Exercise2.19 Consider asimplelinearregressionmodelwithoutanintercept: Yi = xi+i, i = 1, ...,n, where i are i.i.d N (0,2). 1.

Exercise2.19 Consider asimplelinearregressionmodelwithoutanintercept:

Yi = βxi+εi, i = 1, ...,n, where εi are i.i.d N (0,σ2).

1. Find theMLEfor β. Isitunbiased?FinditsMSE.

2. Consider an estimator β˜ =

Σni

=1Yi

Σni

=1 xi

. Isitunbiased?FinditsMSEandcompareitwiththatofthe MLE.

3. Find theMLEfor σ.

4. Find theMLEfor β/σ.

5. ∗DerivetheFisherinformationmatrix I(β,σ).

6. ∗Find theCramer–Raolowerboundforanunbiasedestimatorof β/σ.