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A block bootstrap by hand

StatisticsHard

Returns in time order are $2$ , $6$ , $- 3$ , $- 1$ , $4$ , $4$ . They are cut into $3$ consecutive blocks of length $2$ . A block bootstrap draws $3$ blocks with replacement and joins them. What is the variance of the total of the new series, over all such draws? Answer with a whole number.

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Hint

Work with block totals, and treat the total of the new series as a sum of independent draws of a block total.

Solution

The block totals are $8$ , $- 4$ , $8$ , with mean $\frac{12}{3} = 4$ .
The squared deviations sum to $16 + 64 + 16 = 96$ , so one block total has variance $\frac{96}{3} = 32$ .
The sum of $3$ independent draws has variance $3 \times 32 = 96$ , so the answer is $96$ .

Answer: $96$

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