I am trying to work a problem from the book.
Problem: As an alternative to imposing unbiasedness, an estimator's distribution can be "centered" by requiring that its medium be equal to the unknown parameter $\theta$, if it is, $\hat\theta$ is said to be medium unbiased. Let $Y_1, Y_2,...,Y_n$ be a random sample of size $n$ from the uniform pdf, $f_y(y;\theta) = \frac{1}{\theta}$ , $0 \leq y \leq \theta$. For an arbitrary $n$, is $\hat\theta = \frac{n+1}{n}Y_{max}$ medium unbiased? Is it median unbiased for any value of $n$?
My attempt: Recall $f_{Y_{max}} = nF_Y(y)^{n-1}f_y(y)$
Then we need the cdf. So given $f_y(y;\theta) = \frac{1}{\theta}$ , $0 \leq y \leq \theta$, we have $F_y = \int_{0}^{y} \frac{1}{\theta} dt = \frac{y}{\theta}$ Then $f_{Y_{max}} = nF_Y(y)^{n-1}f_y(y) = n (\frac{y}{\theta})^{n-1}\frac{1}{\theta} = \frac{ny^{n-1}}{\theta^n}$
However, this part is wrong because the the book has $f_{\frac{n+1}{n}Y_{max}}(y) = \frac{n}{n+1}f_{Y_{max}}(\frac{n}{n+1}y) = \frac{n}{n+1}\frac{n}{\theta}\frac{n^{n-1}}{(n+1)^{n-1}}\frac{y^{n-1}}{\theta^{n-1}} = \frac{n^{n+1}}{(n+1)^n}\frac{y^{n-1}}{\theta^n}$
I don't understand how the conclude that.
Can someone please help me? I would really appreciate if someone could write the formula to how the book gets and tell me the the setup of the cdf The book does not have a formula like I tried to write. Instead it just shows $f_{\frac{n+1}{n}Y_{max}}(y) = \frac{n}{n+1}f_{Y_{max}}(\frac{n}{n+1}y) = \frac{n}{n+1}\frac{n}{\theta}\frac{n^{n-1}}{(n+1)^{n-1}}\frac{y^{n-1}}{\theta^{n-1}} = \frac{n^{n+1}}{(n+1)^n}\frac{y^{n-1}}{\theta^n}$. Other than that I understand the rest .
Thank you
Youll want to prove that $\dfrac{Y_{max}}{\theta}\sim Beta(n,1)$ then what is $E\left(\theta\dfrac{Y_{max}}{\theta}\right)$?