Exercise 7.42 established that the optimal weights are q*i = (1/Ï2i)/(j 1/Ï2j). A result due to Tukey

Exercise 7.42 established that the optimal weights are q*i = (1/σ2i)/(ˆ‘j 1/σ2j). A result due to Tukey (see Bloch and Moses 1988) states that if W = ˆ‘i qiWi is an estimator based on another sets of weights qi ‰¥ 0, ˆ‘i gi = 1, then

where λ satisfies (1 + λ)/(l - λ) = bmax/bmin, and bmax and bmin are the largest and smallest of bi = qi/q*i.
(a) Prove Tukey's inequality.
(b) Use the inequality to assess the performance of the usual mean ˆ‘i Wi/k as a function of σ2max / σ2min.

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Var W Var W. S 1-X