˖ڍመڙጲ
http://www.paper.edu.cn
For the missing response data, I(Y
i
− X
T
i
β < 0) can not be observed when δ
i
= 0. Under
MAR assumption, it is easy to see that
E
δ
i
∆
0
(X
i
)
I(Y
i
− X
T
i
β < 0)
= E
I(Y
i
− X
T
i
β < 0)
∆
0
(X
i
)
E(δ
i
|Y
i
, X
i
)
= EI(Y
i
− X
T
i
β < 0),
which suggests a natural way to extend (3) to the missing response case by replacing I(Y
i
−
X
T
i
β < 0) with
δ
i
∆
0
(X
i
)
I(Y
i
− X
T
i
β < 0). Thus we establish the following estimation equation
M
n
(β,
∆) = n
−1
n
i=1
X
i
τ −
δ
i
∆(X
i
)
I(Y
i
− X
T
i
β < 0)
≈ 0, (4)
where
∆(x) =
∑
n
i=1
δ
i
K(
x−X
i
h
n
)
∑
n
j=1
K(
x−X
j
h
n
)
is the Nadaraya-Watson type estimator of ∆
0
(x), K(·) is a kernel
function and h
n
> 0 is a bandwidth sequence converging to zero as n → ∞. Based on (4), we
propose the following quantile objective function
S
n
(β,
∆) = n
−1
n
i=1
δ
i
∆(X
i
)
ρ
τ
(Y
i
− X
T
i
β) +
n
i=1
1 −
δ
i
∆(X
i
)
ρ
τ
(Y
∗
i
− X
T
i
β)
, (5)
where Y
∗
i
is a random imputation of Y
i
and satisfies that Y
∗
i
− X
T
i
β > 0 for all β ∈ B. For
practical application, we set Y
∗
i
= Y
∞
= 100 max{Y
i
} as suggested by Wang and Wang
[31]
.
The standard quantile regression estimator of β
0
is defined by
β
QR
= argmin
β
S
n
(β,
∆). (6)
Next, we define the EL of β
0
. Let g
i
(β) = X
i
[τ −
δ
i
b
∆(X
i
)
I(Y
i
−X
T
i
β < 0)]. From Eg
i
(β
0
) ≈ 0
and using the idea introduced by Owen
[20,32]
, we construct the EL ratio function for β
L(β) = sup
n
i=1
(np
i
) : p
i
≥ 0,
n
i=1
p
i
= 1,
n
i=1
p
i
g
i
(β) = 0
. (7)
By the standard Lagrange multiplier method, the optimal p
i
solving (7) for a given β is
p
i
=
1
n[1 + λ(β)
T
g
i
(β)]
, (8)
where λ(β) is the solution of the following equation
n
−1
n
i=1
g
i
(β)
1 + λ(β)
T
g
i
(β)
= 0. (9)
According to (8) and (9), the empirical log likelihood ratio statistic can be written as
l(β) = 2
n
i=1
log[1 + λ(β)
T
g
i
(β)].
Further, we define the maximum EL estimator of β
0
as
β
EL
= argmin
β
l(β).
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