The problem of estimating a piecewise monotone sequence of normal means is called the nearly isotonic regression. For this problem, an efficient algorithm has been devised by modifying the pool adjacent violators algorithm (PAVA). In this study, we investigate estimation of a piecewise monotone parameter sequence for general one-parameter exponential families such as binomial, Poisson and chi-square. We develop an efficient algorithm based on the modified PAVA, which utilizes the duality between the natural and expectation parameters. We also provide a method for selecting the regularization parameter by using an information criterion. Simulation results demonstrate that the proposed method detects change-points in piecewise monotone parameter sequences in a data-driven manner. Applications to spectrum estimation, causal inference and discretization error quantification of ODE solvers are also presented.

翻译：分段单调正态均值序列的估计问题被称为近单调回归。针对该问题，已有学者通过修正池相邻违规算法（PAVA）设计了高效算法。本研究针对二项分布、泊松分布和卡方分布等一般单参数指数族，探究分段单调参数序列的估计问题。我们基于修正PAVA开发了一种高效算法，该算法利用自然参数与期望参数之间的对偶性。此外，我们还提出了一种使用信息准则选择正则化参数的方法。仿真结果表明，所提方法能够以数据驱动方式检测分段单调参数序列中的变点。同时展示了该方法在频谱估计、因果推断以及常微分方程求解器离散化误差量化中的应用。