10.5.∗General Rank Scores 613can be ignored, so we considerc∗φ=(√
λ 1 λ 2 )−^1 cφ. If we make the change of vari-
ablesu=F(y) and then integrate by parts, we get
c∗φ =∫∞−∞φ′[F(y)]f^2 (y)dy=∫ 10φ′(u)f(F−^1 (u))du=∫ 10φ(u)[
−f′(F−^1 (u))
f(F−^1 (u))]
du. (10.5.25)Recall that the score function∫
φ^2 (u)du= 1. Thus we can state the problem
as
max
φ
c∗φ^2 =max
φ{∫ 10φ(u)[
−
f′(F−^1 (u))
f(F−^1 (u))]
du} 2=⎧
⎪⎨⎪⎩
max
φ{∫
1
0 φ(u)[
−f′(F− (^1) (u))
f(F−^1 (u))
]
du
} 2
∫ 1
0 φ
(^2) (u)du∫^1
0
[
f′(F−^1 (u))
f(F−^1 (u))
] 2
du
⎫
⎪⎬
⎪⎭
∫ 1
0
[
f′(F−^1 (u))
f(F−^1 (u))
] 2
du.
The quantity that we are maximizing in the braces of this last expression, how-
ever, is the square of a correlation coefficient, which achieves its maximum value 1.
Therefore, by choosing the score functionφ(u)=φf(u), where
φf(u)=−κ
f′(F−^1 (u))
f(F−^1 (u))
, (10.5.26)
andκis a constant chosen so that
∫
φ^2 f(u)du= 1, then the correlation coefficient
is 1 and the maximum value is
I(f)=
∫ 1
0
[
f′(F−^1 (u))
f(F−^1 (u))
] 2
du, (10.5.27)
which is Fisher information for the location model. We call the score function given
by (10.5.26) theoptimal score function.
In terms of estimation, ifΔ is the corresponding estimator, then, according tô
(10.5.24), it has the asymptotic variance
τφ^2 =
[
1
I(f)
](
1
n 1
- 1
n 2
)
. (10.5.28)
Thus the estimatorΔ achieves asymptotically the Rao–Cram ́̂ er lower bound; that
is,Δ is an asymptotically efficient estimator of Δ. In terms of asymptotic relativê
efficiency, the ARE between the estimatorΔandthemleofΔis1. Thuswehavê
answered the second question of the first paragraph of this section.
Now we look at some examples. The initial example assumes that the distri-
bution ofεiis normal, which answers the leading question at the beginning of this