
LPOPDFName:
If the BorelTanner distribution is shifted to start at X = 0 and is reparameterized with
the resulting distribution is referred to as the LagrangePoisson distribution (or the Consul generalized Poisson distribution). This distribution has probability mass function
with and denoting the shape parameters. Consul has studied this distribution extensively. In particular, he has studied the effect of allowing to be negative. At this time, Dataplot does not support negative values of . The moments of the LagrangePoisson distribution are
<SUBSET/EXCEPT/FOR qualification> where <x> is a positive integer variable, number, or parameter; <lambda> is a number or parameter in the range (0,1) that specifies the first shape parameter; <theta> is a positive number or parameter that specifies the second shape parameter; <y> is a variable or a parameter where the computed LagrangePoisson pdf value is stored; and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET Y = LPOPDF(X1,0.3,2) PLOT LPOPDF(X,0.3,2) FOR X = 0 1 20
LET THETA = <value> LET LAMBDA = <value> LET Y = LAGRANGE POISSON RANDOM NUMBERS FOR I = 1 1 N
LAGRANGE POISSON PROBABILITY PLOT Y To obtain the method of moments, the method of zero frequency and the mean, and the weighted discrepancies estimates of lambda and theta, enter the command
The method of moments estimates are
with and s^{2} denoting the sample mean and sample variance, respectively. The mean and zero frequency estimates are
with and f_{0} denoting the sample mean and sample frequency at x = 0, respectively. The method of weighted discrepancies (a modification of the maximum likelihood estimates) are the solution to the following equations:
with f(x) and P(x) denoting the frequency at x and the LagrangePoisson probaility mass function value at x, respectively. If you have raw data, Dataplot will automatically bin the data (you can use the CLASS LOWER, CLASS UPPER and CLASS WIDTH commands to specify the binning algorithm). You can generate estimates of lambda and theta based on the maximum ppcc value or the minimum chisquare goodness of fit with the commands
LET THETA2 = <value> LET LAMBDA1 = <value> LET LAMBDA2 = <value> LAGRANGE POISSON KS PLOT Y LAGRANGE POISSON KS PLOT Y2 X2 LAGRANGE POISSON KS PLOT Y3 XLOW XHIGH LAGRANGE POISSON PPCC PLOT Y LAGRANGE POISSON PPCC PLOT Y2 X2 LAGRANGE POISSON PPCC PLOT Y3 XLOW XHIGH The default values of lambda1 and lambda2 are 0.05 and 0.95, respectively. The default values of theta1 and theta2 are 0.5 and 10, respectively. 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 parameter. Also, since the data is integer values, one of the binned forms is preferred for these commands.
Felix Famoye and Carl M. S. Lee (1992), "Estimation of Generalized Poisson Distribution", Communications in Statistics  Simulation, 21(1), pp. 173188. P. C. Consul (1989), "Generalized Poisson Distributions", Dekker, New York.
let theta = 0.4 let lambda = 0.8 let y = lagrange poisson random numbers for i = 1 1 500 . let y3 xlow xhigh = integer frequency table y class lower 1.5 class width 1 let amax = maximum y let amax2 = amax + 0.5 class upper amax2 let y2 x2 = binned y . let k = minimum y lagrange poisson mle y relative histogram y2 x2 limits freeze preerase off line color blue plot lpopdf(x,lambdawd,thetawd) for x = 0 1 amax limits preerase on line color black let lambda = lambdawd let theta = thetawd lagrange poisson chisquare goodness of fit y3 xlow xhigh case asis justification center move 50 97 text Lambda = ^lambdawd, Theta = ^thetawd move 50 93 text Minimum ChiSquare = ^minks, 95% CV = ^cutupp95 . label case asis x1label Lambda y1label Minimum ChiSquare let theta1 = 0.1 let theta2 = 5 let lambda1 = 0.5 let lambda2 = 0.95 lagrange poisson ks plot y3 xlow xhigh let lambda = shape1 let theta = shape2 lagrange poisson chisquare goodness of fit y3 xlow xhigh case asis justification center move 50 97 text Lambda = ^lambda, Theta = ^theta move 50 93 text Minimum ChiSquare = ^minks, 95% CV = ^cutupp95CHISQUARED GOODNESSOFFIT TEST NULL HYPOTHESIS H0: DISTRIBUTION FITS THE DATA ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA DISTRIBUTION: LAGRANGEPOISSON SAMPLE: NUMBER OF OBSERVATIONS = 500 NUMBER OF NONEMPTY CELLS = 13 NUMBER OF PARAMETERS USED = 2 TEST: CHISQUARED TEST STATISTIC = 7.672152 DEGREES OF FREEDOM = 10 CHISQUARED CDF VALUE = 0.339174 ALPHA LEVEL CUTOFF CONCLUSION 10% 15.98718 ACCEPT H0 5% 18.30704 ACCEPT H0 1% 23.20925 ACCEPT H0 CELL NUMBER, LOWER BIN POINT, UPPER BIN POINT, OBSERVED FREQUENCY, AND EXPECTED FREQUENCY WRITTEN TO FILE DPST1F.DATCHISQUARED GOODNESSOFFIT TEST NULL HYPOTHESIS H0: DISTRIBUTION FITS THE DATA ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA DISTRIBUTION: LAGRANGEPOISSON SAMPLE: NUMBER OF OBSERVATIONS = 500 NUMBER OF NONEMPTY CELLS = 13 NUMBER OF PARAMETERS USED = 2 TEST: CHISQUARED TEST STATISTIC = 7.517390 DEGREES OF FREEDOM = 10 CHISQUARED CDF VALUE = 0.324138 ALPHA LEVEL CUTOFF CONCLUSION 10% 15.98718 ACCEPT H0 5% 18.30704 ACCEPT H0 1% 23.20925 ACCEPT H0 CELL NUMBER, LOWER BIN POINT, UPPER BIN POINT, OBSERVED FREQUENCY, AND EXPECTED FREQUENCY WRITTEN TO FILE DPST1F.DAT
Date created: 6/20/2006 