Minimax rates of convergence for nonparametric location-scale models

This paper studies minimax rates of convergence for nonparametric location-scale models, which include mean, quantile and expectile regression settings. Under Hellinger differentiability on the error distribution and other mild conditions, we show that the minimax rate of convergence for estimating the regression function under the squared $L_2$ loss is determined by the metric entropy of the nonparametric function class. Different error distributions, including asymmetric Laplace distribution, asymmetric connected double truncated gamma distribution, connected normal-Laplace distribution, Cauchy distribution and asymmetric normal distribution are studied as examples. Applications on low order interaction models and multiple index models are also given

CNETML: Maximum likelihood inference of phylogeny from copy number profiles of spatio-temporal samples

Phylogenetic trees based on copy number alterations (CNAs) for multi-region samples of a single cancer patient are helpful to understand the spatio-temporal evolution of cancers, especially in tumours driven by chromosomal instability. Due to the high cost of deep sequencing data, low-coverage data are more accessible in practice, which only allow the calling of (relative) total copy numbers due to the lower resolution. However, methods to reconstruct sample phylogenies from CNAs often use allele-specific copy numbers and those using total copy number are mostly distance matrix or maximum parsimony methods which do not handle temporal data or estimate mutation rates. In this work, we developed a new maximum likelihood method based on a novel evolutionary model of CNAs, CNETML, to infer phylogenies from spatio-temporal samples taken within a single patient. CNETML is the first program to jointly infer the tree topology, node ages, and mutation rates from total copy numbers when samples were taken at different time points. Our extensive simulations suggest CNETML performed well even on relative copy numbers with subclonal whole genome doubling events and under slight violation of model assumptions. The application of CNETML to real data from Barrett’s esophagus patients also generated consistent results with previous discoveries and novel early CNAs for further investigations

CNETML: maximum likelihood inference of phylogeny from copy number profiles of multiple samples

Phylogenetic trees based on copy number profiles from multiple samples of a patient are helpful to understand cancer evolution. Here, we develop a new maximum likelihood method, CNETML, to infer phylogenies from such data. CNETML is the first program to jointly infer the tree topology, node ages, and mutation rates from total copy numbers of longitudinal samples. Our extensive simulations suggest CNETML performs well on copy numbers relative to ploidy and under slight violation of model assumptions. The application of CNETML to real data generates results consistent with previous discoveries and provides novel early copy number events for further investigation