VGAM (version 1.1-9)

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 \(y=1,2,\ldots\). 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

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)