skellam (version 0.2.4)

skellam.mle: Maximum Likelihood Estimation for the Skellam Distribution

Description

Estimates the parameters of a Skellam distribution using maximum likelihood.

Usage

skellam.mle(x)

Value

A list with components:

iters

Number of iterations required by nlm.

loglik

Maximized log-likelihood value.

param

Estimated parameters (\(\hat{\lambda}_1\), \(\hat{\lambda}_2\)).

Arguments

x

A vector of integers (positive or negative).

Author

Michail Tsagris

Details

Instead of having to maximize the log-likelihood with respect to both parameters (\(\lambda_1\) and \(\lambda_2\)), the function maximizes with respect to \(\lambda_2\) while setting \(\lambda_1 = \lambda_2 + \bar{x}\). This approach improves computational efficiency. The optimization is performed using nlm as it proved faster than optimise.

References

• Butler, R. (2007) Saddlepoint Approximations with Applications, Cambridge University Press.

• 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.

• 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), 296.

• 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

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

# Larger sample size
x1 <- rpois(10000, 10)
x2 <- rpois(10000, 6)
x <- x1 - x2
skellam.mle(x)