Chapter 7: Problem 9
Consider the situation of the last exercise, but suppose we have the following two independent random samples: (1). \(X_{1}, X_{2}, \ldots, X_{n}\) is a random sample with the common pdf \(f_{X}(x)=\theta^{-1} e^{-x / \theta}\), for \(x>0\), zero elsewhere, and (2). \(Y_{1}, Y_{2}, \ldots, Y_{n}\) is a random sample with common pdf \(f_{Y}(y)=\tau e^{-\tau y}\), for \(y>0\), zero elsewhere. Assume that \(\tau=1 / \theta\) The last exercise suggests that, for some constant \(c, Z=c \bar{X} / \bar{Y}\) might be an unbiased estimator of \(\theta^{2}\). Find this constant \(c\) and the variance of \(Z\). Hint: Show that \(\bar{X} /\left(\theta^{2} \bar{Y}\right)\) has an \(F\) -distribution.
Short Answer
Step by step solution
Key Concepts
These are the key concepts you need to understand to accurately answer the question.