smid09 - E4702 Statistical Inference for FE Professor S Kou...

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E4702. Statistical InferenceforFE .Pro fessorS .Kou . Sample Midterm Solution, Summer 2009, 8/12/2007, 11amm-1:30pm. Closed Book Exam . Total 40 pts. 1. (5 pts) (a) (1 pt) St. Petersburg Stock Exchange in Russia. (b) (2 pts) A natural estimator for μ 1 μ 2 is 1 n 1 n 1 X i =1 X i 1 n 2 n 2 X k =1 Y k . Since samples from X ’s and from Y ’s are independent, we have Var à 1 n 1 n 1 X i =1 X i 1 n 2 n 2 X k =1 Y k ! = à 1 n 1 n 1 X i =1 X i ! + à 1 n 2 n 2 X k =1 Y k ! = σ 2 X n 1 + σ 2 Y n 2 . We can estimate σ 2 X and σ 2 Y by their corresponding sample variances s 2 X and s 2 Y . Therefore, the asymptotic 95% c.i. is given by à 1 n 1 n 1 X i =1 X i 1 n 2 n 2 X k =1 Y k ! ± 1 . 96 s σ 2 X n 1 + σ 2 Y n 2 . (c) (2pts) From the de f nition of variance ( ˆ θ θ )= E [( ˆ θ θ ) 2 ] ( E [ ˆ θ θ ]) 2 . Therefore, rewrite the above equation we have MSE ( ˆ θ E [( ˆ θ θ ) 2 ]= ( ˆ θ θ )+( E [ ˆ θ θ ]) 2 = ( ˆ θ Bias ( ˆ θ )) 2 . The last equality holds because θ is a constant.
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smid09 - E4702 Statistical Inference for FE Professor S Kou...

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