There are N i.i.d. (independent and identically distributed) samples drawn from the distribution with the probability density function: $$ f(x) = \begin{cases} \frac{2x}{\theta^2}, & \text{if $0 \geq x \geq \theta$} \\ 0, & \text{otherwise} \end{cases} $$
Three Researchers want to use three different approaches to estimate parameter $\theta$
- Researcher A estimates $\theta$ by first computing the probability density function of observing the data given $\theta \rightarrow P(X_1,...,X_N; \theta)$, then maximizing this w.rt. $\theta$ (Maximum Likelihood Estimation). Call this estimate of $\theta$ as $R_a$.
- Researcher B estimates the median of this distribution by maximizing the likelihood of the median. Call this estimate as $R_b$
- Researcher C uses the theoretical moment and the sample moment of the data to estimate it. Call this estimate $R_c$.
(1) Please calculate $R_a, R_b, R_c$
(2) Which estimate is better? Please justify your conclusion taking bias and variance into consideration.
Solution: (1) Using similar approach as described by following solution: MLE for Uniform $(0,\theta)$
$$ R_a = X_{(N)} :=\max_{1 \leq i \leq N}X_i $$ $$ R_b = X_{(N)} :=\max_{1 \leq i \leq N}X_i $$ because median = $\frac{\theta}{\sqrt{2}}$ and maximizing Median is same as maximizing $\theta$ or MLE
Now, I want guidance on 2 of the following issues:
If my answers and logic for the first part are correct?
Can you point me in the right direction with the approach for the second part?
