On posterior consistency in nonparametric regression problems

• Published in: Journal of multivariate analysis 2007-11, Vol.98 (10), p.1969-1987
• Main Authors: Choi, Taeryon, Schervish, Mark J.
• Format: Article
• Language: English
• Subjects: Almost sure consistency; Almost sure consistency Differentiable functions Empirical probability measure Hellinger metric In probability metric Sieve; Differentiable functions; Distribution theory; Empirical probability measure; Errors; Exact sciences and technology; Hellinger metric; In probability metric; Linear inference, regression; Mathematics; Multivariate analysis; Nonparametric inference; Normal distribution; Probability and statistics; Probability theory and stochastic processes; Regression analysis; Sciences and techniques of general use; Sieve; Statistics; Studies; Variance analysis
• ISSN: 0047-259X; 1095-7243
• DOI: 10.1016/j.jmva.2007.01.004

We provide sufficient conditions to establish posterior consistency in nonparametric regression problems with Gaussian errors when suitable prior distributions are used for the unknown regression function and the noise variance. When the prior under consideration satisfies certain properties, the crucial condition for posterior consistency is to construct tests that separate from the outside of the suitable neighborhoods of the parameter. Under appropriate conditions on the regression function, we show there exist tests, of which the type I error and the type II error probabilities are exponentially small for distinguishing the true parameter from the complements of the suitable neighborhoods of the parameter. These sufficient conditions enable us to establish almost sure consistency based on the appropriate metrics with multi-dimensional covariate values fixed in advance or sampled from a probability distribution. We consider several examples of nonparametric regression problems.