VGAM (version 1.1-13)

hzeta: Haight's Zeta Family Function

Description

Estimating the parameter of Haight's zeta distribution

Usage

hzeta(lshape = "logloglink", ishape = NULL, nsimEIM = 100)

Value

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

and vgam.

Arguments

lshape

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

ishape,nsimEIM

See CommonVGAMffArguments for more information.

Author

T. W. Yee

Details

The probability function is $$f(y)=(2y-1{)}^{(-\alpha )}-(2y+1{)}^{(-\alpha )},$$ where the parameter $\alpha >0$ and $y=1,2,\dots$. 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 $\alpha \le 1$, and the variance is infinite if $\alpha \le 2$.

References

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

See Also

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

Examples

shape <- exp(exp(-0.1))  # The parameter
hdata <- data.frame(y = rhzeta(n = 1000, shape))
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 shape
c(with(hdata, mean(y)), head(fitted(fit), 1))
summary(fit)