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Dataplot Vol 2 Vol 1

BU3PDF

Name:
    BU3PDF (LET)
Type:
    Library Function
Purpose:
    Compute the Burr type 3 probability density function with shape parameters r and k.
Description:
    The standard Burr type 3 distribution has the following probability density function:

      f(x;r,k) = r*k*x**(r*k-1)/(1+x**k)**(r+1)    x >  0; r, k > 0

    with r and k denoting the shape parameters.

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

      f(x;r,k,loc,scale) = (1/scale)*f((x-loc)/scale;r,k,0,1)

    If X has a Burr type 12 distribution, then 1/X has a Burr type 3 distribution. For this reason, the Burr type 3 distribution is often referred to as the inverse Burr distribution.

    The Burr type 3 distribution is also sometimes referred to as the Dagum type I distribution.

Syntax:
    LET <y> = BU3PDF(<x>,<r>,<k>,<loc>,<scale>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a number, parameter, or variable;
                <y> is a variable or a parameter (depending on what <x> is) where the computed Burr type 3 pdf value is stored;
                <r> is a positive number, parameter, or variable that specifies the first shape parameter;
                <k> is a positive number, parameter, or variable that specifies the second 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 = BU3PDF(0.3,0.2,1.7)
    LET Y = BU3PDF(X,0.5,2.2,0,5)
    PLOT BU3PDF(X,2.3,1.4) FOR X = 0 0.01 5
Note:
    Burr type 3 random numbers, probability plots, and goodness of fit tests can be generated with the commands:

      LET R = <value>
      LET K = <value>
      LET Y = BURR TYPE 3 RANDOM NUMBERS FOR I = 1 1 N
      BURR TYPE 3 PROBABILITY PLOT Y
      BURR TYPE 3 PROBABILITY PLOT Y2 X2
      BURR TYPE 3 PROBABILITY PLOT Y3 XLOW XHIGH
      BURR TYPE 3 KOLMOGOROV SMIRNOV GOODNESS OF FIT Y
      BURR TYPE 3 CHI-SQUARE GOODNESS OF FIT Y2 X2
      BURR TYPE 3 CHI-SQUARE GOODNESS OF FIT Y3 XLOW XHIGH

    The following commands can be used to estimate the r and k shape parameters for the Burr type 3 distribution:

      LET R1 = <value>
      LET R2 = <value>
      LET K1 = <value>
      LET K2 = <value>
      BURR TYPE 3 PPCC PLOT Y
      BURR TYPE 3 PPCC PLOT Y2 X2
      BURR TYPE 3 PPCC PLOT Y3 XLOW XHIGH
      BURR TYPE 3 KS PLOT Y
      BURR TYPE 3 KS PLOT Y2 X2
      BURR TYPE 3 KS PLOT Y3 XLOW XHIGH

    The default values for R1, R2, K1, and K2 are 0.5, 10, 0.5, and 10, respectively.

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

    Since the Burr type 3 distribution can sometimes be heavy-tailed, the following command can be useful before using the BURR TYPE 3 KS PLOT:

      SET PPCC PLOT LOCATION SCALE BIWEIGHT

    This bases the location and scale estimates on the biweight estimates of the fitted line for the underlying probability plot. In the final probability plot, use PPA0BW and PPA1BW as the estimates of location and scale rather than PPA0 and PPA1. Since the ppcc plot is invariant to location and scale, this option does not apply.

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

Default:
    None
Synonyms:
    BURR TYPE III is a synonym for BURR TYPE 3.
Related Commands:
    BU3CDF = Compute the Burr type 3 cumulative distribution function.
    BU3PPF = Compute the Burr type 3 percent point function.
    BU2PDF = Compute the Burr type 2 probability density function.
    BU3PDF = Compute the Burr type 3 probability density function.
    BU4PDF = Compute the Burr type 4 probability density function.
    BU5PDF = Compute the Burr type 5 probability density function.
    BU6PDF = Compute the Burr type 6 probability density function.
    BU7PDF = Compute the Burr type 7 probability density function.
    BU8PDF = Compute the Burr type 8 probability density function.
    BU9PDF = Compute the Burr type 9 probability density function.
    B10PDF = Compute the Burr type 10 probability density function.
    B11PDF = Compute the Burr type 11 probability density function.
    B12PDF = Compute the Burr type 12 probability density function.
    RAYPDF = Compute the Rayleigh probability density function.
    WEIPDF = Compute the Weibull probability density function.
    EWEPDF = Compute the exponentiated Weibull probability density function.
Reference:
    Burr (1942), "Cumulative Frequency Functions", Annals of Mathematical Statistics, 13, pp. 215-232.

    Johnson, Kotz, and Balakrishnan (1994), "Contiunuous Univariate Distributions--Volume 1", Second Edition, Wiley, pp. 53-54.

    Devroye (1986), "Non-Uniform Random Variate Generation", Springer-Verlang, pp. 476-477.

Applications:
    Distributional Modeling
Implementation Date:
    2007/10
Program 1:
     
    LABEL CASE ASIS
    TITLE CASE ASIS
    TITLE OFFSET 2
    .
    MULTIPLOT 4 4
    MULTIPLOT CORNER COORDINATES 0 0 100 95
    MULTIPLOT SCALE FACTOR 4
    .
    LET RVAL = DATA 0.5  1  2  5
    LET KVAL = DATA 0.5  1  2  5
    .
    LOOP FOR IROW = 1 1 4
        LOOP FOR ICOL = 1 1 4
            LET R  = RVAL(IROW)
            LET K = KVAL(ICOL)
            TITLE R = ^r, K = ^k
            PLOT BU3PDF(X,R,K) FOR X = 0.01  0.01  5
        END OF LOOP
    END OF LOOP
    .
    END OF MULTIPLOT
    .
    JUSTIFICATION CENTER
    MOVE 50 97
    TEXT Burr Type 3 Probability Density Functions
        
    plot generated by sample program

Program 2:
     
    let r = 2.1
    let k = 1.3
    let rsav = r
    let ksav = k
    .
    let y = burr type 3 random numbers for i = 1 1 200
    let y = 10*y
    let amax = maximum y
    .
    y1label Correlation Coefficient
    x1label R
    burr type 3 ppcc plot y
    let r = shape1
    let k = shape2
    justification center
    move 50 6
    text Rhat = ^r (R = ^rsav), Khat = ^k (K = ^ksav)
    move 50 2
    text Maximum PPCC = ^maxppcc
    .
    y1label Data
    x1label Theoretical
    char x
    line bl
    burr type 3 prob plot y
    move 50 6
    text Location = ^ppa0, Scale = ^ppa1
    char bl
    line so
    .
    y1label Relative Frequency
    x1label
    relative hist y
    limits freeze
    pre-erase off
    plot bu3pdf(x,r,k,ppa0,ppa1) for x = 0.01 .01 amax
    limits 
    pre-erase on
    .
    let ksloc = ppa0
    let ksscale = ppa1
    burr type 3 kolmogorov smirnov goodness of fit y
        

    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:            BURR TYPE 3
        NUMBER OF OBSERVATIONS              =      200
      
     TEST:
     KOLMOGOROV-SMIRNOV TEST STATISTIC      =   0.3263575E-01
      
        ALPHA LEVEL         CUTOFF              CONCLUSION
                10%       0.086*              ACCEPT 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: 12/17/2007
Last updated: 12/17/2007
Please email comments on this WWW page to alan.heckert@nist.gov.