assignment 5

assignment 5 - 10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1801 Probability and Statistics: Foundations of Actuarial Science Assignment 5 Due Date: December 6, 2010 (Hand in your solutions for Questions 2, 6, 8, 11, 12, 14, 18, 22) 1. Based on a random sample of 2 observations, consider two competing estimators of the population mean μ : 2 1 1 2 1 2 1 ˆ X X + = , 2 1 2 3 2 3 1 ˆ X X + = (a) Are they unbiased? (b) Which estimator is more efficient? How much more efficient? 2. Let be a random sample from a distribution with finite mean n X X X ,..., , 2 1 and finite variance . An estimator of 2 σ in the form n n X c X c X c L + + + = ... 2 2 1 1 is called a linear estimator , where are some known constants. If L is unbiased, then it is called a linear unbiased estimator . A linear unbiased estimator that has the minimum variance among all linear unbiased estimators is call the best linear unbiased estimator (BLUE) . n a a a ,..., , 2 1 (a) Express and in terms of () L E () L Var , , and . 2 n c c c ,..., , 2 1 (b) Show that the sample mean is the BLUE of . 3. An economist gathers a random sample of 500 observations, and loses the records of the last 180. This leaves only 320 observations from which to calculate the sample mean. What is the efficiency of this, relative to what could have been obtained from the whole sample? 4. Let { be a random sample from a distribution with finite mean } n X X X ,..., , 2 1 and finite variance . Let 2 ( = = n i i X X n ) 1 2 1 1 S 2 be the sample variance. Show that if () 1 = S P (i.e. ), then the sample standard deviation S will always underestimate the population standard deviation () 0 S Var . 5. Let { be a random sample from the normal distribution } n X X X ,..., , 2 1 ( ) 2 , σμ N . Find the constant c such that () = = n i i X X n c cS 1 2 1 1 is an unbiased estimator of . P. 1
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10/11 6. Let { be a random sample from the normal distribution } n X X X ,..., , 2 1 ( ) 2 , σμ N . A commonly used unbiased estimator of is the sample variance defined by 2 σ () = = n i i X X n S 1 2 2 1 1 . (a) Find the MSE of . 2 S (b) Another commonly adopted definition of the sample variance is given by () 2 1 2 2 1 1 S n n X X n SD n i i = = = . Find the MSE of as an estimator of and compare it to . Which one is better? 2 SD 2 2 S (c) Suppose we seek for an estimator that takes the form where c is some positive constant. Find the value of c such that has the smallest MSE among all the estimators in the form of . 2 2 ˆ cS = 2 ˆ 2 cS 7. A market survey of young business executives was undertaken to determine what sort of computer would suit a combination of their professional and personal needs. Since those with more children were thought to be more likely to buy a home computer, one of the questions each executive was asked was, “How many children do you have?” Unfortunately, those with more children tend to have less time and inclination to reply to the
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This note was uploaded on 02/01/2012 for the course STAT 1801 taught by Professor Mrchung during the Fall '10 term at HKU.

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assignment 5 - 10/11 THE UNIVERSITY OF HONG KONG DEPARTMENT...

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