Short-tailed Symmetric Distribution Family Function
Fits the short-tailed symmetric distribution of Tiku and Vaughan (1999).
tikuv(d, lmean = "identitylink", lsigma = "loglink", isigma = NULL,
zero = "sigma")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 |
isigma |
Optional initial value for sigma.
A |
zero |
A vector specifying which
linear/additive predictors are modelled as intercept-only.
The values can be from the set {1,2}, corresponding
respectively to mu, sigma.
If |
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 in @misc with
component name d.
Thomas W. Yee
Akkaya, A. D. and Tiku, M. L. (2008). Short-tailed distributions and inliers. Test, 17, 282–296.
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) tdata <- data.frame(y = rtikuv(n = 1000, d = 1, m = m, s = sigma)) tdata <- transform(tdata, sy = sort(y)) fit <- vglm(y ~ 1, tikuv(d = 1), data = tdata, trace = TRUE) coef(fit, matrix = TRUE) (Cfit <- Coef(fit)) with(tdata, mean(y)) ## Not run: with(tdata, hist(y, prob = TRUE)) lines(dtikuv(sy, d = 1, m = Cfit[1], s = Cfit[2]) ~ sy, data = tdata, col = "orange") ## End(Not run)
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