2x [1] Consider the density fx (x; 8) = 122,8 x 1; 0 <...

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2x [1] Consider the density fx (x; 8) = 122,8 ≤ x ≤ 1; 0 < 8 < 1. [a] Find the maximum likelihood estimator (mle) for 8. [b] Clearly, the mle is not unbiased in this situation. Explore asymptotic unbiasedness through simulation. Simulate the performance of this estimator based on random samples of four sizes: 10, 100, 1000, and 10000, for the specific density when & = 0.5. In each sample size instance, repeat the process 10 times and take the mean performance for the estimator based on that sample size. The inverse transformation approach for generating the random deviates for X from Uniform (0, 1) will suffice. [i] Report simulated expected performance(mean) of the estimator for each sample size. [ii] Explain why this suggests asymptotic unbiasedness for your estimator. 2x [1] Consider the density fx (x; 8) = 122,8 ≤ x ≤ 1; 0 < 8 < 1. [a] Find the maximum likelihood estimator (mle) for 8. [b] Clearly, the mle is not unbiased in this situation. Explore asymptotic unbiasedness through simulation. Simulate the performance of this estimator based on random samples of four sizes: 10, 100, 1000, and 10000, for the specific density when & = 0.5. In each sample size instance, repeat the process 10 times and take the mean performance for the estimator based on that sample size. The inverse transformation approach for generating the random deviates for X from Uniform (0, 1) will suffice. [i] Report simulated expected performance(mean) of the estimator for each sample size. [ii] Explain why this suggests asymptotic unbiasedness for your estimator.

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