Let X = (X1, ...,Xn) be asampleofsize n from theuniformdistri- butionontheinterval ( , + ) with

Let X = (X1, ...,Xn) be asampleofsize n from theuniformdistri-

butionontheinterval (μ − δ,μ + δ) with density

with respecttostandardLebesguemeasureon R, where θ = (δ,μ) ∈ Θ = R+ × R with δ and μ both unknown.

a) Determinethemomentestimatorof θ = (δ,μ) based on t(x) =(x, x2) and denote theestimatorby ˜θ = (˜μ, ˜δ);

b) Consideralsotheestimator ˆθ givenby

and showthat ˆμ is anunbiasedestimatorof μ;

c) Consideralsotheestimator ˇμ = med(X1, ...,Xn)
and showthatalsothisisanunbiasedestimatorof μ. Hint: Use that X1, ...,Xn has thesamedistributionas μ + δU1, ...μ + δUn, where Ui is independentand identically uniformlydistributedon (−1, 1);

98 ESTIMATION

d) Comparethevariancesof ˜μ, ˆμ, and ˇμ by simulation;

e) Comparethemeansquareerrorsof ˜δ and ˆδ by simulation.

Transcribed Image Text:

fu,s(x)= 1 == 1(-8,+8) 28