  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.
