The problem of estimating the conditional mean function in a nonparametric regression model is one of the most important in statistical inference. While large sample properties of regression estimators arising in a frequentist approach to the problem have been studied for a long time, the frequentist asymptotic behavior of Bayesian regression estimators has begun to be investigated only in recent years. We consider a random design normal regression model with regression function in an ellipsoidal class in L2. We assign a prior on the given class by putting a prior on the coefficients in a series expansion of the regression function through an orthonormal system. We derive the rate of convergence of the posterior distribution and compare it with the minimax rate under L2-loss for point estimators. We show that the posterior expected regression function attains the minimax rate for L2-distance over ellipsoidal classes, thus providing an optimal Bayesian procedure for regression function estimation.
On Bayesian nonparametric regression function estimation
SCRICCIOLO, CATIA
2005
Abstract
The problem of estimating the conditional mean function in a nonparametric regression model is one of the most important in statistical inference. While large sample properties of regression estimators arising in a frequentist approach to the problem have been studied for a long time, the frequentist asymptotic behavior of Bayesian regression estimators has begun to be investigated only in recent years. We consider a random design normal regression model with regression function in an ellipsoidal class in L2. We assign a prior on the given class by putting a prior on the coefficients in a series expansion of the regression function through an orthonormal system. We derive the rate of convergence of the posterior distribution and compare it with the minimax rate under L2-loss for point estimators. We show that the posterior expected regression function attains the minimax rate for L2-distance over ellipsoidal classes, thus providing an optimal Bayesian procedure for regression function estimation.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.