I have the following question for homework:
- Suppose that $X$ and $Y$ are independent Poisson distributed values with means $\theta$ and $2\theta$, respectively. Consider the combined estimator of $\theta$ $$ \tilde\theta = k_1 X + k_2Y $$ where $k_1$ and $k_2$ are arbitrary constants.
(a) Find the condition on $k_1$ and $k_2$ such that $\tilde \theta$ is an unbiased estimator of $\theta$.
(b) For $\tilde \theta$ unbiased, show that the variance of the estimator is minimized by taking $k_1 = 1/3$ and $k_2 = 1/3$.
(c) Given observations $x$ and $y$ find the maximum likelihood estimate of $\theta$, and hence show that $\tilde\theta$ is also the maximum likelihood estimator.
I know the answer for part a) is $$k_1+ 2k_2 = 1$$
I don't know where to go with part b). So far I have as far as
$$Var(\hat\theta)= (k_1)^2Var(X) + (k_2)^2Var(Y)$$
Any help would be appreciated thanks in advance!