Nonparametric Bayesian regression. (English) Zbl 0608.62052
The paper addresses itself to Bayesian estimation of the function
\[
F(x_ 1,x_ 2)=m+a(x_ 1)+b(x_ 2)+c(x_ 1,x_ 2)
\]
in the model \(y_ i=F(x_{1i},x_{2i})+e_ i\). A prior for F is constructed by putting independent priors on m,a,b, and c. They are normal distribution and Brownian motion. The proposed estimate of F is the limit of Bayesian estimates as Var(m)\(\to \infty.\)
It is demonstrated that the estimator corresponds to the minimum of \(\sum (y_ i-F(x_{1i},x_{2i}))^ 2+P(F)\), where P(F) is a suitable penalty. Asymptotic properties are considered also in the grid case, and relations to variance components analysis are examined.
It is demonstrated that the estimator corresponds to the minimum of \(\sum (y_ i-F(x_{1i},x_{2i}))^ 2+P(F)\), where P(F) is a suitable penalty. Asymptotic properties are considered also in the grid case, and relations to variance components analysis are examined.
Reviewer: R.Schlittgen