hzeta {VGAM} | R Documentation |
Estimating the parameter of Haight's Zeta function.
hzeta(link = "loglog", init.alpha = NULL)
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.
|
init.alpha |
Optional initial value for the (positive) parameter.
The default is to obtain an initial value internally. Use this argument
if the default fails.
|
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 470 of Johnson N. L., Kotz S. and Kemp, A. W. (1993) Univariate Discrete Distributions, 2nd edition, Volume 2, New York: Wiley.
alpha = exp(0.1) # The parameter y = rhzeta(n=400, 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 fitted(fit)[1:4,]