Searching for narrow emission lines in X-ray spectra: Computation and methods

The detection and quantification of narrow emission lines in X-ray spectra is a challenging statistical task. The Poisson nature of the photon counts leads to local random fluctuations in the observed spectrum that often result in excess emission in a narrow band of energy resembling a weak narrow line. From a formal statistical perspective, this leads to a (sometimes highly) multimodal likelihood. Many standard statistical procedures are based on (asymptotic) Gaussian approximations to the likelihood and simply cannot be used in such settings. Bayesian methods offer a more direct paradigm for accounting for such complicated likelihood functions, but even here multimodal likelihoods pose significant computational challenges. The new Markov chain Monte Carlo (MCMC) methods developed in 2008 by van Dyk and Park, however, are able to fully explore the complex posterior distribution of the location of a narrow line, and thus provide valid statistical inference. Even with these computational tools, standard statistical quantities such as means and standard deviations cannot adequately summarize inference and standard testing procedures cannot be used to test for emission lines. In this paper, we use new efficient MCMC algorithms to fit the location of narrow emission lines, we develop new statistical strategies for summarizing highly multimodal distributions and quantifying valid statistical inference, and we extend the method of posterior predictive p-values proposed by Protassov and coworkers to test for the presence of narrow emission lines in X-ray spectra. We illustrate and validate our methods using simulation studies and apply them to the Chandra observations of the high-redshift quasar PG 1634+706.