Bootstrap-determined <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math> values in lattice QCD

We present a general method to determine the probability that stochastic Monte Carlo data, in particular those generated in a lattice QCD calculation, would have been obtained were that data drawn from the distribution predicted by a given theoretical hypothesis. Such a probability, or <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>p</a:mi></a:math>-value, is often used as an important heuristic measure of the validity of that hypothesis. The proposed method offers the benefit that it remains usable in cases where the standard Hotelling <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:msup><c:mi>T</c:mi><c:mn>2</c:mn></c:msup></c:math> methods based on the conventional <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:msup><e:mi>χ</e:mi><e:mn>2</e:mn></e:msup></e:math> statistic do not apply, such as for uncorrelated fits. Specifically, we analyze <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:msup><g:mi>q</g:mi><g:mn>2</g:mn></g:msup></g:math>, defined as the correlated <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:msup><i:mi>χ</i:mi><i:mn>2</i:mn></i:msup></i:math> statistic obtained using an arbitrary covariance matrix estimator, and show how to use the bootstrap as a data-driven method to determine the expected distribution of <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:msup><k:mi>q</k:mi><k:mn>2</k:mn></k:msup></k:math> for a given hypothesis with minimal assumptions. This distribution can then be used to determine the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"><m:mi>p</m:mi></m:math>-value for a fit to the data. We also describe a bootstrap approach for quantifying the impact upon this <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"><o:mi>p</o:mi></o:math>-value of estimating population parameters from a single ensemble of <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"><q:mi>N</q:mi></q:math> samples. The overall method is accurate up to a <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"><s:mn>1</s:mn><s:mo>/</s:mo><s:mi>N</s:mi></s:math> bias which we do not attempt to quantify.