VGAM (version 0.8-1)

zetaff: Zeta Distribution Family Function

Description

Estimates the parameter of the zeta distribution.

Usage

zetaff(link = "loge", earg=list(), init.p = NULL)

Arguments

link

Parameter link function applied to the (positive) parameter $p$. See Links for more choices. Choosing loglog constrains $p>1$, but may fail if the maximum

earg

List. Extra argument for the link. See earg in Links for general information.

init.p

Optional initial value for the parameter $p$. The default is to choose an initial value internally. If converge failure occurs use this argument to input a value.

Value

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

Details

In this long tailed distribution the response must be a positive integer. The probability function for a response $Y$ is $$P(Y=y) = 1/[y^{p+1} \zeta(p+1)],\ \ \ p>0,\ \ \ y=1,2,...$$ where $\zeta$ is Riemann's zeta function. The parameter $p$ is positive, therefore a log link is the default. The mean of $Y$ is $\mu = \zeta(p) / \zeta(p+1)$ provided $p>1$. The variance of $Y$ is $\zeta(p-1) / \zeta(p+1) - \mu^2$ provided $p>2$.

It appears that good initial values are needed for successful convergence. If convergence is not obtained, try several values ranging from values near 0 to values about 10 or more.

References

pp.527-- of Chapter 11 of Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley.

Knight, K. (2000) Mathematical Statistics. Boca Raton: Chapman & Hall/CRC Press.

See Also

zeta, dzeta, hzeta, zipf.

Examples

zdata = data.frame(y = 1:5, w =  c(63, 14, 5, 1, 2)) # Knight, p.304
fit = vglm(y ~ 1, zetaff, zdata, trace=TRUE, wei=w, crit="c")
(phat = Coef(fit)) # 1.682557
with(zdata, cbind(round(dzeta(y, phat) * sum(w), 1), w))

with(zdata, weighted.mean(y, w))
fitted(fit, mat=FALSE)
predict(fit)

# The following should be zero at the MLE:
with(zdata, mean(log(rep(y, w))) + zeta(1+phat, deriv=1)/zeta(1+phat))