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I am trying to understand why the simple bootstrap procedure does not work, but for now I would like to know why we can we write $s^2_n = \frac{1}{n} \sum^n_{i = 1} X^2_i - (\bar{X}_n)^2$ ?

This is a part of a paper by Liu and Singh (1992) Moving blocks jackknifes and bootstrap capture weak dependence.

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    $\begingroup$ This is a standard result, achieved with a few algebraic manipulations. You can check it by noting that both formulas for $s_n^2$ are invariant under permutations of the $X_i$, so it suffices to verify that the coefficients of $X_1^2$ and $X_1X_2$ are the same in both expressions. $\endgroup$
    – whuber
    Apr 16 '15 at 16:29
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    $\begingroup$ If the value of whuber's comment isn't clear to you, take the first form, expand the square, and then simplify and collect terms. $\endgroup$
    – Glen_b
    Apr 16 '15 at 16:34
  • $\begingroup$ $X_1 , X_2 \dots X_n$ are iid? $\endgroup$
    – Monolite
    Apr 16 '15 at 16:39
  • $\begingroup$ @Monolite the iid assumption does not matter for this question, which does not really deal with random variables--it's only about the variance formula. (IID is needed to make sense of the assertion about convergence in the quoted material, though.) $\endgroup$
    – whuber
    Apr 16 '15 at 16:42
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    $\begingroup$ I cannot answer that without seeing exactly what the author says. But comments are no place for follow-up questions. For informal discussions you can try our chat room or--this is probably better--you can formulate a new question about this inconsistency. $\endgroup$
    – whuber
    Apr 17 '15 at 15:35

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