Order Statistics, Unbiasedness, and Expected Values

[Providence88]

Homework Statement

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

Show that \hat{\theta} is an unbiased estimator for \theta

Homework Equations

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

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

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

The Attempt at a Solution

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

Any help? Thanks!

 

[Cyosis]

I am not quite sure what g_n(y) is so you will have to explain why you thought it would equal n y^{n-1}. As for the integral it is not so hard. You can solve it by partial integration.

<br /> \int_{\theta}^{\theta+1} ny(y-\theta)^{n-1}dy=y (y-\theta)^n ]_\theta^{\theta+1}-\int_{\theta}^{\theta+1} (y-\theta)^n dy<br />

 

[Providence88]

Oh, g_{(n)}(y) is the density function for Y_{(n)}=max(Y1, Y2, ..., Yn)

g_{(n)}(y) = n[F(Y)]^{n-1}*f(y), where F(Y) is the distribution function of Y and f(y) is the density function. Since the bounds are theta and theta plus one, I assumed that f(y), by definition, is 1/(theta + one - theta), which equals one. If f(y) = 1, then F(Y) = y + C. I'm starting to think that the plus C would be -(theta).