Some Aspects of Measurement Error in Linear Regression of Astronomical Data

I describe a Bayesian method to account for measurement errors in linear regression of astronomical data. The method allows for heteroscedastic and possibly correlated measurement errors, and intrinsic scatter in the regression relationship. The method is based on deriving a likelihood function for the measured data, and I focus on the case when the intrinsic distribution of the independent variables can be approximated using a mixture of Gaussians. I generalize the method to incorporate multiple independent variables, non-detections, and selection effects (e.g., Malmquist bias). A Gibbs sampler is described for simulating random draws from the probability distribution of the parameters, given the observed data. I use simulation to compare the method with other common estimators. The simulations illustrate that the Gaussian mixture model outperforms other common estimators and can effectively give constraints on the regression parameters, even when the measurement errors dominate the observed scatter, source detection fraction is low, or the intrinsic distribution of the independent variables is not a mixture of Gaussians. I conclude by using this method to fit the X-ray spectral slope as a function of Eddington ratio using a sample of 39 z < 0.8 radio-quiet quasars. I confirm the correlation seen by other authors between the radio-quiet quasar X-ray spectral slope and the Eddington ratio, where the X-ray spectral slope softens as the Eddington ratio increases.Comment: 39 pages, 11 figures, 1 table, accepted by ApJ. IDL routines (linmix_err.pro) for performing the Markov Chain Monte Carlo are available at the IDL astronomy user's library, http://idlastro.gsfc.nasa.gov/homepage.htm

A test for partial correlation with censored astronomical data

A new procedure is presented, which allows, based on Kendall's \tau, to test for partial correlation in the presence of censored data. Further, a significance level can be assigned to the partial correlation -- a problem which hasn't been addressed in the past, even for uncensored data. The results of various tests with simulated data are reported. Finally, we apply this newly developed methodology to estimate the influence of selection effects on the correlation between the soft X--ray luminosity and both total and core radio luminosity in a complete sample of Active Galactic Nuclei

The least squares method in heteroscedastic censored regression models

Consider the heteroscedastic polynomial regression model $Y = \beta_0 + \beta_1X + ... + \beta_pX^p + \sqrt{Var(Y|X)}\epsilon$, where \epsilon is independent of X, and Y is subject to random censoring. Provided that the censoring on Y is 'light' in some region of X, we construct a least squares estimator for the regression parameters whose asymptotic bias is shown to be as small as desired. The least squares estimator is defined as a functional of the Van Keilegom and Akritas (1999) estimator of the bivariate distribution $P(X \leq x, Y \leq y)$, and its asymptotic normality is obtained

A Comparative Study of Two Real Root Isolation Methods

Recent progress in polynomial elimination has rendered the computation of the real roots of ill-conditioned polynomials of high degree (over 1000) with huge coefficients (several thousand digits) a critical operation in computer algebra. To rise to the occasion, the only method-candidate that has been considered by various authors for modification and improvement has been the Collins-Akritas bisection method [1], which is a based on a variation of Vincent’s theorem [2]. The most recent example is the paper by Rouillier and Zimmermann [3], where the authors present&nbsp;“... a new algorithm, which is optimal in terms of memory usage and as fast as both Collins and Akritas’ algorithm and Krandick variant ...” [3] In this paper we compare our own continued fractions method CF [4] (which is directly based on Vincent’s theorem) with the best bisection method REL described in [3]. Experimentation with the data presented in [3] showed that, with respect to time, our continued fractions method CF is by far superior to REL, whereas the two are about equal with respect to space

Estimating Black Hole Masses in Active Galaxies Using the Halpha Emission Line

It has been established that virial masses for black holes in low-redshift active galaxies can be estimated from measurements of the optical continuum strength and the width of the broad Hbeta line. Under various circumstances, however, both of these quantities can be challenging to measure or can be subject to large systematic uncertainties. To mitigate these difficulties, we present a new method for estimating black hole masses. From analysis of a new sample of broad-line active galactic nuclei, we find that Halpha luminosity scales almost linearly with optical continuum luminosity and that a strong correlation exists between Halpha and Hbeta line widths. These two empirical correlations allow us to translate the standard virial mass system to a new one based solely on observations of the broad Halpha emission line.Comment: to appear in Apj; 8 pages; 5 figures; uses emulateapj5.st