02 May Instruction: Show details of your proof and calculation.
Instruction: Show details of your proof and calculation.
Let X , · · · , X
I
~
ID N(?, ?) with ? > 0.
1.Show that the model is a one parameter exponential family. And find the
complete sufficient statistic for ?.
2. Find the MoM
estimator N(?, ? ).
?^1 of ? based on the frist absolute moment of
3. Drive the asymptotic distribution of ?^1 by using CLT and Delta method.
4. Find the MoM estimator ?^2 of ? based on the second moment of N(0, ? ).
5. Drive the asymptotic distribution
of (Hint: E(X
4
) = 3?
4
.)
?^2 by using CLT and Delta method.
6. Compute the ARE of ?^2 with respect to ?^1.
7.Find the MLE of ? and the UMVUE of ?.
8. Derive the UMP test of size a for testing H0 : ? = 1 vs. H1 : ? > 1.
9.Derive the Wilks’s LRT, Wald test and Rao’s score test of size a fortesting
H0 : ? = 1 vs. H1 : ? ƒ= 1.
10.Derive the (1 a) confidence interval of ? by inverting the above Wald
test.
Note that you may need to use the expectation of the so called ?(n) random
variable, which is the square root of the ?
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