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This paper presents rigorous tests of pulsar timing array methods andsoftware, examining their consistency across a wide range of injectedparameters and signal strength. We discuss updates to the 15-year isotropicgravitational-wave background analyses and their corresponding coderepresentations. Descriptions of the internal structure of the flagshipalgorithms \texttt{Enterprise} and \texttt{PTMCMCSampler} are given tofacilitate understanding of the PTA likelihood structure, how models are built,and what methods are currently used in sampling the high-dimensional PTAparameter space. We introduce a novel version of the PTA likelihood that uses atwo-step marginalization procedure that performs much faster when the whitenoise parameters remain fixed. We perform stringent tests of consistency andcorrectness of the Bayesian and frequentist analysis software. For the Bayesiananalysis, we test prior recovery, injection recovery, and Bayes factors. Forthe frequentist analysis, we test that the cross-correlation-based optimalstatistic, when modified to account for a non-negligible gravitational-wavebackground, accurately recovers the amplitude of the background. We alsosummarize recent advances and tests performed on the optimal statistic in theliterature from both GWB detection and parameter estimation perspectives. Thetests presented here validate current and future analyses of PTA data.

The NANOGrav 15-year Gravitational-Wave Background Analysis Pipeline