Question: [1] (30 points) Consider the density fX(x) = 913x2,o [1] (30 points) Consider the density fx(x) = for the random variable X. [a] Develop the

[1] (30 points) Consider the density fX(x) = 913x2,o

[1] (30 points) Consider the density fx(x) = for the random variable X. [a] Develop the maximum likelihood estimator, for theta. [b] Develop the method of moments estimator for theta. [c] Consistent with definition 5.4.2 derive the relative efficiency of the maximum likelihood estimator to the method of moments estimator for theta. [d] Generate random deviates from fx(x) for sample size using by equating the uniform random number r to the cdf and solving for x. For 1000 samples of size compute the mle and method of moments estimator for theta, determine the sample variances for these estimators based on the 1000 samples and report the simulated relative efficiency. [e] Find an unbiased estimator based on 91. [f] Is a sufficient estimator? How do you know without use of the factorization theorem?

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