Working through some homework problems for a Mathematical Statistics course and I'm having a hard time finding good examples in the text to explain some details. This particular problem is:
$Y_1 ... Y_n \sim f_\theta(y) = \frac{6}{\theta^3} y (\theta-y)$
Find a constant a so that $\hat{\theta} = a\overline{Y}$ is unbiased, and show that it is consistent.
I calculated the Method of Moments estimator:
$$ m_1^{\prime} = \frac{6}{\theta^3}\int_0^\infty y^2(\theta - y) dy = \frac{\theta}{2} $$
So, $\hat{\theta}=2\bar{Y} \Rightarrow a=2$
So far so good.
For bias:
$$ B(\hat{\theta}) = E(\hat{\theta}) - \theta $$
$$ B(\hat{\theta}) = E(2\bar{Y}) - \theta $$
$$ B(\hat{\theta}) = 2\mu - \theta $$
And now I'm stuck on what should be obvious. How do I equate $2\mu$ and $ \theta$?
To check consistency, I need bias and variance to be zero, or approach zero as $n \rightarrow \infty$. Bias is above, but I'm stuck calculating variance for $\theta$.
Looking through the text, all the examples and other homework answers make large jumps in logic without any explanation. In cases like this, examples simply state that bias and variance are zero, but there is no work shown for reference.