VGAM (version 0.9-4)

hzeta: Haight's Zeta Family Function

Description

Estimating the parameter of Haight's zeta distribution

Usage

hzeta(link = "loglog", ialpha = NULL, nsimEIM = 100)

Arguments

link

Parameter link function for the parameter. See Links for more choices. Here, a log-log link keeps the parameter greater than one, meaning the mean is finite.

ialpha

Optional initial value for the (positive) parameter. The default is to obtain an initial value internally. Use this argument if the default fails.

nsimEIM

See CommonVGAMffArguments for more information.

Value

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

Details

The probability function is $$f(y) = (2y-1)^{(-\alpha)} - (2y+1)^{(-\alpha)},$$ where the parameter $\alpha>0$ and $y=1,2,\ldots$. The function dhzeta computes this probability function. The mean of $Y$, which is returned as fitted values, is $(1-2^{-\alpha}) \zeta(\alpha)$ provided $\alpha > 1$, where $\zeta$ is Riemann's zeta function. The mean is a decreasing function of $\alpha$. The mean is infinite if $\alpha \leq 1$, and the variance is infinite if $\alpha \leq 2$.

References

Pages 533--4 of Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley.

See Also

Hzeta, zeta, zetaff, loglog, simulate.vlm.

Examples

alpha <- exp(exp(-0.1))  # The parameter
hdata <- data.frame(y = rhzeta(n = 1000, alpha))
fit <- vglm(y ~ 1, hzeta, data = hdata, trace = TRUE, crit = "coef")
coef(fit, matrix = TRUE)
Coef(fit)  # Useful for intercept-only models; should be same as alpha
c(with(hdata, mean(y)), head(fitted(fit), 1))
summary(fit)