An expectiles are a generalization of the mean; expectile is to mean as quantile is to median (see definitions by Wikipedia and SciPy).
But what does that really mean? This tool helps you visualize how the expectile works for different distributions.
Choose a distribution, then drag t ∈ (0, 1) to see the t-expectile.
Distribution
μ = 0
σ = 1.0
shape α = 2.0
σ = 1.0
ν = 5
γ = 1.0
s = 1.0
α = 2.0
β = 2.0
rate λ = 1.0
k = 2.0
scale β = 1.0
b = 1.0
a = 0
b = 1
α = 2.0
a = 0
b = 4
c = 2.0
t = 0.50
t-expectile et0.0000
t-quantile qt0.0000
Balance: t·E[(X−eₜ)₊] = (1−t)·E[(eₜ−X)₊]—
What is an expectile?
The t-expectile et minimises
E[|t − 𝟏(X ≤ μ)| · (X − μ)²] over μ — equivalently, it is the unique solution to
t · E[(X − μ)₊] = (1 − t) · E[(μ − X)₊], where (u)₊ = max(u, 0).
The lower panel plots both sides as equal shaded areas.
At t = ½, et always equals the mean.
Unlike quantiles — which count probability mass — expectiles weight deviations by magnitude.