Exercise7.4 Consider asingleobservation Y N (,2), where || a and is known.We wanttoestimatetheunknown

Exercise7.4 Consider asingleobservation Y ∼ N (μ,σ2), where |μ| ≤ a and σ is known.We wanttoestimatetheunknown μ by alinearestimatoroftheform ˆ μc = cY, c ∈ R.

1. Find theminimaxlinearestimatorw.r.t.thequadraticloss.Isitadmissible(withinthefamilyof linear estimators)?Whatistheminimaxrisk?

2. Is ˆ μ1 =Y an admissibleestimator?

3. Showthattheleastfavorablepriorfor μ is adegeneratetwo-pointprior π

(a) = π(−a) = 0.5.

4. Extend thepreviousparagraphandshowthat,infact,anyprior π(μ) with thesecondmoment Eπμ2 = a2 is leastfavorable.

5. Assume anoninformativeuniformprior U (−a,a) on μ and findtheresultingBayesestimator w.r.t.thequadraticloss.Isitadmissible?

6. Showthatgenerally, ˆ μc is admissiblefor0 ≤ c < 1 (withinafamilyoflinearestimators)andis inadmissible otherwise.

(Hint: consideranyprioron μ with support (−a,a) and finitesecondmoment,andfindthe corresponding Bayeslinearestimator.)