Bayesian Inference for k-Monotone Densities with Applications to Multiple Testing
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Shape restriction, like monotonicity or convexity, imposed on a function of interest, such as a regression or density function, allows for its estimation without smoothness assumptions. The concept of k-monotonicity encompasses a family of shape restrictions, including decreasing and convex decreasing as special cases corresponding to k=1 and k=2. We consider Bayesian approaches to estimate a k-monotone density. By utilizing a kernel mixture representation and putting a Dirichlet process or a finite mixture prior on the mixing distribution, we show that the posterior contraction rate in the Hellinger distance is (n/log n)^- k/(2k + 1) for a k-monotone density, which is minimax optimal up to a polylogarithmic factor. When the true k-monotone density is a finite J_0-component mixture of the kernel, the contraction rate improves to the nearly parametric rate √((J_0 log n)/n). Moreover, by putting a prior on k, we show that the same rates hold even when the best value of k is unknown. A specific application in modeling the density of p-values in a large-scale multiple testing problem is considered. Simulation studies are conducted to evaluate the performance of the proposed method.
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