2005/12/01 by Peter Hall, Joël L. Horowitz, Joel L. Horowitz · 5 citations
Computer Science · Mathematics · #Image and Signal Denoising Methods #Numerical methods in inverse problems #Statistical and numerical algorithms #math.ST #msc:62G08 #msc:62G20 #stat.TH
paper · pdf · doi:10.1214/009053605000000714
published as Annals of Statistics 2005, Vol. 33, No. 6, 2904-2929 · Published at http://dx.doi.org/10.1214/009053605000000714 in the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2005/12/01 · arxiv created 2006/03/06 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We suggest two nonparametric approaches, based on kernel methods and orthogonal series to estimating regression functions in the presence of instrumental variables. For the first time in this class of problems, we derive optimal convergence rates, and show that they are attained by particular estimators. In the presence of instrumental variables the relation that identifies the regression function also defines an ill-posed inverse problem, the “difficulty” of which depends on eigenvalues of a certain integral operator which is determined by the joint density of endogenous and instrumental variables. We delineate the role played by problem difficulty in determining both the optimal convergence rate and the appropriate choice of smoothing parameter.