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In: Statistics and Probability

Consider a large population which has true mean µ and true standard deviation σ. We take...

Consider a large population which has true mean µ and true standard deviation σ. We take a sample of size 3 from this population, thinking of the sample as the RVs X1, X2, X3 where Xi can be considered iid (independent identically distributed). We are interested in estimating µ.

(a) Consider the estimator ˆµ1 = X1 + X2 − X3. Is this estimator biased? Show your work

(b) Find the variance of ˆµ1.

(c) Consider the estimator ˆµ2 = X1+X2+X3 3 . Is this estimator biased? Show your work

(d) Find the variance of ˆµ2. (e) Now, consider the estimator ˆµ3 = X1+2X2+3X3 6 . Is this estimator biased? Show your work

(f) Find the variance of ˆµ3.

(g) Which of these three estimators is preferable? Why?

(h) Now let µ = 100 and σ = 10. Define the estimator ˆµ4 = X1+X2+X3 3 + X1. Compute the MSE for ˆµ4.

(i) Is ˆµ4 ever a better estimator than ˆµ1?

(j) Extra Credit. If the answer to the preceding question is yes, find the range of values of for which this is the case. Assume that is small, so you can drop all terms with an 2 in them

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orchestra answered 2 years ago
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