levy {VGAM} | R Documentation |
Estimates the two parameters of the Levy distribution by maximum likelihood estimation.
levy(delta = NULL, link.gamma = "loge", earg=list(), idelta = NULL, igamma = NULL)
delta |
Location parameter. May be assigned a known value,
otherwise it is estimated (the default).
|
link.gamma |
Parameter link function for the (positive) gamma parameter.
See Links for more choices.
|
earg |
List. Extra argument for the link.
See earg in Links for general information.
|
idelta |
Initial value for the delta parameter
(if it is to be estimated).
By default, an initial value is chosen internally.
|
igamma |
Initial value for the gamma parameter.
By default, an initial value is chosen internally.
|
The Levy distribution is one of three stable distributions whose density function has a tractable form. The formula for the density is
f(y;gamma,delta) = sqrt(gamma / (2 pi)) exp( -gamma / (2(y - delta))) / (y - delta)^{3/2}
where delta<y<Inf and gamma>0. The mean does not exist.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
and vgam
.
If delta is given, then only one parameter is estimated and the default is eta1=log(gamma). If delta is not given, then eta2=delta.
T. W. Yee
Nolan, J. P. (2005) Stable Distributions: Models for Heavy Tailed Data.
The Nolan article is at http://academic2.american.edu/~jpnolan/stable/chap1.pdf.
Documentation accompanying the VGAM package at http://www.stat.auckland.ac.nz/~yee contains further information and examples.
n = 1000 mygamma = 1 # log link ==> 0 is the answer delta = 0 y = delta + mygamma / rnorm(n)^2 # This is Levy(mygamma, delta) # Cf. Table 1.1 of Nolan for Levy(1,0) sum(y > 1) / length(y) # Should be 0.6827 sum(y > 2) / length(y) # Should be 0.5205 fit = vglm(y ~ 1, levy(delta=delta), trace=TRUE) # 1 parameter fit = vglm(y ~ 1, levy(idelta=delta, igamma=mygamma), trace=TRUE) # 2 parameters coef(fit, matrix=TRUE) Coef(fit) summary(fit) weights(fit, type="w")[1:4,]