THE UNDERLYING DISTRIBUTION OF THE OPTICAL TO X-RAY SPECTRAL INDEX FOR QUASARS IN THE PRESENCE OF DETECTION THRESHOLDS

Statistical methods, applied in a standard way to find underlying dist ributions from data subject to large non-random threshold effects, oft en underestimate the uncertainties. We show through simulations that i ncreasing the permissible range of the underlying distribution allows conservative estimates of the uncertainties to be derived. Our simulat ions are roughly matched to measurements of the optical to X-ray spect ral index of optically selected quasars in the presence of thresholds similar to those of flux-limited surveys. For each configuration of th resholds, and several derived parameters of interest, we find the exte nt that it is necessary to increase the permissible range of the under lying distribution so that derived uncertainties are not underestimate d. We apply these results to perform nonparametric fits to the most re cent compilation of Einstein Observatory X-ray measurements of optical ly selected quasars. We verify that the mean optical to X-ray spectral index, alpha(ox), generally increases with 2500 Angstrom spectral lum inosity, l(o), although the dependence is weaker, or possibly absent, for l(o) less than or similar to 10(31) ergs s(-1) Hz(-1). We derive t he intrinsic distribution of alpha(ox) in bands of l(o), together with uncertainties.