10.7. Simple Linear Model 629Differentiating this last expression, we haveμ′T(β)=∑ni=1(xi−x)xi∫∞−∞φ′(F(y+xiβ))f(y+xiβ)f(y)dy,which yields
μ′T(0) =∑ni=1(xi−x)^2∫∞−∞φ′(F(y))f^2 (y)dy. (10.7.12)We need one assumption on thex 1 ,x 2 ,...,xn;namely,n−^1
∑n
i=1(xi−x)(^2) →σ 2
x,
where 0<σ^2 x<∞. Recall that (n−1)−^1 s^2 a→1. Therefore, the efficacy of the
processT(β)isgivenby
cT = lim
n→∞
μ′T(0)
√
nσT(0)
= lim
n→∞
∑n
i=1(xi−x)
2 ∫∞
−∞φ
′(F(y))f (^2) (y)dy
√
n
√
(n−1)−^1 s^2 a
√∑n
i=1(xi−x)^2
= σx
∫∞
−∞
φ′(F(y))f^2 (y)dy. (10.7.13)
Using this, an asymptotic power lemma can be derived for the test based onTφ;
see expression (10.7.17) of Exercise 10.7.6. Based on this, it can be shown that the
asymptotic distribution of the estimatorβ̂φis given by
β̂φhas an approximateN
(
β, τφ^2 /
∑n
i=1
(xi−x)^2
)
distribution, (10.7.14)
where the scale parameterτφisτφ=(
∫∞
−∞φ
′(F(y))f (^2) (y)dy)− (^1). Koul et al. (1987)
developed a consistent estimator of the scale parameterτ, which is the default
estimate in the packageRfit. This can be used to compute a confidence interval
for the slope parameter, as illustrated in Example 10.7.3.
Remark 10.7.2.The least squares (LS) estimates for Model (10.7.1) were discussed
in Section 9.6 in the case that the random errorsε 1 ,ε 2 ,...,εnare iid with aN(0,σ^2 )
distribution. In general, for Model (10.7.1), the asymptotic distribution of the LS
estimator ofβ,sayβ̂LS,is:
β̂
LShas an approximateN
(
β, σ^2 /
∑n
i=1
(xi−x)^2
)
distribution, (10.7.15)
whereσ^2 is the variance ofεi. Based on (10.7.14) and (10.7.15), it follows that the
ARE between the rank-based and LS estimators is given by
ARE(β̂φ,β̂LS)=
σ^2
τφ^2
. (10.7.16)
Hence, if Wilcoxon scores are used, this ARE is the same as the ARE between the
Wilcoxon andt-procedures in the one- and two-sample location models.