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Exercise2.12 1. ShowthattheMLE MLE =Ymax of in theuniformdistribution U (0,) (see Example2.2)is uniformly better(intermsofMSE)thantheMME MME =

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Exercise2.12 1. ShowthattheMLE θˆMLE =Ymax of θ in theuniformdistribution U (0,θ) (see Example2.2)is uniformly better(intermsofMSE)thantheMME θˆMME = 2¯Y (see Example2.10).

(Hint: youfirstneedtoderivethedistributionof Ymax to calculatethe MSE(θˆ ,θ).)

2. Is θˆ unbiased? Ifnot,findanunbiasedestimatorbasedon θˆ .

3. Consider afamilyofestimators aYmax, where a may dependon n butnotonthedata.Whichofthe estimators consideredinthepreviousparagraphsbelongtothisfamily?Findthebest(minimal MSE) estimatorwithinthefamily.

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