
GETPDFName:
with and denoting the shape parameters. The mean and variance of the Geeta distribution are:
^{2} = The Geeta distribution is sometimes parameterized in terms of its mean, , instead of . This results in the following probability mass function:
For this parameterization, the variance is
This probability mass function is also given in the form:
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); <beta> 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 Geeta pdf value is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET Y = GETPDF(X,0.3,1.6) PLOT GETPDF(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 BETA = <value> LET Y = GEETA RANDOM NUMBERS FOR I = 1 1 N
GEETA PROBABILITY PLOT Y
GEETA CHISQUARE GOODNESS OF FIT Y To obtain the method of moment estimates, the mean and ones frequency estimates, and the maximum likelihood estimates of and , enter the command
GEETA MAXIMUM LIKELIHOOD Y2 X2 The method of moments estimators and are:
with and s^{2} denoting the sample mean and sample variance, respectively. The method of ones frequency and sample mean estimate of :
The estimate of is the solution of the equation:
with and n_{1} denoting the sample mean and sample frequency at x = 1, respectively. The maximum likelihood estimate of is:
The estimate of is the solution of the equation:
You can generate estimates of theta (or mu) and beta based on the maximum ppcc value or the minimum chisquare goodness of fit with the commands
LET THETA2 = <value> or
LET MU2 = <value>
LET BETA2 = <value> GEETA KS PLOT Y GEETA KS PLOT Y2 X2 GEETA KS PLOT Y3 XLOW XHIGH GEETA PPCC PLOT Y GEETA PPCC PLOT Y2 X2 GEETA 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 beta1 and beta2 are 1.05 and 5, respectively. Note that when the theta parameterization is used, values of beta that do not lie in the interval 1 ≤ ≤ 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.
Consul (1990), "Geeta Distribution and its Properties", Communications in StatisticsTheory and Methods, 19, pp. 30513068.
set geeta 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, Beta = 1.8 plot getpdf(x,0.3,1.8) for x = 1 1 20 . title Theta = 0.5, Beta = 1.5 plot getpdf(x,0.5,1.5) for x = 1 1 20 . title Theta = 0.7, Beta = 1.2 plot getpdf(x,0.7,1.2) for x = 1 1 20 . title Theta = 0.9, Beta = 1.1 plot getpdf(x,0.9,1.1) for x = 1 1 20 . end of multiplot . justification center move 50 97 text Probability Mass Functions for Geeta Distribution Program 2: SET GEETA DEFINITION MU LET MU = 4.2 LET BETA = 2.2 LET Y = GEETA 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 . GEETA MLE Y RELATIVE HISTOGRAM Y2 X2 LIMITS FREEZE PREERASE OFF LINE COLOR BLUE PLOT GETPDF(X,MUML,BETAML) FOR X = 1 1 AMAX LIMITS PREERASE ON LINE COLOR BLACK LET MU = MUML LET BETA = BETAML GEETA CHISQUARE GOODNESS OF FIT Y3 XLOW XHIGH CASE ASIS JUSTIFICATION CENTER MOVE 50 97 TEXT Mu = ^MUML, Beta = ^BETAML MOVE 50 93 TEXT Minimum ChiSquare = ^STATVAL, 95% CV = ^CUTUPP95 . LABEL CASE ASIS X1LABEL Mu Y1LABLE Minimum ChiSquare GEETA KS PLOT Y3 XLOW XHIGH LET MU = SHAPE1 LET BETA = SHAPE2 GEETA CHISQUARE GOODNESS OF FIT Y3 XLOW XHIGH JUSTIFICATION CENTER MOVE 50 97 TEXT Mu = ^MU, Beta = ^Beta 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: GEETA SAMPLE: NUMBER OF OBSERVATIONS = 500 NUMBER OF NONEMPTY CELLS = 16 NUMBER OF PARAMETERS USED = 2 TEST: CHISQUARED TEST STATISTIC = 11.04658 DEGREES OF FREEDOM = 13 CHISQUARED CDF VALUE = 0.393085 ALPHA LEVEL CUTOFF CONCLUSION 10% 19.81193 ACCEPT H0 5% 22.36203 ACCEPT H0 1% 27.68825 ACCEPT H0 CHISQUARED GOODNESSOFFIT TEST NULL HYPOTHESIS H0: DISTRIBUTION FITS THE DATA ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA DISTRIBUTION: GEETA SAMPLE: NUMBER OF OBSERVATIONS = 500 NUMBER OF NONEMPTY CELLS = 16 NUMBER OF PARAMETERS USED = 2 TEST: CHISQUARED TEST STATISTIC = 9.623067 DEGREES OF FREEDOM = 13 CHISQUARED CDF VALUE = 0.275571 ALPHA LEVEL CUTOFF CONCLUSION 10% 19.81193 ACCEPT H0 5% 22.36203 ACCEPT H0 1% 27.68825 ACCEPT H0
Date created: 8/23/2006 