeze
    pre-erase off
    plot alppdf(x,alpha,ppa0,ppa1) for x = 0.01 .01 amax
    limits 
    pre-erase on
    .
    let ksloc = ppa0
    let ksscale = ppa1
    alpha kolm smir goodness of fit y
        
    plot generated by sample program

    plot generated by sample program

    plot generated by sample program

    plot generated by sample program

                       KOLMOGOROV-SMIRNOV GOODNESS-OF-FIT TEST
      
     NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
     ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
     DISTRIBUTION:            ALPHA
        NUMBER OF OBSERVATIONS              =      200
      
     TEST:
     KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.9138396E-01
      
        ALPHA LEVEL         CUTOFF              CONCLUSION
                10%       0.086*              REJECT H0
                          0.085**
                 5%       0.096*              ACCEPT H0
                          0.095**
                 1%       0.115*              ACCEPT H0
                          0.114**
      
         *  - STANDARD LARGE SAMPLE APPROXIMATION  ( C/SQRT(N) )
        ** - MORE ACCURATE LARGE SAMPLE APPROXIMATION  ( C/SQRT(N + SQRT(N/10)) )
      
        

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