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psiii140a - University of California Department of...

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University of California D. Steigerwald Department of Economics Economics 140A Problem Set III 1. Let Y 1 ,Y 2 ,...,Y n be a random sample from a population with mean µ and variance σ 2 . Thus each Y i is an independently and identically distributed (i.i.d.) random variable with mean µ and variance σ 2 . Consider the following two estimators of µ Y Y n Y iY i i i n i i n i n = = = = = å å å 1 1 1 ~ a. Show that both estimators are linear functions of Y i . b. Show that both estimators are unbiased. c. Compute the variance of each estimator. Which estimator is more efficient? Does your answer depend on the sample size (n)? Explain. d. Now consider the case where Y i is distributed independently with mean µ and variance σ i i n 2 12, , , ..., = (that is, Y i is not identically distributed). Assume that Y i is measured with more and more precision as i increases (i=1,2,...,n). In fact, the following inequality is true: σ σ σ 1 2 2 2 2 > > > ... . n Under these circumstances, which estimator ( Y Y or ~ ) would you choose? Why?
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2. Consider the financial data on four families presented in the table below.
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