hw6_solution - 131B HW#6 solution 8.7 Unbiased Estimators...

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131B HW#6 solution 8.7 Unbiased Estimators 12. (a) Let X denote the value of the characteristic for a person chosen at random from the total population, and let A i denote the event that the person belongs to stratum i ( i = 1 , . . . , k ). Then μ = E ( X ) = k i =1 E ( X | A i ) P ( A i ) = k i =1 μ i p i . Also, E ( μ ) = k i =1 p i E ( ¯ X i ) = k i =1 p i μ i = μ. (b) Since the samples are taken independently of each other, the variables ¯ X 1 , . . . , ¯ X k are independent. Therefore, var(ˆ μ ) = k i =1 p 2 i var( ¯ X i ) = k i =1 p 2 i σ 2 i n i . Hence, the values of n 1 , . . . , n k must be chosen to minimize v = k i =1 ( p i σ i ) 2 n i , subject to the constraint that k i =1 n i = n . If we let n k = n - k 1 i =1 n i , then ∂v ∂n i = - ( p i σ i ) 2 n 2 i + ( p k σ k ) 2 n 2 k for i = 1 , . . . , k - 1 . When each of these partial derivatives is set equal to 0, it is found that n i / ( p i σ i ) has the same value for i = 1 , . . . , k . Therefore, n i = cp i σ i for some constant c . It follows that n = k j =1 h j = c k j =1 p j σ j . Hence, c = n/ k j =1 p j σ j and, in turn, n i = np i σ i
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