Master of Applied Statistics
Comprehensive Exam: Theory
January 2016
Directions: this is a closed book exam with a three-hour time limit. Attached you will find three pages of formulas and tables for the t, 2, and F distributions. You may use a non-programmable, non-graphing calculator. Answer only five of the six questions.
1. Let Y1,Y2,…,Yn be independent and identically distributed as N(0,2) for some 0.
a)Find the MLE for 2. Justify.
b)Why is an MLE(in general) usually considered to be a good estimator for a parameter?
c)Find the UMVUE for 2. Justify.
2. The Beta-Binomial model. Let X1,X2,…,Xnbe independent Bernoulli random variables
with probability of success (i.e. of being 1) given by p.
a)Assume the prior distribution of p is given by abeta(; ) distribution. Derive the posterior distribution of p.
b)Romney (1999) looks at the level of consensus among 24 Guatemalan women on whetherthey think polio is non-contagious. The survey data are given below:
1 1 1 1 0 1 1 0 1 0 1 1 1 0 1 1 1 1 1 1 0 0 0 1
where 1 indicates that the respondent believes polio to be non-contagious and 0 indicates thatthe respondent believes polio to be contagious. Let p denote the probability of each womanbelieving polio to be non-contagious. Apply the beta-binomial model with a beta(1; 1) prior to find the posterior distribution of p.
c)For the above data, find a Bayes point estimate and a 90% credible interval for p.
3. Suppose X ~ exponential(1) and Y ~ exponential(2), where X and Y are independent and 1, 2 >0. Define Z = min(X,Y) and let W = 1 if X < Y, and W = 0 otherwise.
a)Derive the probability mass function of W.
b)What is the distribution of Z? justify your answer.
c) Consider two simple random samples (X1,X2,…Xn) from exponential(1) and (Y1,Y2,…Yn) from exponential(2), and the two samples are independent of each other. Define Zi = min(Xi,Yi) for i = 1,2,…,n. What is the distribution of ? Also, describe its limiting distribution under appropriate standardization.
4. A retailer buys items from a supplier; each item can be either acceptable or defective,
andseparate items are independent.
a)Suppose the probability of each item being defective is 0.1. What is the probability that
there are 6 defective items in a lot of 25?
b)Suppose there are 6 defective items in a lot of 25. If 7 items are randomly sampled withoutreplacement from the lot, what is the probability of finding no defective item among the 7items?
c)The lot will be unacceptable if more than 5 items are defective. Suppose the retailerselects randomly K items and decides to accept the lot if there is no defective item in thesample. How large does K have to be to ensure that the probability that the retailer acceptsan unacceptable lot is less than 0.10?
5. Let Y1,Y2,…,Yn be independent and identically distributed as Poisson() for some >0.
a)Show that the most powerful level- test of H0: =1 vs. Ha: =2 rejects H0 when Yi c for some c.
b)Argue that your test from part (a) is UMP for testing H0: =1 vs. Ha: > 1.
c)One can test H0: =0 vs. Ha: 0 for any choice of the GLR test rejects H0ifYi c1 or Yi c2 for c1, c2 depending on 0,, and n. Assuming that you have a computer program that can find c1 and c2 for every choice of 0, , and n. How might you construct a conservative 95% confidence region for given a value for Yi ?
6. Suppose X1~ gamma(1, 1) and X2 ~ gamma(2, 1), where X1 and X2are independent and 1, 2 > 0. Justify your answers to each of the following.
a)Define V = X1 + X2. What is the distribution of V?
b)Define U = X1/(X1 + X2). What is the distribution of U?
c)What is the conditional distribution of U given V?
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