Haight's Zeta Family Function

Share:

Description

Estimating the parameter of Haight's zeta distribution

Usage

1
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.

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

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

1
2
3
4
5
6
7
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)

Want to suggest features or report bugs for rdrr.io? Use the GitHub issue tracker.