612 Nonparametric and Robust Statistics10.5.2 Estimating Equations Based on General Scores
Suppose we are using the scoresaφ(i)=φ(i/(n+ 1)) discussed in Section 10.5.1.
Recall that the mean of the test statisticWφis 0. Hence the corresponding estimator
of Δ solves the estimating equations
Wφ(Δ)̂ ≈ 0. (10.5.22)By Theorem 10.5.1,Wφ(Δ) is a decreasing step function of Δ. Furthermore, thê
maximum value is positive and the minimum value is negative (only degenerate
cases would result in one or both of these as 0); hence, the solution to equation
(10.5.22) exists. BecauseWφ(Δ) is a step function, it may not be unique. When̂
it is not unique, though, as with Wilcoxon and median procedures, there is an
interval of solutions, so the midpoint of the interval can be chosen. This is an
easy equation to solve numerically because simple iterative techniques such as the
bisection method or the method of false position can be used; see the discussion on
page 210 of Hettmansperger and McKean (2011). The asymptotic distribution of
the estimator can be derived using the asymptotic power lemma and is given by
Δ̂φhas an approximateN(
Δ,τφ^2(
1
n 1 +1
n 2))
distribution, (10.5.23)where
τφ=[∫∞−∞φ′[F(y)]f^2 (y)dy]− 1. (10.5.24)
Hence the efficacy can be expressed ascφ=√
λ 1 λ 2 τφ−^1. As Exercise 10.5.9 shows,
the parameterτφis a scale parameter. Since the efficacy iscφ=√
λ 1 λ 2 τφ−^1 ,the
efficacy varies inversely with scale. This observation proves helpful in the next
subsection.10.5.3 Optimization:BestEstimates..................
We can now answer the questions posed in the first paragraph. For a given pdf
f(x), we show that in general we can select a score function that maximizes the
power of the test and minimizes the asymptotic variance of the estimator. Under
certain conditions we show that estimators based on this optimal score function
have the same efficiency as maximum likelihood estimators (mles); i.e., they obtain
the Rao–Cram ́er Lower Bound.
As above, letX 1 ,...,Xn 1 be a random sample from the continuous cdfF(x)
with pdff(x). LetY 1 ,...,Yn 2 be a random sample from the continuous cdfF(x−Δ)
with pdff(x−Δ). The problem is to chooseφto maximize the efficacycφgiven in
expression (10.5.20). Note that maximizing the efficacy is equivalent to minimizing
the asymptotic variance of the corresponding estimator of Δ.
For a general score functionφ(u), consider its efficacy given by expression
(10.5.20). Without loss of generality, the relative sample sizes in this expression