2 This is a more general version of Problem C.1. Let Y1, Y2, ..., Y, be n pairwise uncorrelated random variables with common mean m and common variance o. Let Y denote the sample average. (i) Define the class of linear estimators of u by W. = a,Y, + a,Y, + ..+ a,Y. where the a, are constants. What restriction on the a, is needed for W to be an unbiased estimator of u? (ii) Find Var( W.). Math Refresher C Fundamentals of Mathematical Statistics 745 For any numbers a1, az. ..., a, the following inequality holds: (a, + a, + .. + a,}/n s a + a + + a. Use this, along with parts (i) and (ii), to show that Var( Wa) = Var(Y) whenever W is unbiased, so that Y is the best linear unbiased estimator. [Hint: What does the inequality become when the a, satisfy the restriction from part (i)?] (iii)
2 This is a more general version of Problem C.1. Let Y1, Y2, ..., Y, be n pairwise uncorrelated random variables with common mean m and common variance o. Let Y denote the sample average. (i) Define the class of linear estimators of u by W. = a,Y, + a,Y, + ..+ a,Y. where the a, are constants. What restriction on the a, is needed for W to be an unbiased estimator of u? (ii) Find Var( W.). Math Refresher C Fundamentals of Mathematical Statistics 745 For any numbers a1, az. ..., a, the following inequality holds: (a, + a, + .. + a,}/n s a + a + + a. Use this, along with parts (i) and (ii), to show that Var( Wa) = Var(Y) whenever W is unbiased, so that Y is the best linear unbiased estimator. [Hint: What does the inequality become when the a, satisfy the restriction from part (i)?] (iii)
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![2 This is a more general version of Problem C.1. Let Y1, Y2, ..., Y, be n pairwise uncorrelated random
variables with common mean m and common variance o. Let Y denote the sample average.
(i) Define the class of linear estimators of u by
W. = a,Y, + a,Y, + ..+ a,Y.
where the a, are constants. What restriction on the a, is needed for W to be an unbiased estimator of u?
(ii) Find Var( W.).
Math Refresher C Fundamentals of Mathematical Statistics 745
For any numbers a1, az, ..., a, the following inequality holds:
(a, + a, + ... + a,}/n s a + a + + a. Use this, along with parts (i) and (ii), to show
that Var( Wa) = Var(Y) whenever W is unbiased, so that Y is the best linear unbiased estimator.
[Hint: What does the inequality become when the a, satisfy the restriction from part (i)?]
(iii)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7316bc90-902c-436d-9111-7217d6db79ee%2Fd05d0f9e-71de-4217-a4d6-828167194582%2Fgi2nwoo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2 This is a more general version of Problem C.1. Let Y1, Y2, ..., Y, be n pairwise uncorrelated random
variables with common mean m and common variance o. Let Y denote the sample average.
(i) Define the class of linear estimators of u by
W. = a,Y, + a,Y, + ..+ a,Y.
where the a, are constants. What restriction on the a, is needed for W to be an unbiased estimator of u?
(ii) Find Var( W.).
Math Refresher C Fundamentals of Mathematical Statistics 745
For any numbers a1, az, ..., a, the following inequality holds:
(a, + a, + ... + a,}/n s a + a + + a. Use this, along with parts (i) and (ii), to show
that Var( Wa) = Var(Y) whenever W is unbiased, so that Y is the best linear unbiased estimator.
[Hint: What does the inequality become when the a, satisfy the restriction from part (i)?]
(iii)
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