Mathematical Statistics—Homework Assignment #3

Due Monday, February 1 (beginning of class)

Important Reminders

Please respect me, your classmates, and yourself by taking the Honor Code very seriously. Your grade will depend on both the content and exposition of your answers. That is, be sure your logic is clear, you defend all your steps (unless they are, for example, obvious algebra steps), your solutions read smoothly (even if using symbols—they should still read like an English sentences), and that one of your peers could read and understand your solutions without asking any additional questions.

Okay-to-work-together Problems (3 problems)

Chapter 7: 15, 29

Additional Problem:

Suppose are a random sample from a distribution with p.d.f., .

a. Show that the method-of-moments estimator of is .

b. Show that the maximum-likelihood estimator of is .

c. It is difficult to find the distribution of the method-of-moments estimator (although we could use simulation to estimate the sampling distribution), but we can determine the distribution of the maximum-likelihood estimator. Use the moment-generating-function technique to determine the distribution, including parameters, of the maximum-likelihood estimator (use the definition of the m.g.f. and then the definition of expectation—you should recognize the m.g.f. you get in the end).

d. Using your result in part c, show that i) the maximum-likelihood estimator, , is an unbiased estimator of and ii) that the variance of goes to 0 as the sample size goes to infinity. [This property of unbiased estimators (variance converging to 0 in the limit) is called “consistency”—that is, is a “consistent” estimator of .] Note: You can use results we know about the means and variances of common distributions. That is, you don’t need to re-derive the mean of a p.d.f. for which we’re familiar.

Work-alone Problems (5 problems)

Chapter 7: 20, 23, 30

Chapter 8: 10, 11