  Motivated by the problem of estimating the bacterial growth rates for genome
assemblies from shotgun metagenomic data, we consider the permuted monotone
matrix model $Y=\Theta\Pi+Z$, where $Y\in \mathbb{R}^{n\times p}$ is observed,
$\Theta\in \mathbb{R}^{n\times p}$ is an unknown approximately rank-one signal
matrix with monotone rows, $\Pi \in \mathbb{R}^{p\times p}$ is an unknown
permutation matrix, and $Z\in \mathbb{R}^{n\times p}$ is the noise matrix. This
paper studies the estimation of the extreme values associated to the signal
matrix $\Theta$, including its first and last columns, as well as their
difference. Treating these estimation problems as compound decision problems,
minimax rate-optimal estimators are constructed using the spectral column
sorting method. Numerical experiments through simulated and synthetic
microbiome metagenomic data are presented, showing the superiority of the
proposed methods over the alternatives. The methods are illustrated by
comparing the growth rates of gut bacteria between inflammatory bowel disease
patients and normal controls.
