1. ## unbiased estimators

Q: Suppose that $\displaystyle Y_{1},Y_{2},Y_{3}$ denote a random sample from an exponential distribution with density function

$\displaystyle f(y)=\frac{1}{\theta}e^{-\frac{y}{\theta}}$ for $\displaystyle y>0$ and $\displaystyle 0$ otherwise.

Consider that following estimators of $\displaystyle \theta$.

$\displaystyle \theta\hat\\=\frac{Y_{1}+2Y_{2}}{3}$

$\displaystyle \theta\hat\\=\overline{Y}$

Note: both thetas directly above should have hats on them.

I completely stuck on this one. Do I figure the distribution of each function of theta hat and then take the expectation?

Help getting started on either one would be great.

Thank you

2. IS there a question here?
They both are unbiased.
BY inspection $\displaystyle E(Y_i)=\theta$
Since the coefficients sum to one, both are unbiased.
Next you should obtain the variances and see that the sample mean has a smaller variance.

$\displaystyle V(\bar Y)={\theta^2\over n}$ where n=3.

and $\displaystyle V(\hat\theta_2)= {1\over 9}\theta^2+ {4\over 9}\theta^2 = {5\over 9}\theta^2$