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Unformatted text preview: Second Edition 10-5 The first column is nn-1 x1 s followed by nn-1 x2 s followed by, . . ., followed by nn-1 xn s The second column is nn-2 x1 s followed by nn-2 x2 s followed by, . . ., followed by nn-2 xn s, repeated n times The third column is nn-3 x1 s followed by nn-3 x2 s followed by, . . ., followed by nn-3 xn s, repeated n2 times . . . The nth column is 1 x1 followed by 1 x2 followed by, . . ., followed by 1 xn , repeated nn-1 times So now it is easy to see that each column in the data array has mean x, hence the entire bootstrap data set has mean x. Appealing to the 33 3 data array, we can write the numerator of the variance of the bootstrap means as
3 3 3 i=1 j=1 k=1 1 (xi + xj + xk ) - x 3
3 3 2 = 1 32 1 32 3 [(xi - x) + (xj - x) + (xk - x)] i=1 j=1 k=1 3 3 3 2 = (xi - x)2 + (xj - x)2 + (xk - x)2 , i=1 j=1 k=1 because all of the cross terms are zero (since they are the sum of deviations from the mean). Summing up and collecting terms shows that 1 32
3 3 3 3 (xi - x)2 + (xj - x)2 + (xk - x)2 = 3 i=1 j=1 k=1 i=1 (xi - x)2 , and thus the average of the variance of the bootstrap means is 3
3 i=1 (xi 33 - x)2 which is the usual estimate of the variance of X if we divide by n instead of n - 1. The general result should now be clear. The variance of the bootstrap means is
n n n i1 =1 i2 =1 = 1 n2 in =1 n n 1 (xi + xi2 + + xin ) - x n 1
n 2 i1 =1 i2 =1 in =1 (xi1 - x)2 + (xi2 - x)2 + + (xin - x)2 , since all of the cross terms are zero. Summing and collecting terms shows that the sum is n n nn-2 i=1 (xi - x)2 , and the variance of the bootstrap means is nn-2 i=1 (xi - x)2 /nn = n 2 2 (xi - x) /n . i=1 ^ ^ 10.15 a. As B Var () = Var (). B ^ b. Each VarBi () is a sample variance, and they are independent so the LLN applies and 1 m
m ^ m ^ ^ Var i () EVar () = Var (), B B
i=1 where the last equality follows from Theorem 5.2.6(c). ...
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This note was uploaded on 02/03/2012 for the course STA 1014 taught by Professor Dr.hackney during the Spring '12 term at UNF.
- Spring '12