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# EUPOISSON procedure

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.

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```
Updated on May 26, 2023