2. Let Y, Y2, Y3, Y and Y5 be iid (, 0). Let = ...

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2. Let Y₁, Y2, Y3, Y₁ and Y5 be iid (µ, 0²). Let Ý = ¹ Σ1 Yt. t=1 (a) What are the expected value and variance of Y? (b) Now, consider a different estimator of µ: W What are the expected value and variance of W? (e) Find Var (Wa). (f) For any - numbers ai, (c) Which estimator of μ do you prefer? Fully justify your answer. n i=1 Let Y₁, Y2, Y3,,, Yn ~ iid (µ, 0²) and let Ỹ = ½ Σ₁ Yį. As we discussed in class, if more than one estimators are unbiased, we can compare their variances (in general, MSE) to pick a better point estimator. One related comcept is the best linear unbiased estimator (BLUE). (d) Define the class of linear estimator of µ by Y₁+ Y₂+ i = 1, where as are constants. What restriction on the as is need for W₁ to be an unbiased estimator of u? ... n Y3 + W₁ = Σa;Y₁ a i=1 Y₁₁ 2 n, the following inequality holds n n 1 (₂) se Ai a². n i=1 i=1 Use this (and above results) to show Ỹ is the best linear unbiased estimator (BLUE). 2. Let Y₁, Y2, Y3, Y₁ and Y5 be iid (µ, 0²). Let Ý = ¹ Σ1 Yt. t=1 (a) What are the expected value and variance of Y? (b) Now, consider a different estimator of µ: W What are the expected value and variance of W? (e) Find Var (Wa). (f) For any - numbers ai, (c) Which estimator of μ do you prefer? Fully justify your answer. n i=1 Let Y₁, Y2, Y3,,, Yn ~ iid (µ, 0²) and let Ỹ = ½ Σ₁ Yį. As we discussed in class, if more than one estimators are unbiased, we can compare their variances (in general, MSE) to pick a better point estimator. One related comcept is the best linear unbiased estimator (BLUE). (d) Define the class of linear estimator of µ by Y₁+ Y₂+ i = 1, where as are constants. What restriction on the as is need for W₁ to be an unbiased estimator of u? ... n Y3 + W₁ = Σa;Y₁ a i=1 Y₁₁ 2 n, the following inequality holds n n 1 (₂) se Ai a². n i=1 i=1 Use this (and above results) to show Ỹ is the best linear unbiased estimator (BLUE).

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