Calculates expected values of the upper parts of Poisson distributions (R.W. Payne).
||Boundary of upper part of distribution|
||Means of the distributions|
||Saves the expected values|
||Saves the cumulative upper probabilities|
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
An important use of
EUPOISSON is to provide estimates for censored Poisson observations for the
HGTOBITPOISSON procedures. Option:
The expected value for each mean is given by the sum, for
n running from
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.
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