Consider the the linear regression model Yi = β xi + ei , where the numbers x1, . . . , xn are known, the independent random variables e1, . . . , en have the N(0, σ 2 ) distribution, and the parameters β and σ 2 are unknown.

1) Find the maximum likelihood estimator for β .

My answer:

Sxy/Sxx where Sxx is the sum of squares (xi^2) and Sxy is the sum of the yis and xis.

2) State the Gauss–Markov theorem in the context of this model.

My answer: the Maximum likelihood estimator above is the best unlinear unbiased estimator for β. (Ie it has the smallest covariance of any other estimator)

3)Write down the distribution of an arbitrary linear estimator for β . Hence show that there exists a linear, unbiased estimator βb for β such that Eβ, σ 2 [(βb − β)^ 4 ] 6 Eβ, σ 2[(βe − β)^4 4 ] for all linear, unbiased estimators βe.

I am confused about what the distribution of an arbitary estimator is - I know it will be normal but I am not sure about the parameters; Is the estimator a linear function of y?

Please help!

Reference : Q19 from this document http://www.maths.cam.ac.uk/sites/www.maths.cam.ac.uk/files/pre2014/undergrad/pastpapers/2010/Part_IB/PaperIB_1.pdf


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