X7p5D - RESEARCH ASSISTANTS You ask two research assistants...

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You ask two research assistants to obtain estimates X 1 and X 2 of the population mean μ of a random variable X . You know from past experience that both of them will obtain unbiased estimates, but one of the assistants is less careful than the other, and the variance of X 2 about the mean will be three times the variance of X 1 . You have to come up with a single figure yourself. Should you take the average of X 1 and X 2 , or discard X 2 and take X 1 as your estimate, or do something else? 1 RESEARCH ASSISTANTS
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2 We will suppose that our estimate, which we will denote Z , should be a linear combination of the estimates of the research assistants. We will derive the optimal values of λ 1 and 2 . 2 2 1 1 ) ( σ μ = = X X E 2 2 2 3 ) ( 2 = = X X E 2 2 1 1 X X Z + = RESEARCH ASSISTANTS
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3 We will first derive the condition that λ 1 and 2 must satisfy for Z to be an unbiased estimator of μ . We use Expected Value Rule 1 to decompose the expected value expression. 2 2 1 1 ) ( σ = = X X E 2 2 2 3 ) ( 2 = = X X E 2 2 1 1 X X Z + = 1 if ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 2 1 1 = + = + = + = + = + = + = X E X E X E X E X X E Z E RESEARCH ASSISTANTS
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4 Next we use Expected Value Rule 2 to bring λ 1 and 2 out of their terms. We are told that E ( X 1 ) and E ( X 2 ) are both equal to μ . 2 2 1 1 ) ( σ = = X X E 2 2 2 3 ) ( 2 = = X X E 2 2 1 1 X X Z + = 1 if ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 2 1 1 = + = + = + = + = + = + = X E X E X E X E X X E Z E RESEARCH ASSISTANTS
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5 Thus the condition for unbiasedness is that λ 1 and 2 should sum to 1. The sample mean, with 1 = 2 = 0.5, satisfies this condition, but that does not mean that it is optimal. 1 if ) ( ) ( ) ( ) ( ) ( ) ( ) ( 2 1 2 1 2 1 2 2 1 1 2 2 1 1 2 2 1 1 = + = + = + = + = + = + = μ X E X E X E X E X X E Z E 2 2 1 1 ) ( σ = = X X E 2 2 2 3 ) ( 2 = = X X E 2 2 1 1 X X Z + = RESEARCH ASSISTANTS
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6 We want our estimator to be as efficient as possible, so we will choose λ 1 and 2 so that its population variance is minimized.
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This note was uploaded on 05/26/2010 for the course ECON 301 taught by Professor Öcal during the Spring '10 term at Middle East Technical University.

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X7p5D - RESEARCH ASSISTANTS You ask two research assistants...

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