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 |
earg |
List. Extra argument for the link.
See |
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 - δ)^{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.
nn = 1000; delta = 0 mygamma = 1 # log link ==> 0 is the answer ldat = data.frame(y = delta + mygamma/rnorm(nn)^2) # Levy(mygamma, delta) # Cf. Table 1.1 of Nolan for Levy(1,0) with(ldat, sum(y > 1) / length(y)) # Should be 0.6827 with(ldat, sum(y > 2) / length(y)) # Should be 0.5205 fit = vglm(y ~ 1, levy(delta=delta), ldat, trace=TRUE) # 1 parameter fit = vglm(y ~ 1, levy(idelta=delta, igamma=mygamma), ldat, trace=TRUE) # 2 parameters coef(fit, matrix=TRUE) Coef(fit) summary(fit) head(weights(fit, type="w"))