Optimality

Then
(3.5)
Local asymptotic minimax risk/asymmetric loss 317
1
1 − W 2 (Z− β0(W))
|W 2 (Z− β0(W))− | = |
r(W, σ)
Now, for any large a, b > 0, we have
(3.6)
� b
−b
la(ξ(z, w, u)−θ)ψ(θ|z, w)dθ
� b
= la(ξ(z, w, u)−y−
−b
≤
W − 1 2 (Z−β0(W))
σ 2
r(W, σ)
M
σ 2 r(W, σ)
1
w 2 (z− β0(w))
)φy(0,
r(w, σ)
|
= M
σ 2 W + 1
1
r(w, σ) )dy,
≤ ɛ
2 .
where y = θ− w 1 2 (z−β0(w))
r(w,σ) . Now, since θ|z, w∼N( w 1 2 (z−β0(w)) 1
r(w,σ) , r(w,σ) ), we have
1
1
y|z, w∼N(0, r(w,σ) −w− 2 β0(w)| > ɛ
2 .
Hence due to the nature of the loss function, for a given w > 0, we can have, from
(3.6),
(3.7)
r(w,σ) ). It can be seen that|ξ(z, w, u)− w 1 2 (z−β0(w))
� b
−b
la(ξ(z, w, u)−
≥
≥
� √ b
− √ b
� √ b
− √ b
1
w 2 (z− β0(w))
− y)φy(0,
r(w, σ)
1 − 1
la(w 2 β0(w)−y)φy(0,
r(w, σ) )dy
1
r(w, σ) )dy
1 −
la(w 2 β(a, ˜
1
b, λ, w)−y)φy(0, )dy + δ
r(w, σ)
= ˜ h( ˜ β(a, b, λ, w)) + δ,
where δ > 0 depends only on ɛ but not on a, b, σ2 and (3.7) holds for sufficiently
≤ w≤m).
large a, b, σ 2 (here λ = 1
wσ 2→ 0 as σ 2 →∞ and 1
m
(3.8)
A simple calculation yields
˜h( ˜ β(a, b, λ, w))
=
≥
� √ b
− √ b
� √ b
− √ b
= h(β0(w))−
1 −
la(w 2 β(a, ˜
1
b, λ, w)−y)φy(0,
r(w, σ) )dy
1 −
la(w 2 β(a, ˜ b, λ, w)−y)φy(0, 1
� √ b
− √ b
1 −
la(w
w
y2
)(1− )dy
σ2 2 ˜ β(a, b, λ, w)−y) y 2
1
φy(0,
σ2 w )dy.