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Dataplot Vol 2 Vol 1

CONPDF

Name:
    CONPDF (LET)
Type:
    Library Function
Purpose:
    Compute the Consul probability mass function.
Description:
    The Consul distribution has the following probability mass function:

      p(x;theta,m) = (1/x)*(m*x x-1)*theta**(x-1)*(1-theta)**(m*x-x+1) x = 1, ...; 0 < theta < 1; 1 <= m <= 1/theta

    with theta and m denoting the shape parameters. The case where m = 1 reduces to the geometric distribution. For this reason, the Consul distribution is also referred to as the generalized geometric distribution.

    The mean and variance of the Consul distribution are:

      mu = 1/(1 - theta*m)

      sigma2 = m*theta*(1 - theta)/(1 - m*theta)**3

    The Consul distribution is also parameterized in terms of its mean, mu, which results in the following probability mass function:

      p(x;mu,m) = (1/x)*(m*x x-1)* ((mu-1)/(m*mu))**(x-1)*(1-(mu-1)/(m*mu))**(m*x-x+1) x = 1, ...; mu > 1; m >= 1

    theta can be expressed in terms of mu:

      theta = (mu-1)/(m*mu)

    For this parameterization, the mean and variance are:

      mu

      sigma2 = mu*(mu-1)*(m*mu-mu+1)/m

    Dataplot supports both parameterizations (see the Note section below).

Syntax:
    LET <y> = CONPDF(<x>,<shape>,<m>)
                            <SUBSET/EXCEPT/FOR qualification>
    where <x> is a positive integer variable, number, or parameter;
                <shape> is a number, parameter, or variable that specifies the valuie of theta (or mu);
                <m> is a number, parameter, or variable that specifies the second shape parameter;
                <y> is a variable or a parameter (depending on what <x> is) where the computed Consul pdf value is stored;
                and where the <SUBSET/EXCEPT/FOR qualification> is optional.
Examples:
    LET A = CONPDF(3,0.5,1.4)
    LET Y = CONPDF(X,0.3,1.6)
    PLOT CONPDF(X,0.3,1.6) FOR X = 1 1 20
Note:
    For a number of commands utilizing the Consul distribution, it is convenient to bin the data. There are two basic ways of binning the data.

    1. For some commands (histograms, maximum likelihood estimation), bins with equal size widths are required. This can be accomplished with the following commands:

        LET AMIN = MINIMUM Y
        LET AMAX = MAXIMUM Y
        LET AMIN2 = AMIN - 0.5
        LET AMAX2 = AMAX + 0.5
        CLASS MINIMUM AMIN2
        CLASS MAXIMUM AMAX2
        CLASS WIDTH 1
        LET Y2 X2 = BINNED

    2. For some commands, unequal width bins may be helpful. In particular, for the chi-square goodness of fit, it is typically recommended that the minimum class frequency be at least 5. In this case, it may be helpful to combine small frequencies in the tails. Unequal class width bins can be created with the commands

        LET MINSIZE = <value>
        LET Y3 XLOW XHIGH = INTEGER FREQUENCY TABLE Y

      If you already have equal width bins data, you can use the commands

        LET MINSIZE = <value>
        LET Y3 XLOW XHIGH = COMBINE FREQUENCY TABLE Y2 X2

      The MINSIZE parameter defines the minimum class frequency. The default value is 5.

Note:
    To use the MU parameterization, enter the command

      SET CONSUL DEFINITION MU

    To restore the THETA parameterization, enter the command

      SET CONSUL DEFINITION THETA
Note:
    You can generate Consul random numbers, probability plots, and chi-square goodness of fit tests with the following commands:

      LET N = VALUE
      LET THETA = <value> (or LET MU = <value>)
      LET M =
      LET Y = CONSUL RANDOM NUMBERS FOR I = 1 1 N

      CONSUL PROBABILITY PLOT Y
      CONSUL PROBABILITY PLOT Y2 X2
      CONSUL PROBABILITY PLOT Y3 XLOW XHIGH

      CONSUL CHI-SQUARE GOODNESS OF FIT Y
      CONSUL CHI-SQUARE GOODNESS OF FIT Y2 X2
      CONSUL CHI-SQUARE GOODNESS OF FIT Y3 XLOW XHIGH

    To obtain the method of moment estimates, the mean and ones frequency estimates, and the maximum likelihood estimates of mu and beta, enter the command

      CONSUL MAXIMUM LIKELIHOOD Y
      CONSUL MAXIMUM LIKELIHOOD Y2 X2

    The moment estimates of mu and m are:

      muhat = xbar

      mhat = xbar*(xbar-1)^2/[xbar^2*(xbar-1) - s^2]

    with xbar and s2 denoting the sample mean and sample variance, respectively.

    The method of ones frequency and sample mean estimate of mu is:

      muhat = xbar

    The estimate of m is the solution of the equation:

      m*LOG(1 - (xbar - 1)/(m*xbar)) - LOG(n(1)/n)

    with xbar and n1 denoting the sample mean and sample frequency at x = 1, respectively.

    The maximum likelihood mean estimate of mu is:

      muhat = xbar

    The estimate of m is the solution of the equation:

      (m*xbar - xbar + 1)/(m*xbar) - EXP{-((1/(n*xbar))*SUM[x-2 to k][SUM[i=0 to x-2][x*n(x)/(m*x - i)]]}

    You can generate estimates of theta (or mu) and m based on the maximum ppcc value or the minimum chi-square goodness of fit with the commands

      LET THETA1 = <value>
      LET THETA2 = <value>

    or

      LET MU1 = <value>
      LET MU2 = <value>

      LET M1 = <value>
      LET M2 = <value>
      CONSUL KS PLOT Y
      CONSUL KS PLOT Y2 X2
      CONSUL KS PLOT Y3 XLOW XHIGH
      CONSUL PPCC PLOT Y
      CONSUL PPCC PLOT Y2 X2
      CONSUL PPCC PLOT Y3 XLOW XHIGH

    The default values of theta1 and theta2 are 0.05 and 0.95, respectively. The default values for mu1 and mu2 are 1 and 5, respectively. The default values for m1 and m2 are 1.05 and 5, respectively. Note that when the theta parameterization is used, values of m that do not lie in the interval 1 ≤ m ≤ 1/theta are skipped.

    Due to the discrete nature of the percent point function for discrete distributions, the ppcc plot will not be smooth. For that reason, if there is sufficient sample size the KS PLOT (i.e., the minimum chi-square value) is typically preferred. However, it may sometimes be useful to perform one iteration of the PPCC PLOT to obtain a rough idea of an appropriate neighborhood for the shape parameters since the minimum chi-square statistic can generate extremely large values for non-optimal values of the shape parameters. Also, since the data is integer values, one of the binned forms is preferred for these commands.

Default:
    None
Synonyms:
    None
Related Commands:
    CONCDF = Compute the Consul cumulative distribution function.
    CONPPF = Compute the Consul percent point function.
    GEOPDF = Compute the geometric probability mass function.
    GETPDF = Compute the Geeta probability mass function.
    GLSPDF = Compute the generalized logarithmic series probability mass function.
    DLGPDF = Compute the logarithmic series probability mass function.
    YULPDF = Compute the Yule probability mass function.
    ZETPDF = Compute the Zeta probability mass function.
    BGEPDF = Compute the beta geometric probability mass function.
    POIPDF = Compute the Poisson probability mass function.
    BINPDF = Compute the binomial probability mass function.
    INTEGER FREQUENCY TABLE = Generate a frequency table at integer values with unequal bins.
    COMBINE FREQUENCY TABLE = Convert an equal width frequency table to an unequal width frequency table.
    KS PLOT = Generate a minimum chi-square plot.
    MAXIMUM LIKELIHOOD = Perform maximum likelihood estimation for a distribution.
Reference:
    Consul and Famoye (2006), "Lagrangian Probability Distribution", Birkhauser, chapter 8.

    Famoye (1997), "Generalized Geometric and Some of its Applications", Journal of Mathematical Sciences, 8, pp. 1-13.

    Consul (1990), "New Class of Location-Parameter Discrete Probability Distributions and Their Chaacterizations", Communications in Statistics--Theory and Methods, 19, pp. 4653-4666.

Applications:
    Distributional Modeling
Implementation Date:
    2006/8
Program 1: 
     
set consul definition theta
title size 3
tic label size 3
label size 3
legend size 3
height 3
x1label displacement 12
y1label displacement 15
.
multiplot corner coordinates 0 0 100 95
multiplot scale factor 2
label case asis
title case asis
case asis
tic offset units screen
tic offset 3 3
title displacement 2
y1label Probability Mass
x1label X
.
ylimits 0 1
major ytic mark number 6
minor ytic mark number 3
xlimits 0 20
line blank
spike on
.
multiplot 2 2
.
title Theta = 0.3, M = 1.8
plot conpdf(x,0.3,1.8) for x = 1 1 20
.
title Theta = 0.5, M = 1.5
plot conpdf(x,0.5,1.5) for x = 1 1 20
.
title Theta = 0.7, M = 1.2
plot conpdf(x,0.7,1.2) for x = 1 1 20
.
title Theta = 0.9, M = 1.1
plot conpdf(x,0.9,1.1) for x = 1 1 20
.
end of multiplot
.
justification center
move 50 97
text Probability Mass Functions for Consul Distribution
    plot generated by sample program

Program 2: 
     
SET CONSUL DEFINITION MU
LET MU   = 4.2
LET M = 2.2
LET Y = CONSUL RANDOM NUMBERS FOR I = 1 1 500
.
LET Y3 XLOW XHIGH = INTEGER FREQUENCY TABLE Y
CLASS LOWER 0.5
CLASS WIDTH 1
LET AMAX = MAXIMUM Y
LET AMAX2 = AMAX + 0.5
CLASS UPPER AMAX2
LET Y2 X2 = BINNED Y
.
CONSUL MLE Y
RELATIVE HISTOGRAM Y2 X2
LIMITS FREEZE
PRE-ERASE OFF
LINE COLOR BLUE
PLOT CONPDF(X,MUML,MML) FOR X = 1  1  AMAX
LIMITS
PRE-ERASE ON
LINE COLOR BLACK
LET MU    = MUML
LET M  = MML
CONSUL CHI-SQUARE GOODNESS OF FIT Y3 XLOW XHIGH
CASE ASIS
JUSTIFICATION CENTER
MOVE 50 97
TEXT Mu = ^MUML, M = ^MML
MOVE 50 93
TEXT Minimum Chi-Square = ^STATVAL, 95% CV = ^CUTUPP95
.
LABEL CASE ASIS
X1LABEL Mu
Y1LABEL Minimum Chi-Square
CONSUL KS PLOT Y3 XLOW XHIGH
LET MU = SHAPE1
LET M  = SHAPE2
CONSUL CHI-SQUARE GOODNESS OF FIT Y3 XLOW XHIGH
JUSTIFICATION CENTER
MOVE 50 97
TEXT Mu = ^MU, M = ^M
MOVE 50 93
TEXT Minimum Chi-Square = ^MINKS, 95% CV = ^CUTUPP95
    plot generated by sample program 
                       CHI-SQUARED GOODNESS-OF-FIT TEST
  
 NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
 ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
 DISTRIBUTION:            CONSUL (GENERALIZED GEOMETRIC)
  
 SAMPLE:
    NUMBER OF OBSERVATIONS      =      500
    NUMBER OF NON-EMPTY CELLS   =       17
    NUMBER OF PARAMETERS USED   =        2
  
 TEST:
 CHI-SQUARED TEST STATISTIC     =    12.56680
    DEGREES OF FREEDOM          =       14
    CHI-SQUARED CDF VALUE       =    0.439119
  
    ALPHA LEVEL         CUTOFF              CONCLUSION
            10%       21.06414               ACCEPT H0
             5%       23.68479               ACCEPT H0
             1%       29.14124               ACCEPT H0
    plot generated by sample program 
                       CHI-SQUARED GOODNESS-OF-FIT TEST
  
 NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
 ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
 DISTRIBUTION:            CONSUL (GENERALIZED GEOMETRIC)
  
 SAMPLE:
    NUMBER OF OBSERVATIONS      =      500
    NUMBER OF NON-EMPTY CELLS   =       17
    NUMBER OF PARAMETERS USED   =        2
  
 TEST:
 CHI-SQUARED TEST STATISTIC     =    12.12600
    DEGREES OF FREEDOM          =       14
    CHI-SQUARED CDF VALUE       =    0.403815
  
    ALPHA LEVEL         CUTOFF              CONCLUSION
            10%       21.06414               ACCEPT H0
             5%       23.68479               ACCEPT H0
             1%       29.14124               ACCEPT H0

Date created: 8/23/2006
Last updated: 8/23/2006
Please email comments on this WWW page to alan.heckert@nist.gov.