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, rrvglm 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} ζ (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.465--471, Chapter 11 of Johnson NL, Kotz S, and Kemp AW (1993) Univariate Discrete Distributions, 2nd ed. New York: Wiley.

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

Examples

Run this code

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

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

# MLE should satisfy the following:
mean(log(rep(y, w))) + zeta(1+phat, deriv=1)/zeta(1+phat) # Should be 0

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