# Calculate the constants and the MSE from two estimators related to a uniform distribution

Consider a simple random sample $$X_{1},X_{2},\ldots,X_{n}$$ whose distribution is given by $$X\sim U(0,\theta)$$. Moreover, consider the estimators $$\hat{\theta}_{1} = c_{1}\overline{X}$$ and $$\hat{\theta}_{2} = c_{2}X_{(n)} = c_{2}\max{X_{1},X_{2},\ldots,X_{n})}$$.

(a) Find $$c_{1}$$ and $$c_{2}$$ such that $$\hat{\theta}_{1}$$ and $$\hat{\theta}_{2}$$ becomes unbiased.

(b) Find both Mean Squared Errors.

MY ATTEMPT

The estimator $$\hat{\theta}_{1}$$ is unbiased iff $$\textbf{E}(\hat{\theta}_{1}) = \theta_{1}$$, which is equivalent to \begin{align*} \textbf{E}(\hat{\theta}_{1}) = \textbf{E}(c_{1}\overline{X}) = c_{1}\textbf{E}(\overline{X}) = \frac{c_{1}\theta}{2} = \theta \Longleftrightarrow c_{1} = 2 \end{align*}

As to the second case, I am unable to work with $$X_{(n)}$$. Could someone help me?

(b) Since $$\hat{\theta}_{1}$$ is unbiased for $$c_{1} = 2$$, we conclude that $$\text{MSE}(\hat{\theta}_{1}) = \text{Var}(\hat{\theta}_{1})$$. Consequently, \begin{align*} \text{Var}(\hat{\theta}_{1}) = \text{Var}(2\overline{X}) = 4\text{Var}(\overline{X}) = 4\times\frac{\theta^{2}}{12n} = \frac{\theta^{2}}{3n} \end{align*}

The same problem applies to $$\text{MSE}(\hat{\theta}_{2})$$. Could someone help me with this as well?

• Please add the self-study tag. Can you find the CDF of $X_{(n)}$? Commented Mar 26, 2019 at 23:47

$$F_{X_{(n)}}(x)=P(X_{(n)}\leq x)=P(\max \{X_1,\cdots, X_n\} \leq x)=P(X_1 \leq x,\cdots ,X_n\leq x)=(P(X_1\leq x))^n=(F_{X_1}(x))^n=(\frac{x}{\theta})^n$$
so $$f_{X_{(n)}}(x)=\frac{nx^{n-1}}{\theta^n} \hspace{1cm} 0\leq x \leq \theta$$
now can you calculate $$E(c X_{(n)})$$ and $$Var(c X_{(n)})$$ ?