hzeta {VGAM} | R Documentation |
Estimating the parameter of Haight's Zeta function.
hzeta(link = "loglog", earg=list(), ialpha = NULL, nsimEIM=100)
link |
Parameter link function for the parameter.
See |
earg |
List. Extra argument for the link.
See |
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 |
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.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
and vgam
.
T. W. Yee
Page 533 of Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley.
alpha = exp(exp(-0.1)) # The parameter hdata = data.frame(y = rhzeta(n = 1000, alpha)) fit = vglm(y ~ 1, hzeta, hdata, trace = TRUE, crit = "c") 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)