1. Suppose that a family of four wants to estimate the number of goldfish in its pond (there are only goldfish in the pond). Call this number N. The family decides to use a capture/recapture strategy for collecting data. They plan to capture a random sample of Afish, mark them so they can be identified later, and then release them back into the pond. After allowing sufficient time for these fish to circulate with the other fish, they will then capture a new, independent random sample of ngoldfish and observe how many of those ngoldfish are marked. Let the random variable Xrepresent the number of goldfish captured in the second sample that are marked as having also been caught in the first sample.
(a) Derive an expression for the method-of-moments estimator of N, in terms of A, n, and X.
(b) Now suppose that the family captures 10 goldfish in the first sample and 12 goldfish in the second sample, of which 5 had been marked. Determine the methodof-moments estimate of N.
2. Reconsider the previous problem. Again suppose that the family captures 10 goldfish in the first sample and 12 goldfish in the second sample, of which 5 had been marked.
(a) Write out the likelihood function for N, and produce a sketch (you may use software for the sketch, excel will work).
(b) Determine the maximum likelihood estimate of N(again, you may use software to choose the maximum likelihood value from part (a)).
3. Let Y1, ..., Ynconstitute a random sample from a population with probability density function g(y|?) = (?+ 1)(y?) for 0 1, where the parameter ?is greater than -1.
(a) Determine the maximum likelihood estimator of ?.
(b) Determine the method-of-moments estimator of ?.
4. Let the random variable Yibe the number of typographical errors on a page of a 400-page book (for i = 1, ... , 400), and suppose that the Yi0sare independent and identically distributed according to a Poisson distribution with parameter ?. Let the random variable Xbe the number of pages of this book that contain at least one typographical error. Suppose that you are told the value of Xbut are not told anything about the values of Yi.
(a) Determine the maximum likelihood estimator of ?through the following steps:
i. Identify the probability distribution of X.
ii. Write out the probability function of Xin terms of the parameter ?.
iii. Write out the likelihood function of ?and also the log likelihood function of?.
iv. Take the derivative of the log likelihood function with respect to?.
v. Set that derivative equal to zero and solve for ?.
(b) Suppose that the data reveal that 25 of the 400 pages contain a typographical error. What is the maximum likelihood estimate of ??
(c) Show that you could have used the invariance property of maximum likelihood estimators to determine the MLE of ?.