Astrostatistics: Goodness-of-Fit and All That!

We consider the problem of fitting a parametric model to an astronomical dataset. This occurs when fitting simple heuristic models (e.g. powerlaw relation in a bivariate scatter diagram) or complex astrophysical models (e.g. thermal plasma model of an X-ray spectrum). After regression procedures have found the ‘best’ fit, the chi-squared (χ2) or Kolmogorov-Smirnov (K-S) statistics are often used to evaluate confidence limits for the parameters and goodness-of-fit probabilities. However, these procedures have mathematical limitations and biases which are often not fully recognized among astronomers. Here we offer a recently developed approach that works under very general conditions (e.g. correlated parameter estimators). We combine K-S statistics with bootstrap resampling to achieve unbiased parameter confidence bands. This method can be extended using the Kullback-Leibler distance measure to discriminate goodness-of-fit between models. This method is unusual in that it can treat different families of models, as well as nested models within one family. This is an example of how contemporary statistics can address methodological issues that often confront astronomers. Penn State has recently created a Center for Astrostatistics to facilitate development and promulgation of statistical expertise for astronomy and related observational sciences. We are developing tutorials in methodology and software, promoting cross-disciplinary research, providing Web resources, and otherwise serving the statistical needs of astronomers. 1. The Problem of Model Selection and Fitting The aim of model fitting is to provide most parsimonious ‘best fit of a parametric model to data. It might be a simple, heuristic model to phenomenological relationships between observed properties in a sample of astronomical objects. Examples include characterizing the Fundamental Plane of elliptical galaxies or the power law index of solar flare energies. Perhaps more important are complex nonlinear models based on our astrophysical understanding of the observed phenomenon. Here, if the model family truly represents the underlying phenomenon, the fitted parameters give insights into sizes, masses, compositions, temperatures, geometries, and evolution of astronomical objects. Examples of astrophysical modeling include: