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I ran across a new paper from the Berkeley NLP group on statistical testing, An Empirical Investigation of Statistical Significance in NLP.

There is pseudocode for computing a p-value in the paper, basically, the idea is that the sample set of $x_1,x_2,...,x_N$ are sampled with replacement from data $x$. Then

$\text{p-value} = \text{count}(\delta(x_i) > 2\delta(x))/N$, where $\delta(x_i)$ is a metric gain.

I could understand the formula for computing the p-value in Koehn's paper Statistical significance tests for machine translation evaluation, in which:

$\text{p-value} = \text{count}(\delta_a(x_i) < \delta_b(x_i))/N$, where $\delta_a$ and $\delta_b$ are the metric gain for system $a$ and $b$ respectively.

Is there any explanation or reference for the formula $\text{p-value} = \text{count}(\delta(x_i) > 2\delta(x))/N$. The authors also noted that if the mean of $\delta(x_i)$ is $\delta(x)$ and $\delta(x_i)$ is symmetric, then both formulas above are equivalent.

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As far as I understand from looking at section 2, the authors seem to explain their rationale for the bootstrap test as follows-

"the $x_i$ were sampled from $x$, and so their average $\delta(x_i)$ won’t be zero like the null hypothesis demands; the average will instead be around $\delta(x)$... The solution is a re-centering of the mean – we want to know how often $A$ does more than $\delta(x)$ better than expected. We expect it to beat $B$ by $\delta(x)$. Therefore, we count up how many of the $x_i$ have $A$ beating $B$ by at least $\delta(x)$."

The authors want to test if the gain is non-zero so they write the p-value as $\delta(x_i) < 2\delta(x)$ , which could be re-written as $0 < 2\delta(x) - \delta(x_i)$; because $E[\delta(x_i)]=\delta(x)$ the R.H.S. of the inequality then becomes $\delta(x)$, which is the $H_0$ they were seeking to reject.

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