This is an exam. You can use any written material, either in books or online but you can not discuss it with ANYONE.
Do as much as possible analytically, otherwise use numerical method or simulation.

Below is a sample from a population with density \(f(x\vert a)=a/x^{a+1};x>1, a>1\)

1 1.02 1.03 1.03 1.03 1.03 1.03 1.04 1.05 1.05 1.05 1.06 1.06 1.07 1.07 1.08 1.09 1.09 1.09 1.1 1.11 1.11 1.12 1.13 1.13 1.14 1.15 1.15 1.16 1.16 1.16 1.17 1.2 1.2 1.21 1.21 1.22 1.23 1.23 1.23 1.24 1.24 1.24 1.24 1.24 1.25 1.25 1.26 1.28 1.34 1.35 1.41 1.43 1.43 1.43 1.44 1.44 1.44 1.44 1.44 1.45 1.45 1.46 1.47 1.48 1.48 1.51 1.51 1.53 1.55 1.56 1.58 1.58 1.61 1.65 1.66 1.69 1.71 1.73 1.75 1.77 1.78 1.8 1.84 1.87 1.88 1.93 1.97 2.02 2.12 2.26 2.3 2.52 2.62 2.7 2.73 3.1 3.74 3.82 4.29

You can generate your own samples with

rmid=function(n,a) runif(n)^(-1/a)

  1. Find the method of moments and the maximum likelihood estimators of a.

  2. Find the bias of the mle.

  3. Is the mle a consistent estimator of a?

  4. Using the p value approach do the likelihood ratio test at the 5% level to test

\[H_0:a=2\text{ vs. }H_a:a\ne 2\] based on the large sample approximation of the likelihood ratio test

  1. Not using the p-value approach do the likelihood ratio test at the 5% level to test

\[H_0:a=2\text{ vs. }H_a:a\ne 2\] without using the large sample approximation of the likelihood ratio test.

  1. Find the Bayesian point estimator of a using the posterior mean and the prior \(\pi(a)=1/a;a>1\)