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 \(\theta\).

max{L(\(\theta\)|0} = 1/3, f(0|1)=1/3, so mle of \(\theta\) is 1
max{L(\(\theta\)|1} = 1/3, f(1|1)=1/3, so mle of \(\theta\) is 1
max{L(\(\theta\)|2} = 1/4, f(2|2)=f(2|3)=1/4, so mle of \(\theta\) is 2 or 3
max{L(\(\theta\)|3} = 1/2, f(3|3)=1/2, so mle of \(\theta\) is 3
max{L(\(\theta\)|4} = 1/4, f(4|3)=1/4, so mle of \(\theta\) is 3

Problem 2

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

  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 \(\beta\)?

\(L(\beta) = 0\) for \(\beta<\max\{x\}=x_{[n]}\). If \(\beta>\max\{x\}=x_{[n]}\)

\(L(\beta) = \alpha^n/\beta^{n\alpha}\prod x_i^{\alpha-1}\)

so L is strictly decreasing for \(\beta>\max\{x\}=x_{[n]}\), and so argmax{L(\(\beta\)|x)} = x[n]

  1. Is the mle of \(\beta\) a sufficient statistic?

and T=x[n] is a sufficient statistic for \(\beta\).

  1. Is T unbiased?

so T is not unbiased (but it is asymptotically unbiased)

  1. Is T consistent?

because P(X[n]<\(\beta\)+\(\epsilon\))=P(X[n]<\(\beta\))=1

so T is consistent

  1. Show that the Rao-Cramer lower bound for \(\beta\) does not hold.