(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let Y1, Y2, ..., Yn denote a random sample from the uniform distribution on the interval ([tex]\theta[/tex], [tex]\theta + 1[/tex]). Let [tex]\hat{\theta} = Y_{(n)} - \frac{n}{n+1}[/tex]

Show that [tex]\hat{\theta}[/tex] is an unbiased estimator for [tex]\theta[/tex]

2. Relevant equations

Well, to check for unbiasedness, E([tex]\hat{\theta}[/tex]) should = [tex]\theta[/tex].

The difficulty for me arises when calculating [tex]g_{(n)}(y)[/tex], needed to find E[[tex]\hat{\theta}[/tex]]. The interval ([tex]\theta[/tex], [tex]\theta + 1[/tex]) seems to make this integral very complicated:

[tex]E[\hat{\theta}][/tex] = [tex]\int^{\theta + 1}_{\theta} yg_{(n)}(y)[/tex]

3. The attempt at a solution

I attempted to find [tex]g_{(n)}(y)[/tex], which I thought to be [tex]ny^{n-1}[/tex], but according to our solutions manual, it's actually [tex]n(y-\theta)^{n-1}[/tex], which I have no idea how that is concluded. And even if that is the true value of [tex]g_{(n)}(y)[/tex], the integral is still looking very daunting.

Any help? Thanks!

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# Order Statistics, Unbiasedness, and Expected Values

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