LPOPDF

LPOPDF (LET)

Library Function

Compute the Lagrange-Poisson probability mass function.

Given a single queue with random arrival times of customers at constant rate l, constant service time , and k initial customers, the Borel-Tanner distribution is the distribution of the total number of customers served before the queue vanishes. The distribution is parameterized with = l.

If the Borel-Tanner distribution is shifted to start at X = 0 and is reparameterized with

= k



the resulting distribution is referred to as the Lagrange-Poisson 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 Lagrange-Poisson distribution are

mean	=
variance	=
skewness	=
kurtosis	=

LET <y> = LPOPDF(<x>,<lambda>,<theta>)>
                        <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 Lagrange-Poisson pdf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

LET A = LPOPDF(3,0.5,3)
LET Y = LPOPDF(X1,0.3,2)
PLOT LPOPDF(X,0.3,2) FOR X = 0 1 20

For a number of commands utilizing the Lagrange-Poisson 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.

You can generate Lagrange-Poisson random numbers, probability plots, and chi-square goodness of fit tests with the following commands:
LET N = VALUE
LET THETA = <value>
LET LAMBDA = <value>
LET Y = LAGRANGE POISSON RANDOM NUMBERS FOR I = 1 1 N


LAGRANGE POISSON PROBABILITY PLOT Y
LAGRANGE POISSON PROBABILITY PLOT Y2 X2
LAGRANGE POISSON PROBABILITY PLOT Y3 XLOW XHIGH
LAGRANGE POISSON CHI-SQUARE GOODNESS OF FIT Y LAGRANGE POISSON CHI-SQUARE GOODNESS OF FIT Y2 X2 LAGRANGE POISSON CHI-SQUARE GOODNESS OF FIT Y3 XLOW XHIGH

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

LAGRANGE POISSON MAXIMUM LIKELIHOOD Y LAGRANGE POISSON MAXIMUM LIKELIHOOD Y2 X2

The method of moments estimates are





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

The mean and zero frequency estimates are





with and f₀ 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 Lagrange-Poisson 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 chi-square goodness of fit with the commands

LET THETA1 = <value>
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 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 parameter. Also, since the data is integer values, one of the binned forms is preferred for these commands.

None

None

LPOCDF	= Compute the Lagrange-Poisson cumulative distribution function.
LPOPPF	= Compute the Lagrange-Poisson percent point function.
BTAPDF	= Compute the Borel-Tanner probability mass function.
LOSPDF	= Compute the lost games probability mass function.
POIPDF	= Compute the Poisson probability mass function.
HERPDF	= Compute the Hermite probability mass function.
BINPDF	= Compute the binomial probability mass function.
NBPDF	= Compute the negative binomial probability mass function.
GEOPDF	= Compute the geometric 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.

Johnson, Kotz, and Kemp (1992), "Univariate Discrete Distributions", Second Edition, Wiley, pp. 394-400.

Felix Famoye and Carl M. -S. Lee (1992), "Estimation of Generalized Poisson Distribution", Communications in Statistics -- Simulation, 21(1), pp. 173-188.

P. C. Consul (1989), "Generalized Poisson Distributions", Dekker, New York.

Distributional Modeling

2006/6

 
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
pre-erase off
line color blue
plot lpopdf(x,lambdawd,thetawd) for x = 0 1 amax
limits
pre-erase on
line color black
let lambda = lambdawd
let theta = thetawd
lagrange poisson chi-square goodness of fit y3 xlow xhigh
case asis
justification center
move 50 97
text Lambda = ^lambdawd, Theta = ^thetawd
move 50 93
text Minimum Chi-Square = ^minks, 95% CV = ^cutupp95
.
label case asis
x1label Lambda
y1label Minimum Chi-Square
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 chi-square goodness of fit y3 xlow xhigh
case asis
justification center
move 50 97
text Lambda = ^lambda, Theta = ^theta
move 50 93
text Minimum Chi-Square = ^minks, 95% CV = ^cutupp95
    

    
       
    
    
                   CHI-SQUARED GOODNESS-OF-FIT TEST
  
 NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
 ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
 DISTRIBUTION:            LAGRANGE-POISSON
  
 SAMPLE:
    NUMBER OF OBSERVATIONS      =      500
    NUMBER OF NON-EMPTY CELLS   =       13
    NUMBER OF PARAMETERS USED   =        2
  
 TEST:
 CHI-SQUARED TEST STATISTIC     =    7.672152
    DEGREES OF FREEDOM          =       10
    CHI-SQUARED 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.DAT
    
    
       
    
    
                   CHI-SQUARED GOODNESS-OF-FIT TEST
  
 NULL HYPOTHESIS H0:      DISTRIBUTION FITS THE DATA
 ALTERNATE HYPOTHESIS HA: DISTRIBUTION DOES NOT FIT THE DATA
 DISTRIBUTION:            LAGRANGE-POISSON
  
 SAMPLE:
    NUMBER OF OBSERVATIONS      =      500
    NUMBER OF NON-EMPTY CELLS   =       13
    NUMBER OF PARAMETERS USED   =        2
  
 TEST:
 CHI-SQUARED TEST STATISTIC     =    7.517390
    DEGREES OF FREEDOM          =       10
    CHI-SQUARED 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