Quantile/expectile regression, and extreme data analysis

Asymmetric MLE
Asymmetric MLE III
Notation
t Notation
Comments
Y
Response. Has mean µ, cdf F (y), pdf f (y)
Q Y (τ) = τ-quantile of Y 0 < τ < 1
ξ(τ) = ξ τ = τ-quantile Koenker and Bassett (1978), ξ( 1 ) = median
2
µ(ω) = µ ω = ω-expectile 0 < ω < 1, µ( 1 ) = µ, Newey and Powell (1987)
2
bξ(τ), bµ(ω)
Sample quantiles and expectiles
centile
Same as quantile and percentile here
regression quantile Koenker and Bassett (1978)
regression expectile Newey and Powell (1987)
regression percentile All forms of asymmetric fitting, Efron (1992)
ρ τ (u) = u · (τ − I (u < 0)) Check function corresponding to ξ(τ)
ρ [2]
ω (u) = u 2 · |ω − I (u < 0)| Check function corresponding to µ(ω)
u + = max(u, 0)
Positive part of u
u − = min(u, 0)
Negative part of u
© T. W. Yee (University of Auckland) Quantile/expectile regression, and extreme data analysis 18 July 2012 @ Cagliari 46/101/