VGAM (version 1.0-4)

hzeta: Haight's Zeta Family Function

Description

Estimating the parameter of Haight's zeta distribution

Usage

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

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.

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

# NOT RUN {
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)
# }