Maximum likelihood estimation of the shape parameter of a univariate skew-normal distribution.
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
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
argument is used.
An object of class
The object is used by modelling functions such as
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.
It is a good idea to use several different initial values to ensure that the global solution is obtained.
This family function will be modified (hopefully soon) to handle a location and scale parameter too.
Thomas W. Yee
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.
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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)) ## Not run: 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")) ## End(Not run) 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)
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