{"paper":{"title":"Distribution of the least-squares estimators of a single Brownian trajectory diffusion coefficient","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cond-mat.stat-mech","authors_text":"Carlos Mejia-Monasterio, David S. Dean, Denis Boyer, Gleb Oshanin","submitted_at":"2013-01-18T13:14:11Z","abstract_excerpt":"In this paper we study the distribution function $P(u_{\\alpha})$ of the estimators $u_{\\alpha} \\sim T^{-1} \\int^T_0 \\, \\omega(t) \\, {\\bf B}^2_{t} \\, dt$, which optimise the least-squares fitting of the diffusion coefficient $D_f$ of a single $d$-dimensional Brownian trajectory ${\\bf B}_{t}$. We pursue here the optimisation further by considering a family of weight functions of the form $\\omega(t) = (t_0 + t)^{-\\alpha}$, where $t_0$ is a time lag and $\\alpha$ is an arbitrary real number, and seeking such values of $\\alpha$ for which the estimators most efficiently filter out the fluctuations. W"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.4374","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}