SOLN_3b

L2 is best most efficient to compute the mean of a

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Unformatted text preview: σ2 )= (iii) var( L3 ) = σ2 ( ⋅ + ⋅ 4 35 4 105 105 Þ 3. L2 is best (most efficient) ˆ To compute the mean of a observe that 2 éæ 1 öù ˆ E (a ) = E (Wn2 ) = E êç å i Wi ÷ ú øú êè n ë û 1é 2 = 2 E å i Wi + å i å j ≠i WW j ù i ë û n 1 2 = 2 é å i E (Wi ) + å i å j ≠i E (WW j ) ù i û në But by independence, if µ = E (W ) and σ 2 = var(W ) , then E (WiW j ) = E (Wi ) E (W j ) = µ 2 , and σ 2 = E (W 2 ) − µ 2 Þ µ 2 = E (W 2 ) − σ 2 So recalling that a = E (W 2 ) , we see that 1 [nE (W 2 ) + n(n − 1) µ 2 ] 2 n 1 n −1 [ E (W 2 ) − σ 2 ] = E (W 2 ) + n n n −1 2 n −1 2 σ = a− σ <a = E (W 2 ) − n n ˆ E (a) = ˆ Þ a is biased, and always underestimates a. O...
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