VGAM (version 1.1-12)

skewnormal: Univariate Skew-Normal Distribution Family Function

Description

Maximum likelihood estimation of the shape parameter of a univariate skew-normal distribution.

Usage

skewnormal(lshape = "identitylink", ishape = NULL, nsimEIM = NULL)

Value

An object of class "vglmff" (see vglmff-class). The object is used by modelling functions such as vglm, and vgam.

Arguments

lshape, ishape, nsimEIM

See Links and CommonVGAMffArguments.

Author

Thomas W. Yee

Warning

It is well known that the EIM of Azzalini's skew-normal distribution is singular for skewness parameter tending to zero, and thus produces influential problems.

Details

The univariate skew-normal distribution has a density function that can be written $$f(y) = 2 \, \phi(y) \, \Phi(\alpha y)$$ where \(\alpha\) is the shape parameter. Here, \(\phi\) is the standard normal density and \(\Phi\) its cumulative distribution function. When \(\alpha=0\) the result is a standard normal distribution. When \(\alpha=1\) it models the distribution of the maximum of two independent standard normal variates. When the absolute value of the shape parameter increases the skewness of the distribution increases. The limit as the shape parameter tends to positive infinity results in the folded normal distribution or half-normal distribution. When the shape parameter changes its sign, the density is reflected about \(y=0\).

The mean of the distribution is \(\mu=\alpha \sqrt{2/(\pi (1+\alpha^2))}\) and these are returned as the fitted values. The variance of the distribution is \(1-\mu^2\). The Newton-Raphson algorithm is used unless the nsimEIM argument is used.

References

Azzalini, A. A. (1985). A class of distributions which include the normal. Scandinavian Journal of Statistics, 12, 171--178.

Azzalini, A. and Capitanio, A. (1999). Statistical applications of the multivariate skew-normal distribution. Journal of the Royal Statistical Society, Series B, Methodological, 61, 579--602.

See Also

skewnorm, uninormal, foldnormal.

Examples

sdata <- data.frame(y1 = rskewnorm(nn <- 1000, shape = 5))
fit1 <- vglm(y1 ~ 1, skewnormal, data = sdata, trace = TRUE)
coef(fit1, matrix = TRUE)
head(fitted(fit1), 1)
with(sdata, mean(y1))
if (FALSE)  with(sdata, hist(y1, prob = TRUE))
x <- with(sdata, seq(min(y1), max(y1), len = 200))
with(sdata, lines(x, dskewnorm(x, shape = Coef(fit1)), col = "blue")) 

sdata <- data.frame(x2 = runif(nn))
sdata <- transform(sdata, y2 = rskewnorm(nn, shape = 1 + 2*x2))
fit2 <- vglm(y2 ~ x2, skewnormal, data = sdata, trace = TRUE, crit = "coef")
summary(fit2)