The Skew-t (sT) Distribution Sample Clauses

The Skew-t (sT) Distribution. The sT distribution, introduced by Xxxxxxxx and Xxxxxxxxx (2003)[11], simultaneously allows for both heavier tails and skewness. This allows for more robust fitting in the presence of outlying observations, while accounting for asymmetry in each component distribution. A random variable T is said to follow the sT distribution sT (µ, σ2, λ, ν), with skewness parameter λ ∈ R and degrees of freedom ν ∈ (0, ∞), if it has the following representation Z 2 T = µ + σ√τ , Z ∼ sN (µ, σ , λ), τ ∼ Γ(ν/2, ν/2), Z⊥τ, (3.35) However, it is computationally more convenient to use the following representation of the sT model, due to Azzalini (1986)[10] and Xxxxx (1986)[56]. Adopting the stochastically equivalent representation of Z ∼ sN (µ, σ2, λ) given by Z = δλ|U1| + λ √ 1 − δ2 U2, where δλ = λ/√1 + λ2, and U1 and U2 are independent N (0, 1) random variables, we get the following hierarchical representation of sT (µ, σ2, λ, ν) given by T|γ, τ ∼ N µ + δλγ, 1 δ2 − λ σ2 , τ γ|τ ∼ TN σ2 0, τ ; (0, ∞) , (3.36) τ ∼ Γ(ν/2, ν/2), where, TN (µ, σ2; (a, b)) represents the the truncated normal distribution with N (µ, σ2) lying within the truncated interval (a, b).
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