(3) Suppose you have an independent sample of two observations, denoted y₁ and y2, from a population of interest. Further, suppose that E(yi) = μ and Var(yi) = o², i = 1, 2. Consider the following estimator of μ: û = cy₁ + dy2, for some given constants c and d that you are able to choose. Think about this question as deciding how to weight the observations y₁ and y2 (by choosing c and d) when estimating μ. (3a) Under what condition will be an unbiased estimator of ? (Your answer will state a restriction on the constants c and d in order for the estimator to be unbiased). (3b) Given your answer in (3a), solve for d in terms of c and substitute that result back into the expression for above. Note that the resulting estimator, now a function of c only, is unbiased. Once you have made this substitution, what is the variance of û in terms of o² and c? 2 3 (3c) What is the value of c that minimizes the variance expression in (3b)? (You will need to take a derivative to determine the optimal c). Can you provide any intuition for this result?
(3) Suppose you have an independent sample of two observations, denoted y₁ and y2, from a population of interest. Further, suppose that E(yi) = μ and Var(yi) = o², i = 1, 2. Consider the following estimator of μ: û = cy₁ + dy2, for some given constants c and d that you are able to choose. Think about this question as deciding how to weight the observations y₁ and y2 (by choosing c and d) when estimating μ. (3a) Under what condition will be an unbiased estimator of ? (Your answer will state a restriction on the constants c and d in order for the estimator to be unbiased). (3b) Given your answer in (3a), solve for d in terms of c and substitute that result back into the expression for above. Note that the resulting estimator, now a function of c only, is unbiased. Once you have made this substitution, what is the variance of û in terms of o² and c? 2 3 (3c) What is the value of c that minimizes the variance expression in (3b)? (You will need to take a derivative to determine the optimal c). Can you provide any intuition for this result?
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
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Problem 1P
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Transcribed Image Text:(3) Suppose you have an independent sample of two observations, denoted y₁ and y2, from
a population of interest. Further, suppose that E(yi) = μ and Var(yi) = o2, i = 1, 2.
Consider the following estimator of μ:
û = cy₁ + dy2,
for some given constants c and d that you are able to choose. Think about this question as
deciding how to weight the observations y₁ and y2 (by choosing c and d) when estimating μ.
(3a) Under what condition will be an unbiased estimator of u? (Your answer will state a
restriction on the constants c and d in order for the estimator to be unbiased).
(3b) Given your answer in (3a), solve for d in terms of c and substitute that result back into
the expression for above. Note that the resulting estimator, now a function of c only, is
unbiased.
Once you have made this substitution, what is the variance of û in terms of o2 and c?
2/3
3
(3c) What is the value of c that minimizes the variance expression in (3b)? (You will need to
take a derivative to determine the optimal c). Can you provide any intuition for this result?
(3d) Re-derive the variance in part b, but this time suppose that Var(y₁) = o² and Var(y₂) =
30². If the variances are unequal in this way (and y2 is much "noisier"), what is the value of
c that minimizes the variance expression? Comment on any intuition behind your result.
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