LongbeforeRonaldFisher proposedthemaximumlikelihoodmethodin1922,in1894Karl Pearsonproposedthe methodofmoments. Thisexpressesthemeanoftheprobability distribution intermsoftheparameterandequatesittothesamplemean.Fortwoparameters, youequatethefirsttwomoments.Let Y1, ...,Yn be n independentrandomvariablesfromthe uniform distribution(2.4)over(0, ). (a)

LongbeforeRonaldFisher proposedthemaximumlikelihoodmethodin1922,in1894Karl Pearsonproposedthe methodofmoments. Thisexpressesthemeanoftheprobability distribution intermsoftheparameterandequatesittothesamplemean.Fortwoparameters, youequatethefirsttwomoments.Let Y1, ...,Yn be n independentrandomvariablesfromthe uniform distribution(2.4)over(0, θ).

(a) Showthatthemethodofmomentsestimateis ˜θ = 2y.

(b) Let y(1) ≤ y(2) ≤ ⋯ ≤ y(n) denote theorderedobservations, called orderstatistics.

Explain whythelikelihoodfunctionis ℓ(θ) = 1~θn for θ ≥ y(n). ExplainwhytheML estimate ˆθ = y(n).

(c) Reportthetwoestimatesforthesampledata {1, 2, 9}. Explainwhythemethodofmo-

mentsestimatedoesnotmakesense.(FordistributionsforwhichMLandmethodof momentsestimatorsdiffer,FishershowedthattheMLestimatorismoreefficient.)

(d) ThisisararecaseinwhichtheMLestimator ˆθ doesnothavealarge-samplenormal distribution, becauseofviolatingtheregularityconditionthatthepossiblevaluesfor y cannot dependonthevalueof θ. Showthatfor 0 < y < θ, the cdf of ˆθ is F(y) = (y~θ)n and its pdf is f(y) = (nyn−1)~θn.

(e) Showthatthemethodofmomentsestimator ˜θ is unbiasedbut E(ˆθ) = [n~(n + 1)]θ, illustrating thatanunbiasedestimatorneednotbebetterthanabiasedestimator.

(f)IfyouweretousethebootstrapwiththeMLestimator ˆθ = Y(n), describethelikely appearanceofthebootstrapdistribution.Howwouldthepercentile-basedconfidencein-

tervalperformforestimating θ?