Calculation of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:mi>p</mml:mi> </mml:math> -values for quadratic statistics in pulsar timing arrays

Pulsar Timing Array (PTA) projects have reported various lines of evidence suggesting the presence of a stochastic gravitational wave (GW) background in their data. One key line of evidence involves a detection statistic sensitive to interpulsar correlations, such as those induced by GWs. A <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>p</a:mi> </a:math> -value is then calculated to assess how unlikely it is for the observed signal to arise under the null hypothesis <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msub> <c:mi>H</c:mi> <c:mn>0</c:mn> </c:msub> </c:math> , purely by chance. However, PTAs cannot empirically draw samples from <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:msub> <e:mi>H</e:mi> <e:mn>0</e:mn> </e:msub> </e:math> . As a workaround, various techniques are used in the literature to approximate <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>p</g:mi> </g:math> -values under <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>H</i:mi> <i:mn>0</i:mn> </i:msub> </i:math> . One such technique, which has been characterized as a model-independent method, is the use of “scrambling” transformations that modify the data to cancel out pulsar correlations, thereby simulating realizations from <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msub> <k:mi>H</k:mi> <k:mn>0</k:mn> </k:msub> </k:math> . In this work, scrambling methods and the detection statistic are investigated from first principles. The <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:mi>p</m:mi> </m:math> -value methodology that is discussed is general, but the discussions regarding a specific detection statistic apply to the detection of a stochastic background of gravitational waves with PTAs. All methods in the literature to calculate <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>p</o:mi> </o:math> -values for such a detection statistic are rigorously analyzed, and analytical expressions are derived for the distribution of the detection statistic and the corresponding <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>p</q:mi> </q:math> -values. All this leads to the conclusion that scrambling methods are model independent and thus not completely empirical. With a single realization of data our results are necessarily always model dependent, which any analysis will need to accept. Instead of scrambling approaches, rigorous Bayesian and Frequentist <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:mi>p</s:mi> </s:math> -value calculation methods are advocated, the evaluation of which depend on the generalized <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msup> <u:mi>χ</u:mi> <u:mn>2</u:mn> </u:msup> </u:math> distribution. This view is consistent with the posterior predictive <w:math xmlns:w="http://www.w3.org/1998/Math/MathML" display="inline"> <w:mi>p</w:mi> </w:math> -value approach that is already in the literature. Efficient expressions are derived to evaluate the generalized <y:math xmlns:y="http://www.w3.org/1998/Math/MathML" display="inline"> <y:msup> <y:mi>χ</y:mi> <y:mn>2</y:mn> </y:msup> </y:math> distribution of the detection statistic on real data. It is highlighted that no Frequentist <ab:math xmlns:ab="http://www.w3.org/1998/Math/MathML" display="inline"> <ab:mi>p</ab:mi> </ab:math> -values have been calculated correctly in the PTA literature to date.