Abstract
The skew-normal (SN) distribution is a generalization of the normal distribution, where a shape parameter is added to adopt skewed forms. The SN distribution has some of the properties of a univariate normal distribution, which makes it very attractive from a practical standpoint; however, it presents some inference problems. Specifically, the maximum likelihood estimator for the shape parameter tends to infinity with a positive probability. A new Bayesian approach is proposed in this paper which allows to draw inferences on the parameters of this distribution by using improper prior distributions in the ``centered parametrization'' for the location and scale parameter and a Beta-type for the shape parameter. Samples from posterior distributions are obtained by using the Metropolis-Hastings algorithm. A simulation study shows that the mode of the posterior distribution appears to be a good estimator in terms of bias and mean squared error. A comparative study with similar proposals for the SN estimation problem was undertaken. Simulation results provide evidence that the proposed method is easier to implement than previous ones. Some applications and comparisons are also included.
Highlights
The skew-normal distribution is a three parameter class of distribution with location, scale and shape parameters, and it contains the normal distribution when the shape parameter equals zero
A continuous random variable Z is said to obey the skew-normal law with shape parameter λ ∈ R and it is denoted by SN (λ) if its density function is: fZ (z; λ) = 2φ (z) Φ(λz)I(−∞,∞)(z), (1)
Y = ξ + ωZ, with ξ ∈ R, ω ∈ R+, Y is said to have a skew-normal distribution with location-scale (ξ, ω) parameters and shape parameter λ, and it is denoted by Y ∼ SND(ξ, ω, λ)
Summary
The skew-normal distribution is a three parameter class of distribution with location, scale and shape parameters, and it contains the normal distribution when the shape parameter equals zero. From a classical point of view, there are at least two reasons to consider the “centered parametrization”: i) it provides a more practical interpretation of the parameters, and ii) it solves the well known problems of the likelihood function under direct parametrization. Arellano-Valle & Azzalini (2008) stated that the standard likelihood based methods and the Bayesian methods are problematic when they are applied to inference on the parameters in the direct paramaerization near λ = 0 This is due to the fact that the direct paramerization is not numerically suitable for estimation.
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