zetaff {VGAM} | R Documentation |
Estimates the parameter of the zeta distribution.
zetaff(link = "loge", earg=list(), init.p = NULL)
link |
Parameter link function applied to the (positive) parameter p.
See Links for more choices.
Choosing loglog constrains p>1, but
may fail if the maximum likelihood estimate is less than one.
|
earg |
List. Extra argument for the link.
See earg in Links for general information.
|
init.p |
Optional initial value for the parameter p.
The default is to choose an initial value internally.
If converge failure occurs use this argument to input a value.
|
In this long tailed distribution the response must be a positive integer. The probability function for a response Y is
P(Y=y) = 1/(y^(p+1) zeta(p+1)), p>0, y=1,2,...
where zeta is Riemann's zeta function. The parameter p is positive, therefore a log link is the default. The mean of Y is mu = zeta(p)/zeta(p+1) provided p>1. The variance of Y is zeta(p-1) / zeta(p+1) - mu^2 provided p>2.
It appears that good initial values are needed for successful convergence. If convergence is not obtained, try several values ranging from values near 0 to values about 10 or more.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
rrvglm
and vgam
.
The zeta
function may be used to
compute values of the zeta function.
T. W. Yee
pp.527– of Chapter 11 of Johnson N. L., Kemp, A. W. and Kotz S. (2005) Univariate Discrete Distributions, 3rd edition, Hoboken, New Jersey: Wiley.
Knight, K. (2000) Mathematical Statistics. Boca Raton: Chapman & Hall/CRC Press.
Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.
y = 1:5 # Knight, p.304 w = c(63, 14, 5, 1, 2) fit = vglm(y ~ 1, zetaff, trace=TRUE, wei=w, crit="c") (phat = Coef(fit)) # 1.682557 cbind(dzeta(y, phat) * sum(w), w) weighted.mean(y, w) fitted(fit, mat=FALSE) predict(fit) # MLE should satisfy the following: mean(log(rep(y, w))) + zeta(1+phat, deriv=1)/zeta(1+phat) # Should be 0