COVERAGE OF CREDIBLE INTERVALS IN NONPARAMETRIC MONOTONE REGRESSION

For nonparametric univariate regression under a monotonicity constraint on the regression function f, we study the coverage of a Bayesian credible interval for f (x(0)), where x(0) is an interior point. Analysis of the posterior becomes a lot more tractable by considering a projection-posterior distribution based on a finite random series of step functions with normal basis coefficients as a prior for f. A sample f from the resulting conjugate posterior distribution is projected on the space of monotone increasing functions to obtain a monotone function f* closest to f, inducing the projection-posterior. We use projection-posterior samples to obtain credible intervals for f (x(0)). We obtain the asymptotic coverage of the credible interval thus constructed and observe that it is free of nuisance parameters involving the true function. We observe a very interesting phenomenon that the coverage is typically higher than the nominal credibility level, the opposite of a phenomenon observed by Cox (Ann. Statist. 21 (1993) 903-923) in the Gaussian sequence model. We further show that a recalibration gives the right asymptotic coverage by starting from a lower credibility level that can be explicitly calculated.

Automated summary

This study investigates the coverage of a Bayesian credible interval for a regression function under a monotonicity constraint, using a projection-posterior distribution for analysis. Sample projections onto the space of monotone increasing functions are used to obtain credible intervals for a specific point. The study also examines the phenomenon of higher coverage compared to nominal credibility levels, with a proposed recalibration method for achieving the right asymptotic coverage.