hzeta {VGAM}R Documentation

Haight's Zeta Family Function

Description

Estimating the parameter of Haight's Zeta function.

Usage

hzeta(link = "loglog", earg=list(), 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.
earg List. Extra argument for the link. See earg in Links for general information.
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.

Details

The probability function is

f(y) = (2y-1)^(-alpha) - (2y+1)^(-alpha),

where the parameter alpha>0 and y=1,2,.... 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 <= 1, and the variance is infinite if alpha <= 2.

Value

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

Author(s)

T. W. Yee

References

Page 533 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.

Examples

alpha = exp(exp(-0.1))  # The parameter
y = rhzeta(n=1000, alpha) # Generate some hzeta random variates
fit = vglm(y ~ 1, hzeta, trace = TRUE, crit="c")
coef(fit, matrix=TRUE)
Coef(fit)  # Useful for intercept-only models; should be same as alpha
c(mean(y), fitted(fit)[1,])
summary(fit)

[Package VGAM version 0.7-7 Index]