
CONPDFName:
with 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:
^{2} = The Consul distribution is also parameterized in terms of its mean, , which results in the following probability mass function:
can be expressed in terms of : For this parameterization, the mean and variance are:
^{2} = Dataplot supports both parameterizations (see the Note section below).
<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.
LET Y = CONPDF(X,0.3,1.6) PLOT CONPDF(X,0.3,1.6) FOR X = 1 1 20
To restore the THETA parameterization, enter the command
LET THETA = <value> (or LET MU = <value>) LET M = LET Y = CONSUL RANDOM NUMBERS FOR I = 1 1 N
CONSUL PROBABILITY PLOT Y
CONSUL CHISQUARE GOODNESS OF FIT Y 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 Y2 X2 The moment estimates of and m are:
with and s^{2} denoting the sample mean and sample variance, respectively. The method of ones frequency and sample mean estimate of is:
The estimate of m is the solution of the equation:
with and n_{1} denoting the sample mean and sample frequency at x = 1, respectively. The maximum likelihood mean estimate of is:
The estimate of m is the solution of the equation:
You can generate estimates of (or ) and m based on the maximum ppcc value or the minimum chisquare goodness of fit with the commands
LET THETA2 = <value> or
LET MU2 = <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 parameterization is used, values of m that do not lie in the interval 1 ≤ m ≤ 1/ 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 chisquare 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 chisquare statistic can generate extremely large values for nonoptimal values of the shape parameters. Also, since the data is integer values, one of the binned forms is preferred for these commands.
Famoye (1997), "Generalized Geometric and Some of its Applications", Journal of Mathematical Sciences, 8, pp. 113. Consul (1990), "New Class of LocationParameter Discrete Probability Distributions and Their Chaacterizations", Communications in StatisticsTheory and Methods, 19, pp. 46534666.
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 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 PREERASE OFF LINE COLOR BLUE PLOT CONPDF(X,MUML,MML) FOR X = 1 1 AMAX LIMITS PREERASE ON LINE COLOR BLACK LET MU = MUML LET M = MML CONSUL CHISQUARE GOODNESS OF FIT Y3 XLOW XHIGH CASE ASIS JUSTIFICATION CENTER MOVE 50 97 TEXT Mu = ^MUML, M = ^MML MOVE 50 93 TEXT Minimum ChiSquare = ^STATVAL, 95% CV = ^CUTUPP95 . LABEL CASE ASIS X1LABEL Mu Y1LABEL Minimum ChiSquare CONSUL KS PLOT Y3 XLOW XHIGH LET MU = SHAPE1 LET M = SHAPE2 CONSUL CHISQUARE GOODNESS OF FIT Y3 XLOW XHIGH JUSTIFICATION CENTER MOVE 50 97 TEXT Mu = ^MU, M = ^M MOVE 50 93 TEXT Minimum ChiSquare = ^MINKS, 95% CV = ^CUTUPP95 CHISQUARED GOODNESSOFFIT 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 NONEMPTY CELLS = 17 NUMBER OF PARAMETERS USED = 2 TEST: CHISQUARED TEST STATISTIC = 12.56680 DEGREES OF FREEDOM = 14 CHISQUARED CDF VALUE = 0.439119 ALPHA LEVEL CUTOFF CONCLUSION 10% 21.06414 ACCEPT H0 5% 23.68479 ACCEPT H0 1% 29.14124 ACCEPT H0 CHISQUARED GOODNESSOFFIT 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 NONEMPTY CELLS = 17 NUMBER OF PARAMETERS USED = 2 TEST: CHISQUARED TEST STATISTIC = 12.12600 DEGREES OF FREEDOM = 14 CHISQUARED 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 