Estimating boundary curves has many applications such as economics, climate science, and medicine. Bayesian trend filtering has been developed as one of locally adaptive smoothing methods to estimate the non-stationary trend of data. This paper develops a Bayesian trend filtering for estimating the boundary trend. To this end, the truncated multivariate normal working likelihood and global-local shrinkage priors based on the scale mixtures of normal distribution are introduced. In particular, well-known horseshoe prior for difference leads to locally adaptive shrinkage estimation for boundary trend. However, the full conditional distributions of the Gibbs sampler involve high-dimensional truncated multivariate normal distribution. To overcome the difficulty of sampling, an approximation of truncated multivariate normal distribution is employed. Using the approximation, the proposed models lead to an efficient Gibbs sampling algorithm via the P\'olya-Gamma data augmentation. The proposed method is also extended by considering a nearly isotonic constraint. The performance of the proposed method is illustrated through some numerical experiments and real data examples.

翻译：边界曲线的估计在经济学、气候科学和医学等诸多领域具有广泛应用。贝叶斯趋势滤波作为一种局部自适应平滑方法，已被用于估计数据的非平稳趋势。本文提出了一种用于估计边界趋势的贝叶斯趋势滤波方法。为此，引入了截断多元正态工作似然函数和基于正态尺度混合分布的全局-局部收缩先验。特别地，针对差分项采用著名的马蹄形先验，可实现对边界趋势的局部自适应收缩估计。然而，吉布斯采样器的全条件分布涉及高维截断多元正态分布。为克服采样困难，采用截断多元正态分布的近似方法。基于该近似，所提模型通过Pólya-Gamma数据增广实现了高效吉布斯采样算法。本文还通过考虑近乎单调约束对该方法进行了扩展。通过数值实验和真实数据实例验证了所提方法的有效性。