Let x1,x2... xn be i.i.d.l ognormal random variables, with parameters mu and sigma^2. That is, ln(xi)~ N(mu, sigma^2).

1. Find the maximum likelihood estimators for mu and sigma^2.
2. Is the m.l.e for mu unbiased?
3. Is the m.l.e for mu consistent?
4. Is the m.l.e for mu efficient?
5. Is the m.l.e for mu sufficient?
6. What is the sampling distribution of the m.l.e for mu?
7. Use the sampling distribution to derive a 100(1- alpha)% confidence interval for mu. ( Assume n is large and sigma^2 is unknown)
8. Suppose a test for the null hypothesis mu=2.5 versus the alternative hypothesis mu not equal to 2.5 is desired. ( Assume n is large and sigma^2 is unknown). Propose a test statistic using the likelihood ratio technique and describe a rule for when the null hypothesis will be rejected if the significance level of the test is to be 0.05.
9. a) Use a data to obtain 95% confidence interval using your formula from Number 7. Does your interval contain the correct value?