Foul or Fair?

This paper gives a short overview of Monte Carlo studies on the usefulness of Heckman?s (1976, 1979) two?step estimator for estimating a selection model. It shows that exploratory work to check for collinearity problems is strongly recommended before deciding on which estimator to apply. In the absence of collinearity problems, the full?information maximum likelihood estimator is preferable to the limited?information two?step method of Heckman, although the latter also gives reasonable results. If, however, collinearity problems prevail, subsample OLS (or the Two?Part Model) is the most robust amongst the simple?to? calculate estimators. --

Quantile Regression Under Random Censoring

No abstract.

The bootstrap -A review

The bootstrap, extensively studied during the last decade, has become a powerful tool in different areas of Statistical Inference. In this work, we present the main ideas of bootstrap methodology in several contexts, citing the most relevant contributions and illustrating with examples and simulation studies some interesting aspects

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

The X-ray Energy Dependence of the Relation between Optical and X-ray Emission in Quasars

We develop a new approach to the well-studied anti-correlation between the optical-to-X-ray spectral index, alpha_ox, and the monochromatic optical luminosity, l_opt. By cross-correlating the SDSS DR5 quasar catalog with the XMM-Newton archive, we create a sample of 327 quasars with X-ray S/N > 6, where both optical and X-ray spectra are available. This allows alpha_ox to be defined at arbitrary frequencies, rather than the standard 2500 Angstroms and 2 keV. We find that while the choice of optical wavelength does not strongly influence the alpha_ox-l_opt relation, the slope of the relation does depend on the choice of X-ray energy. The slope of the relation becomes steeper when alpha_ox is defined at low (~ 1 keV) X-ray energies. This change is significant when compared to the slope predicted by a decrease in the baseline over which alpha_ox is defined. The slopes are also marginally flatter than predicted at high (~ 10 keV) X-ray energies. Partial correlation tests show that while the primary driver of alpha_ox is l_opt, the Eddington ratio correlates strongly with alpha_ox when l_opt is taken into account, so accretion rate may help explain these results. We combine the alpha_ox-l_opt and Gamma -L_bol/L_Edd relations to naturally explain two results: 1) the existence of the Gamma-l_x relation as reported in Young et al. (2009) and 2) the lack of a Gamma-l_opt relation. The consistency of the optical/X-ray correlations establishes a more complete framework for understanding the relation between quasar emission mechanisms. We also discuss two correlations with the hard X-ray bolometric correction, which we show correlates with both alpha_ox and Eddington ratio. This confirms that an increase in accretion rate correlates with a decrease in the fraction of up-scattered disk photons.Comment: 27 pages preprint style, 10 figures. New material added to section 4.3 (Fig. 11) regarding hard X-ray bolometric corrections. Accepted for publication in Astrophysical Journa