statistics.nb

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The chi - square  statistic follows the following distribution (see \
Wikipedia) :\
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k is the number of degrees of freedom, which in our case is the number of \
fitted parameters. We are fitting one parameter, < \[Sigma]v >, for each \
value of mass, so in our case k=1.\
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Say we discover dark matter.  Wilks' theorem tells us that 2 \[CapitalDelta] \
ln L (which is a function of  < \[Sigma]v >) follows the chi - square \
distribution, so if I want to find what value of 2 \[CapitalDelta] ln L \
corresponds to 95% exclusion, I can integrate the chi-square distribution and \
find where that integral = 0.95\.00\
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So values of < \[Sigma]v > that give 2 \[CapitalDelta] ln L > 3.84 are \
excluded at 95 % CL. These will be values of < \[Sigma]v > that are either \
too big or too small relative to the actual value.\
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However, we have not discovered dark matter. Instead we are just setting an \
upper limit on the cross section. So to find the values that are excluded at \
95 % CL, we have to find where the previous integral is actually equal to 0.90\
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Which is the number you are familiar with. The idea here is that we can only \
exclude values of < \[Sigma]v > that are too large, not too small as before, \
so the \[OpenCurlyDoubleQuote]missing\[CloseCurlyDoubleQuote] 5% comes from \
the fact that the low values are now allowed. \
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I' m hiding alot in this discussion -- mainly how this is all ultimately \
derived from Gaussian statistics, making the whole one - sided limit argument \
a bit more sensible, but that is talked about a bit in the review I sent you, \
I think...\
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