Question

In: Statistics and Probability

1. Consider the following weighted averages of independent random variables X1, X2, X3, all with mean...

1. Consider the following weighted averages of independent random variables X1, X2, X3, all with mean u and variance σ^2

θ1 = 1/3(X1) + 1/3(X2) + 1/3(X3)

θ2 = 1/4(X1) + 2/4(X2) + 1/4(X3)

θ3 = 2/5(X1) + 2/5(X2) + 2/5(X3)

a) Find E[θ1], E[θ2], E[θ3]

b) Are θ1, θ2 and θ3 unbiased for u? Explain

c) Find the variance for θ1, θ2 and θ3

d) If you had to use one of the above estimators, which would you pick? Explain

Solutions

Expert Solution

a)

       

                                                                                                       

        

                                                                     

         

                                                                                                         

b)

    .

c) We will use the fact that are independent here.

                                                             

                                                               

  

                                                                 

d) We would like to choose an unbiased estimator with the least variance.

The estimator has variance , while has variance

Since, the variance for is lesser than that of we will prefer the estimator . You can also note that the variance of is greater than , and we always want an estimator with the least variance.

         

orchestra answered 2 years ago
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