cSFM (version 1.1)

Reparameterization: Reparameterize Skewed Normal Parameterized using Shape and Skewness.

Description

The transformation functions used to reparameterize skewned normal from shape (direct parameter) to skewness (central parameter), and vice visa.

Usage

skewness.cp(alpha)
shape.dp(gamma)
D.gamma(alpha)

Arguments

alpha

shape parameters

gamma

skewness parameters

Value

skewness.cp(alpha) gives the skewness corresponding to the shape alpha; shape.dp(gamma) gives the shape value corresponding to the skewness gamma.D.gamma(alpha) gives the first and second derivative of skewness wrt. the shape alpha.

Details

For skewed normal distributions, there is a one-to-one mapping from the shape to the skewness, regardless of the other parameters such as mean and variances. The parameters (mean, variance, skewness) are called centeral parameters (cp), while (location, scale, shape) are called direct parameters (dp). When estimating model parameters, skewness (central parameter) is more stable than the shape parameter. Note that the skewness for a skewed normal is bounded.

References

[1]. Azzalini, A. (1985). A class of distributions which includes the normal ones. Scandinavian journal of statistics, 171-178.

See Also

D.SN, cp2dp, dp2cp

Examples

gamma1 <- skewness.cp(10) # the skewness when the shape is 10
alpha1 <- shape.dp(gamma1) # the shape when the skewenss is gamma1; should be 10
ret <- D.gamma(10) # the derivatives of the skewness as a function of the shape