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:
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:
Dataplot supports both parameterizations (see the Note section below).
Dataplot computes the cumulative distribution function using the following recurrence relation given on page 320 of Consul and Famoye:
where . If the parameterization is used, it is converted to the parameterization and the above recurrence relation is applied.
The Consul percent point function is computed by summing the cumulative distribution function until the specified probability is obtained.
where <p> is a positive integer variable, number, or parameter in the range (0,1);
<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 ppf value is stored;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.
LET Y = CONPPF(P,0.3,1.6)
PLOT CONPPF(P,0.3,1.6) FOR P = 0 0.01 0.99
To restore the THETA parameterization, enter the command
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.
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 x1label Probability y1label X . xlimits 0 1 major xtic mark number 6 minor xtic mark number 3 . multiplot 2 2 . title Theta = 0.3, M = 1.8 plot conppf(p,0.3,1.8) for p = 0 0.01 0.99 . title Theta = 0.5, M = 1.5 plot conppf(p,0.5,1.5) for p = 0 0.01 0.99 . title Theta = 0.7, M = 1.2 plot conppf(p,0.7,1.2) for p = 0 0.01 0.99 . title Theta = 0.9, M = 1.1 plot conppf(p,0.9,1.1) for p = 0 0.01 0.99 . end of multiplot . justification center move 50 97 text Percent Point Functions for Consul Distribution
Date created: 8/23/2006