{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Second Midterm 2006

# Second Midterm 2006 - AEB 6933 Mathematical II I Assuming...

This preview shows pages 1–3. Sign up to view the full content.

AEB 6933 Mathematical Statistics Examination II November 13, 2006 I. Assuming that the data in Column 2 of Table 1 are distributed Pareto 1 ; , , k m m m k x f x k x k x x x with 5 m x : a. Estimate the shape parameter k using maximum likelihood. (15 points) b. What is the variance of the estimate of k and why? (15 points) c. Test 0 : 0.75 H k against 1 : 0.75 H k . (15 points) II. Using the data in Column 5 of Table 1 and assuming 2 ~ , y N , test for 0 : 6.0 H against 1 : 6.0 H . How does the test change if the variance is known and equal to 2.5? (15 points) III. Suppose we start with a prior distribution ( 1) g p p p for the Bernoulli distribution. The posterior distribution then becomes 1 1 1 0 1 1 1 1 1 1 24 1 1 3 2 x x x x x x p p p p f p x p p p p dp p p p x x Implying 2 2 12 1 for 0 12 1 for 1 p p x f p x p p x Compute the expected value of p for each possible outcome of x . What is the probability of 0.5 p if 1 x ? (20 points) IV. When are ordinary least squares estimates also the maximum likelihood estimators? (20 points)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Table 1. Empirical Data Observation x ln x 2 x y 2 y (1) (2) (3) (4) (5) (6) 1 10.015 2.304 100.291 3.523 12.415 2 46.011 3.829 2116.968 3.722 13.852 3 5.760 1.751 33.174 7.036 49.504 4 97.633 4.581 9532.111 6.384 40.749 5 605.676 6.406 366843.391 3.721 13.849 6 17.319 2.852 299.964 2.845 8.094 7 8.300 2.116 68.894 3.192 10.192 8 37.797 3.632 1428.610 6.620 43.828 9 5.805 1.759 33.698 6.292 39.589 10 48.033 3.872 2307.185 0.759 0.576 11 8.091 2.091 65.465 2.003 4.013 12 386.114
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}