1% ============================================================================2% UQO Working Paper No. 28 — Beyond One-Step Distributed Maximum Likelihood3% Preamble = UQO template (Recherche-UQO/HARMONISATION.md) + amsthm environments + algorithmic4% ============================================================================5\documentclass[12pt,letterpaper]{article}6\usepackage[utf8]{inputenc}7\usepackage[T1]{fontenc}8\usepackage[english]{babel}9\usepackage[letterpaper,top=1in,bottom=1in,left=1in,right=1in]{geometry}10\usepackage{mathptmx}11\usepackage{microtype}12\usepackage{setspace}13\onehalfspacing14\usepackage{amsmath,amssymb,amsthm}15\usepackage{mathtools}16\theoremstyle{plain}17\newtheorem{theorem}{Theorem}18\newtheorem{proposition}{Proposition}19\newtheorem{lemma}{Lemma}20\newtheorem{corollary}{Corollary}21\theoremstyle{definition}22\newtheorem{assumption}{Assumption}23\newtheorem{definition}{Definition}24\theoremstyle{remark}25\newtheorem{remark}{Remark}26\newtheorem{example}{Example}27\usepackage{booktabs}28\usepackage{tabularx}29\usepackage{multirow}30\usepackage{threeparttable}31\usepackage{longtable}32\usepackage{siunitx}33\sisetup{input-symbols = {()}, table-format = -1.3, parse-numbers = false}34\usepackage{graphicx}35\graphicspath{{figures/}}36\usepackage{subcaption}37\usepackage{float}38\usepackage{placeins}39\usepackage[dvipsnames]{xcolor}40\usepackage[colorlinks=true,linkcolor=NavyBlue,citecolor=NavyBlue,urlcolor=NavyBlue,breaklinks=true]{hyperref}41\usepackage[nameinlink,capitalise,noabbrev]{cleveref}42\crefname{assumption}{Assumption}{Assumptions}\Crefname{assumption}{Assumption}{Assumptions}43\usepackage[authoryear,round,semicolon]{natbib}44\bibliographystyle{aer}45\usepackage[font=small,labelfont=bf,labelsep=period,justification=justified,singlelinecheck=false]{caption}46\usepackage{fancyhdr}47\pagestyle{fancy}48\fancyhf{}49\renewcommand{\headrulewidth}{0pt}50\fancyfoot[C]{\thepage}51\usepackage{titlesec}52\titleformat{\section}{\large\bfseries}{\thesection.}{0.5em}{}53\titleformat{\subsection}{\normalsize\bfseries}{\thesubsection.}{0.5em}{}54\titleformat{\subsubsection}{\normalsize\itshape}{\thesubsubsection.}{0.5em}{}55\usepackage[toc,page]{appendix}56\usepackage{enumitem}57\setlist{nosep,leftmargin=*}58\usepackage{algorithm}59\usepackage{algpseudocode}60\usepackage{tikz}61\usetikzlibrary{arrows.meta,positioning,calc}6263% --- notation macros64\newcommand{\E}{\mathbb{E}}65\newcommand{\Var}{\operatorname{Var}}66\newcommand{\Cov}{\operatorname{Cov}}67\newcommand{\R}{\mathbb{R}}68\newcommand{\N}{N}69\newcommand{\dd}{\mathrm{d}}70\newcommand{\norm}[1]{\lVert #1 \rVert}71\newcommand{\abs}[1]{\lvert #1 \rvert}72\newcommand{\thetahat}{\widehat{\theta}}73\newcommand{\thetaN}{\widehat{\theta}_N}74\newcommand{\pilot}[1]{\theta^{(0)}_{#1}}75\newcommand{\iter}[2]{\theta^{(#2)}_{#1}}76\newcommand{\op}{O_p}77\newcommand{\smallop}{o_p}78\newcommand{\pto}{\xrightarrow{\;p\;}}79\newcommand{\dto}{\Rightarrow}80\newcommand{\alphafr}[2]{\alpha(#1,#2)}81\DeclareMathOperator*{\argmax}{arg\,max}82\DeclareMathOperator{\tr}{tr}83\DeclareMathOperator{\diag}{diag}84\newcommand{\Jr}[1]{J_{#1}}85\newcommand{\Amp}{A_r}86\newcommand{\betar}{\beta_r}8788% ============================================================================89% METADATA90% ============================================================================91\newcommand{\WPnumber}{28}92\newcommand{\WPtitle}{Beyond One-Step Distributed Maximum Likelihood: Local Bias Cancellation, Global Synchronization, and the Price of Inexact Curvature}93\newcommand{\WPsubtitle}{A joint order--round frontier, its sharpness in an exact model, and measured communication costs}94\newcommand{\WPdate}{October 2026}95\newcommand{\WPversion}{1.0}96\newcommand{\WPabstract}{%97 \input{sections/abstract}%98}99\newcommand{\WPkeywords}{distributed estimation, maximum likelihood, split-sample jackknife, one-step Newton estimator, communication complexity, asymptotic equivalence, Monte Carlo}100\newcommand{\WPjel}{C13, C15, C55, C63}101\newcommand{\WPauthor}{Simon-Pierre Boucher\\[0.15cm]{\normalsize with the AI co-author designation GPT-6.1 (prior exploratory draft; see the disclosure on page~\pageref{sec:provenance})}}102\newcommand{\WPaffiliation}{%103 Département des sciences administratives\\104 Université du Québec en Outaouais%105}106\newcommand{\WPemail}{simon-pierre.boucher@uqo.ca}107\newcommand{\WPaddress}{%108 Gatineau -- Pavillon Alexandre-Taché\\109 283, boulevard Alexandre-Taché\\110 Gatineau, Québec, Canada J9A 1L8%111}112\hypersetup{pdftitle={Beyond One-Step Distributed Maximum Likelihood: Local Bias Cancellation, Global Synchronization, and the Price of Inexact Curvature}, pdfauthor={Simon-Pierre Boucher}}113114\input{tables/numbers}115116\begin{document}117\input{sections/titlepage}118\input{sections/provenance}119\input{sections/introduction}120\input{sections/literature}121\input{sections/environment}122\input{sections/construction}123\input{sections/results}124\input{sections/exactmodel}125\input{sections/inexact}126\input{sections/costmodel}127\input{sections/montecarlo}128\input{sections/benchmarks}129\input{sections/robustness}130\input{sections/discussion}131\input{sections/conclusion}132133\newpage134\bibliography{references}135136\begin{appendices}137\input{appendices/proofs}138\input{appendices/exponential}139\input{appendices/algorithms}140\input{appendices/replication}141\end{appendices}142\end{document}143