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 boundary trend. To this end, the truncated multivariate normal working likelihood and global-local shrinkage priors based on 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 Pólya-Gamma data augmentation. The proposed method is also extended by considering nearly isotonic constraint. The performance of the proposed method is illustrated through some numerical experiments and real data examples.

估计边界曲线有许多应用，例如经济学、气候科学和医学。贝叶斯趋势滤波已被发展为局部自适应平滑方法之一，用于估计数据的非平稳趋势。本文开发了一种用于估计边界趋势的贝叶斯趋势过滤。为此，引入了基于正态分布尺度混合的截断多元正态工作似然和全局局部收缩先验。特别是，众所周知的差异马蹄先验导致了边界趋势的局部自适应收缩估计。然而，吉布斯采样器的完整条件分布涉及高维截断多元正态分布。为了克服采样的困难，采用截断多元正态分布的近似。使用该近似，所提出的模型通过 Pólya-Gamma 数据增强产生有效的吉布斯采样算法。所提出的方法还通过考虑近等渗约束进行了扩展。通过一些数值实验和真实数据示例说明了所提出方法的性能。