tikuv {VGAM} | R Documentation |
Fits the short-tailed symmetric distribution of Tiku and Vaughan (1999).
tikuv(d, lmean="identity", lsigma="loge", emean=list(), esigma=list(), isigma=NULL, zero=2)
d |
The d parameter. It must be a single numeric value less than 2.
Then h=2-d>0 is another parameter.
|
lmean, lsigma |
Link functions for the mean and standard
deviation parameters of the usual univariate normal distribution
(see Details below).
They are mu and sigma respectively.
See Links for more choices.
|
emean, esigma |
List. Extra argument for each of the links.
See earg in Links for general information.
|
isigma |
Optional initial value for sigma.
A NULL means a value is computed internally.
|
zero |
An integer-valued vector specifying which
linear/additive predictors are modelled as intercepts only.
The values must be from the set {1,2} corresponding
respectively to mu, sigma.
If zero=NULL then all linear/additive predictors are modelled as
a linear combination of the explanatory variables.
For many data sets having zero=2 is a good idea.
|
The short-tailed symmetric distribution of Tiku and Vaughan (1999) has a probability density function that can be written
f(y) = (K/(sqrt(2*pi)*sigma)) * [1 + (1/(2*h)) * ((y-mu)/sigma)^2]^2 * exp( -0.5 * (y-mu)^2/ sigma^2)
where h=2-d>0, K is a function of h, -Inf < y < Inf, sigma > 0. The mean of Y is E(Y) = mu and this is returned as the fitted values.
An object of class "vglmff"
(see vglmff-class
).
The object is used by modelling functions such as vglm
,
and vgam
.
Under- or over-flow may occur if the data is ill-conditioned,
e.g., when d is very close to 2 or approaches -Inf
.
The density function is the product of a univariate normal
density and a polynomial in the response y.
The distribution is bimodal if d>0, else is unimodal.
A normal distribution arises as the limit as d approaches
-Inf, i.e., as h approaches Inf.
Fisher scoring is implemented.
After fitting the value of d
is stored as @misc\$d
.
Thomas W. Yee
Akkaya, A. D. and Tiku, M. L. (2006) Short-tailed distributions and inliers. Test, 15(2), in press.
Tiku, M. L. and Vaughan, D. C. (1999) A family of short-tailed symmetric distributions. Technical report, McMaster University, Canada.
m = 1.0; sigma = exp(0.5) sy = sort(y <- rtikuv(n=1000, d=1, m=m, s=sigma)) fit = vglm(y ~ 1, fam=tikuv(d=1), trace=TRUE) coef(fit, mat=TRUE) (Cfit = Coef(fit)) mean(y) ## Not run: hist(y, prob=TRUE) lines(sy, dtikuv(sy, d=1, m=Cfit[1], s=Cfit[2]), col="red") ## End(Not run)