Semiparametric Bivariate Density Estimation with Irregularly Truncated Data

This work develops an estimator for the bivariate density given a sample of data truncated to a non-rectangular region. Such inference problems occur in various fields; the motivating application here was a problem in astronomy. The approach is semiparametric, combining a nonparametric local likelihood density estimator with a simple parametric form to account for the dependence of the two random variables. Large sample theory for M-estimators is utilized to approximate the distribution for the estimator. A method is described for approximating the integrated mean squared error of the estimator; smoothing ∗Chad Schafer is Visiting Assistant Professor, Department of Statistics, Carnegie Mellon University, Pittsburgh, PA 15213 (email: [email protected]). Research supported by NSF Grants #0434343 and #0240019. The author acknowledges Chris Genovese and Larry Wasserman for many helpful discussions. Funding for the SDSS and SDSS-II has been provided by the Alfred P. Sloan Foundation, the Participating Institutions, the National Science Foundation, the U.S. Department of Energy, the National Aeronautics and Space Administration, the Japanese Monbukagakusho, the Max Planck Society, and the Higher Education Funding Council for England. The SDSS Web Site is http://www.sdss.org/.