skellam (version 0.2.3)

skellam.mle: MLE of the Skellam distribution

Description

MLE of the Skellam distribution.

Usage

skellam.mle(x)

Value

A list including:

iters

The number of iterations required by "nlm".

loglik

The maximised log-likelihood value.

param

The estimated parameters, \(\hat{\lambda}_1\) and \(\hat{\lambda}_2\).

Arguments

x

A vector of integers, positive or negative.

Author

Michail Tsagris

Details

Instead of having to maximise the log-likelihood with respect to the two parameters, \(\lambda_1\) and \(\lambda_2\), we maximise with respect to \(\lambda_2\) and then \(\lambda_1 = \lambda_2 + \bar{x}\). This makes it faster. The command "nlm" is used to optimise the log-likelihood as it proved to be faster than the "optimise".

References

Butler, R. (2007) Saddlepoint Approximations with Applications, Cambridge University Press, Cambridge & New York, p.17.

Johnson, N. L. (1959) On an extension of the connection between Poisson and \(chi^2\) distributions. Biometrika 46, 352--362.

Johnson, N. L.; Kotz, S.; Kemp, A. W. (1993) Univariate Discrete Distributions, 2nd ed., John Wiley and Sons, New York, pp.190-192.

Skellam, J. G. (1946) The frequency distribution of the difference between two Poisson variates belonging to different populations. Journal of the Royal Statistical Society, series A 109/3, 26.

Strackee, J.; van der Gon, J. J. D. (1962) The frequency distribution of the difference between two Poisson variates. Statistica Neerlandica 16/1, 17-23.

Abdulhamid, A. A.; Maha, A. O. (2010) On The Poisson Difference Distribution Inference and Applications. BULLETIN of the Malaysian Mathematical Sciences Society, 33/1, 17--45.

Wikipedia. Skellam distribution https://en.wikipedia.org/wiki/Skellam_distribution

Examples

require('skellam')

x1 <- rpois(1000, 10)
x2 <- rpois(1000, 6)
x <- x1 - x2
skellam.mle(x)

x1 <- rpois(10000, 10)
x2 <- rpois(10000, 6)
x <- x1 - x2
skellam.mle(x)