Name:______Student ID:______May 19, 2008
FINAL EXAMINATION Statistics 200b D. R. Brillinger
There are 5 questions. Answer 4 in the space provided. IF MORE THAN 4 ANSWERED, FIRST FOUR NOT CROSSED OUT WILL BE GRADED. Answer in the space provided. Show your work. The text, the class notes on the web, and an unsophisticated calculator may be used. Nothing more. Please no questions to the proctor. If you are not sure of the meaning of a question, set down a reasonable interpretation, and provide your answer.You have 3 hours for the exam.
Question 1. Listed below are scores on a midterm.
37, 32, 33, 34, 34, 34, 36, 36 , 36, 37, 37, 38, 38, 39, 39, 38, 39, 35, 33
1 a). Prepare a useful stem-and-leaf of these values.
1 b). Draw a boxplot of these data
1 c). How would you estimate the spread/scale of these data? Give a reason for your answer.
1 d). Give some reasons for preferring to use the median of these data rather than their average in certain situations.
Question 2. Independent observations Yj , j=1,..., n, have Poisson distributions with means mj , where log(mj) = hj, and hj is the linear predictor b0 + b1xj. The xj are known values, not all the same. The n by 2 matrix X is defined to have jth row [1 xj].
a) Set down the likelihood equations for the estimates b0 and b1 and show that, for some function s(.), they can be written in the form
XTs(b0 ,b1) = 0
b) Defining the function s(.) specifically, what general function important to likelihood analysis, is it a particular case of? Indicate some properties of that function.
c) How would you solve for b0 and b1?
d) What is the large sample distribution of (b0 , b1)? Give reasons for your answer.
Question 3. Let ygr , g=1,...,G, r=1,...,R, be independent normal random variables with means mg and common variance s2 .
3 a) Set down the log-likelihood function for this situation.
3b) Find minimal sufficient statistics for the mg‘s and the common variance s2 . Say what you are doing.
3 c) Prove that the maximum likelihood estimate of s2 is statistically independent of those of the mg‘s.
3 d). What is the asymptotic, when R ® ¥, null distribution of the likelihood ratio test of the hypothesis that the mg are all equal?
4. Suppose that random variables Ygj , j=1,..., n , g=1,2 are independent and that they satisfy the normal linear model with E(Ygj ) = xgTb .
4 a) Write down the covariate/explanatory variable matrix, X, for this model.
4 b) Using the standard notation for X and Y, evaluate XTX and XTY.
4 c) Defining the vector Z to have gth element ng-1 åj Ygj g=1,2, show that a weighted least squares estimate based on Z, and an unweighted one based on Y give the same parameter estimates and confidence intervals, when s2 is known. Discuss the result.
4 d) Indicate how the residuals for the two setups differ, and say which is preferable for model checking. Give reasons for your answer.
5. Counts x , y , x are observed from a trinomial density
Pr(X = x, Y = y, Z = z) = m! p1x p2y p3z /(x!y!z!) , x,y,z = 0,...,m , x+y+z = m
with m a non-negative integer, where 0 < p1, p2, p3 <1 and p1+p2+p3 = 1.
5 a). Suppose that p1 = p2 = p . Evaluate the maximum likelihood estimates of p and p3 .
5 b). What is the distribution of the m.l.e. of p?
5 c). How would you approximate the distribution of the difference of the estimates of p and p3 if m were large?
5 d). Indicate briefly how your answers change if p1 ¹ p2 .