Calculates expected values of the upper parts of Poisson distributions (R.W. Payne).
Option
BOUND = scalar |
Boundary of upper part of distribution |
Parameters
MEANS = variates or scalars |
Means of the distributions |
EXPECTEDVALUES = variates or scalars |
Saves the expected values |
CUPROBABILITIES = variates or scalars |
Saves the cumulative upper probabilities |
Description
EUPOISSON calculates expected values of the upper parts of Poisson distributions. The calculation is for all values greater than or equal to the value specified by the BOUND option. The MEANS parameter specifies a variate containing means of the distributions, or it can be set to a scalar if there is a single distribution. The expected values can be saved in either a variate or a scalar (to match the type of MEANS) by the EXPECTEDVALUES parameter. The corresponding cumulative upper probabilities can similarly be saved by the CUPROBABILITIES parameter.
An important use of EUPOISSON is to provide estimates for censored Poisson observations for the RTOBITPOISSON, GLTOBITPOISSON and HGTOBITPOISSON procedures. Option: BOUND.
Parameters: MEANS, EXPECTEDVALUES, CLPROBABILITIES.
Method
The expected value for each mean is given by the sum, for n running from BOUND to nmax, of
n * PRPOISSON(n; mean) / CUPOISSON(BOUND-1; mean)
The upper limit nmax is taken as the highest value for which the Poisson probability is greater than or equal to 1.0 E-14. For means up to 1300, this is calculated by a quartic polynomial. A Normal approximation is used the values greater than that.
Problems arise when the cumulative upper probability becomes very low. The calculation becomes unreliable, due to numerical round-off, for values less than 1.0 E-10. A missing value is then returned for the expected value, and a zero value for the cumulative upper probability.
See also
Directive: DISTRIBUTION.
Procedures: ELPOISSON, RTOBITPOISSON, GLTOBITPOISSON, HGTOBITPOISSON.
Commands for: Basic and nonparametric statistics.
Example
CAPTION 'EUPOISSON example',\
'Expected values for means 101-120 with a lower bound of 100.';\
STYLE=meta,plain
VARIATE [VALUES=101...120] Means
EUPOISSON [BOUND=100] Means; EXPECTEDVALUES=Expected; CUPROBABILITIES=Cupr
PRINT Means,Expected,Cupr; DECIMALS=0,3,4