  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 results 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 et al. (2002) 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 PG1634+706.
