Skellam Distribution Family Function

Estimates the two parameters of a Skellam distribution by maximum likelihood estimation.

Usage

skellam(lmu1 = "loglink", lmu2 = "loglink", imu1 = NULL, imu2 = NULL,
        nsimEIM = 100, parallel = FALSE, zero = NULL)

Arguments

lmu1, lmu2

Link functions for the $\mu_1$ and $\mu_2$ parameters. See Links for more choices and for general information.

imu1, imu2

Optional initial values for the parameters. See CommonVGAMffArguments for more information. If convergence failure occurs (this VGAM family function seems to require good initial values) try using these arguments.

nsimEIM, parallel, zero

See CommonVGAMffArguments for information. In particular, setting parallel=TRUE will constrain the two means to be equal.

Details

The Skellam distribution models the difference between two independent Poisson distributions (with means $\mu_{j}$, say). It has density function $$f(y;\mu_1,\mu_2) = \left( \frac{ \mu_1 }{\mu_2} \right)^{y/2} \, \exp(-\mu_1-\mu_2 ) \, I_{|y|}( 2 \sqrt{ \mu_1 \mu_2}) $$ where $y$ is an integer, $\mu_1 > 0$, $\mu_2 > 0$. Here, $I_v$ is the modified Bessel function of the first kind with order $v$.

The mean is $\mu_1 - \mu_2$ (returned as the fitted values), and the variance is $\mu_1 + \mu_2$. Simulated Fisher scoring is implemented.

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm and vgam.

Warning

This VGAM family function seems fragile and very sensitive to the initial values. Use very cautiously!!

References

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, 296.

Note

Numerical problems may occur for data if $\mu_1$ and/or $\mu_2$ are large.

Examples

if (FALSE) { # \dontrun{
sdata <- data.frame(x2 = runif(nn <- 1000))
sdata <- transform(sdata, mu1 = exp(1 + x2), mu2 = exp(1 + x2))
sdata <- transform(sdata, y = rskellam(nn, mu1, mu2))
fit1 <- vglm(y ~ x2, skellam, data = sdata, trace = TRUE, crit = "coef")
fit2 <- vglm(y ~ x2, skellam(parallel = TRUE), data = sdata, trace = TRUE)
coef(fit1, matrix = TRUE)
coef(fit2, matrix = TRUE)
summary(fit1)
# Likelihood ratio test for equal means:
pchisq(2 * (logLik(fit1) - logLik(fit2)),
       df = df.residual(fit2) - df.residual(fit1), lower.tail = FALSE)
lrtest(fit1, fit2)  # Alternative
} # }