recipEnzpois: First Reciprocal Moment of the Truncated Poisson Distribution

In spatstat.random: Random Generation Functionality for the 'spatstat' Family

First Reciprocal Moment of the Truncated Poisson Distribution

Description

Computes the first reciprocal moment (first negative moment) of the truncated Poisson distribution (the Poisson distribution conditioned to have a nonzero value).

Usage

  recipEnzpois(mu, exact=TRUE)

Arguments

mu	The mean of the original Poisson distribution. A single positive numeric value, or a vector of positive numbers.
exact	Logical value specifying whether to use the exact analytic formula if possible.

Details

This function calculates the expected value of 1/N given N > 0, where N is a Poisson random variable with mean \mu.

If the library gsl is loaded, and if exact=TRUE (the default), then the calculation uses the exact analytic formula

\nu = \frac{e^{-\mu}}{1- e^{-\mu}} \left( Ei(\mu) - \log \mu - \gamma \right)

(see e.g. Grab and Savage, 1954) where \nu is the desired reciprocal moment, and

Ei(x) = \int_{-\infty}^x t e^{-t} dt

is the first exponential integral, and \gamma \approx 0.577 is the Euler-Mascheroni constant.

If gsl is not loaded, or if exact=FALSE is specified, the value is computed approximately (and more slowly) by summing over the possible values of N up to a finite limit.

Value

A single numerical value or a numeric vector.

Author(s)

\adrian

.

References

Grab, E.L. and Savage, I.R. (1954) Tables of the expected value of 1/X for positive Bernoulli and Poisson variables. Journal of the American Statistical Association 49, 169–177.

See Also

rpoisnonzero

Examples

  if(require(gsl)) {
    v <- recipEnzpois(10)
    print(v)
  }
  recipEnzpois(10, exact=FALSE)

spatstat.random documentation built on Sept. 30, 2024, 9:46 a.m.