Consider the example where researchers want to estimate 1/ using a random sample from N(, o)....

 

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Consider the example where researchers want to estimate 1/μ using a random sample from N(μ, o²). They use 1/X, the inverse of the sample mean, for their estimator, and estimate the variance using the delta method. This is an asymptotic variance, meaning it's approximate, with the approximation getting better when the sample size increases. (a) Do simulations with n = 30, µ = 5 with o² = 1 to compare the variance of 1/X with the calculated asymptotic variance. (b) Use your simulations results from (a) to make density histograms of √(1/X 1/μ) and superimpose a mean zero normal density with vari- ance o/. Is the approximated density close to the true density? Tek (c) Do simulations with n = 30, μ = 1/2 with o² 1 to compare the variance of 1/X with the calculated asymptotic variance. S (d) Repeat (c) with n = 100. Is this large enough? What about n 1000? (e) For n = 100 and n = 1000, use your simulation results from (b) and (c) to make density histograms of √n(1/X 1/μ) and superimpose a mean zero normal density with variance σ²/^. Consider the example where researchers want to estimate 1/μ using a random sample from N(μ, o²). They use 1/X, the inverse of the sample mean, for their estimator, and estimate the variance using the delta method. This is an asymptotic variance, meaning it's approximate, with the approximation getting better when the sample size increases. (a) Do simulations with n = 30, µ = 5 with o² = 1 to compare the variance of 1/X with the calculated asymptotic variance. (b) Use your simulations results from (a) to make density histograms of √(1/X 1/μ) and superimpose a mean zero normal density with vari- ance o/. Is the approximated density close to the true density? Tek (c) Do simulations with n = 30, μ = 1/2 with o² 1 to compare the variance of 1/X with the calculated asymptotic variance. S (d) Repeat (c) with n = 100. Is this large enough? What about n 1000? (e) For n = 100 and n = 1000, use your simulation results from (b) and (c) to make density histograms of √n(1/X 1/μ) and superimpose a mean zero normal density with variance σ²/^.