Skewnormal

Skewnormal is a family of probability distributions that extends the normal distribution to accommodate skewness. It is parameterized by location ξ, scale ω > 0, and shape α ∈ R, commonly denoted SN(ξ, ω, α).

The probability density function is f(x) = 2 φ((x-ξ)/ω) Φ( α (x-ξ)/ω ) / ω, where φ and Φ denote the standard normal pdf and cdf, respectively.

The cumulative distribution function is F(x) = Φ((x-ξ)/ω) - 2 T( - (x-ξ)/ω, α ), where T is Owen's T function.

Special cases: when α = 0 the skewnormal reduces to the normal distribution N(ξ, ω^2). A common stochastic representation uses δ = α / sqrt(1+α^2); if Z0 and Z1 are independent standard normals, then Y = ξ + ω (δ |Z0| + sqrt(1-δ^2) Z1) follows SN(ξ, ω, α).

Moments: for the standardized form (ξ = 0, ω = 1) the mean is E[X] = δ sqrt(2/π), the variance Var[X] = 1 - 2 δ^2 / π, and skewness is γ1 = ((4-π) δ^3) / (1 - 2 δ^2 / π)^(3/2), where δ = α / sqrt(1+α^2). In general, E[X] = ξ + ω δ sqrt(2/π), Var[X] = ω^2 (1 - 2 δ^2 / π).

Estimation and use: the skewnormal provides a flexible alternative to the normal for data with asymmetry. Parameters are typically estimated by maximum likelihood, sometimes with EM-type algorithms. It is unimodal for all α and has finite moments for all α. Applications span econometrics, finance, engineering, and environmental sciences.