Question

R1. Suppose X is a continuous RV with E(X-μ and Var(X-σ2 where both μ and σ are unknown. Note that X may not be a normal distribution. Show that X is an asymptotically unbiased estimator for μ2. (This problem does not require the computer.) R2. Let X ~ N(μ 10.82). Following up on R1, we will be approximating μ2, which we can see should be 100, For now, let the sample size be n 3. Pick 3 random numbers from X, compute X', and repeat the process a total of 50000 times. Plot a smooth version of the histogram of these 50000 values for X the plot (density (.. command in R will be useful. Now find the average of your 50000 values and make a vertical dotted line in R at this number (match the color of your curve). You have just made a rough picture of the density function for X (when n 3) and identified its (approximate) expected value. Now you should repeat this process for n 6, n10 and n 50. Plot all four of these curves (use different colors) and vertical lines on the same set of axes choosing limits on the axes so the picture looks nice. Finally, put a dark solid vertical line at 2100. You should be able to see the asymptotic unbiasedness in the picture. Your answer is your code and a rough sketch of the plot you have created

Answer 1

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