Convergence rates for Bayesian estimation and testing in monotone regression

Shape restrictions such as monotonicity on functions often arise naturally in statistical modeling. We consider a Bayesian approach to the estimation of monotone regression function and testing for monotonicity. construct prior distribution using piecewise constant functions. For estimation, imposing heights these steps is sensible, but resulting posterior harder analyze theoretically. “projection-posterior” approach, where conjugate normal used, constraint imposed samples by projection map onto space show that contracts at optimal rate n−1∕3 under L1-metric nearly empirical Lp-metrics 0<p≤2. The projection-posterior also computationally more convenient. test hypothesis probability shrinking neighborhood set has universal consistency property obtain separation which ensures power approaches one.