The Underlying Distribution of Optical to X-Ray Spectral Index for Quasars in the Presence of Detection Thresholds

Statistical methods, applied in a standard way to find underlying distributions from data subject to large non-random threshold effects, often underestimate the uncertainties. We show through simulations that increasing the permissible range of the underlying distribution allows conservative estimates of the uncertainties to be derived. Our simulations are roughly matched to measurements of the optical to X-ray spectral index of optically selected quasars in the presence of thresholds similar to those of flux-limited surveys. For each configuration of thresholds, and several derived parameters of interest, we find the extent that it is necessary to increase the permissible range of the underlying distribution so that derived uncertainties are not underestimated. We apply these results to perform nonparametric fits to the most recent compilation of Einstein Observatory X-ray measurements of optically selected quasars. We verify that the mean optical to X-ray spectral index, α_οχο, although the dependence is weaker, or possibly absent, for l_ο ≲ 10³¹ ergs s^-1 Hz^-1. We derive the intrinsic distribution of α_οχ in bands of l_ο, together with uncertainties.