Chapter 5

Exercises: 5.1, 5.2, 5.3(a)

Also:

Suppose $X_1, X_2, \ldots, X_n$ is an iid random sample from a uniform distribution on the interval $[0, \theta]$. Let $Y = X_{(n)}$ be the maximum of $X_1, X_2, \ldots, X_n$.

1. Use the fact that $P(Y \le y) = P(X_1 \le y, X_2 \le y, \ldots, X_n \le y)$ to find the cumulative distribution function of $Y$.
2. Use the previous result to find the probability density function of $Y$.
3. Is $Y$ an unbiased estimator for $\theta$? If not, find a constant $k$ such that $W = kY$ is an unbiased estimator for $Y$.
4. Find the variance of $W$.
5. Compare the variance of $W$ with the variance of the method of moments estimator $2\bar{X}$.