Question

In: Statistics and Probability

Consider the model: yi = βxi + ei, i = 1,...,n where E(ei) = 0 and...

Consider the model:

yi = βxi + ei, i = 1,...,n

where E(ei) = 0 and Variance(ei) = σ2 and ei(s) are non-correlated errors.

a) Obtain the minimum-square estimator for β and propose an unbiased estimator for σ2.

b) Specify the approximate distribution of the β estimator.

c) Specify an approximate confidence interval for the parameter β with confidence

coefficient γ, 0 < γ < 1.

Expert Solution

a) We have

To obtain the minimum-square esimator we use,

We minimise Q to get a least- square estimator/ Minimum-square estimator.

Replacing by , we get

Now we take the derivative of Q with respect to , set it equal to 0 and solve the resulting equation to get a minimum.

The above steps will result in the least square estimator ,

The mean square error of wil be its unbiased estimator.

b) will approximately follow a Normal distribution with parameters and .

c) A 95% Confidence Interval of is given by

orchestra answered 2 years ago

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