Problem 1

We have one observation from the random variable with \(f(x|\theta)\) where \(\theta \in \{1,2,3\}\) and

Find the mle of \(\theta\).

Problem 2

we have a sample of size n from a rv with distribution given by

where of \(\alpha>1\) and \(\beta>0\).

  1. If we know \(\beta\), what is the mle of \(\alpha\)?

In parts b-e assume we know \(\alpha\)

  1. What is the mle of \(\beta\)?
  2. is the mle of \(\beta\) a sufficient statistic?
  3. Is the mle of \(\beta\) unbiased?
  4. Is the mle of \(\beta\) consistent?
  5. Show that the Rao-Cramer lower bound for \(\beta\) does not hold.

Hint: consider \(S(\pmb{x})=c x_{[n]}\) where c is such that S is an unbiased estimator of \(\beta\).