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Nonparametric binary regression: A Bayesian approach. (English) Zbl 0797.62031

Let be given a binary random variable \(\eta\) depending on a whole sequence of binary covariates \(\xi = (\xi_ k)\) and define \(f(\xi): = P(\eta = 1 \mid \xi)\) which can be considered as a function mapping [0,1] into itself. Let be given \(2^ n\) data with independent \(\eta_ k\) having covariates covering all possible patterns of the first \(n\) covariates, the remaining covariates are assumed to be uniform and independent.
Defining a prior distribution \(\sum^ \infty_{k=1} \pi_ kw_ k/ \sum w_ \nu\) where \(\pi_ k\) is uniform on the set of functions \(f\) which depend on the first \(k\) covariates only and \(w_ k>0\) for infinitely many \(k\) are weights for the dimensions, the authors discuss the consistency of the Bayes estimates for \(f\). If the weights \(w_ k\) are decreasing rapidly enough there is consistency, otherwise the function \(f \equiv 1/2\) is not estimated consistently. In any case, if \(f\) depends only on finitely many covariates the corresponding order can be estimated consistently. Although the model is very specific this is an interesting paper.

MSC:

62G07 Density estimation
62A01 Foundations and philosophical topics in statistics
62G05 Nonparametric estimation
62E20 Asymptotic distribution theory in statistics
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