TY - JOUR

T1 - Asymptotics of sums of regression residuals under multiple ordering of regressors

AU - Chebunin, M. G.

AU - Kovalevskii, A. P.

N1 - Funding Information: Chebunin, M.G., Kovalevskii, A.P., Asymptotics of sums of regression residuals under multiple ordering of regressors. © 2021 Chebunin M.G., Kovalevskii A.P. The work is supported by Mathematical Center in Akademgorodok under agreement No. 075-15-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation. Received June, 14, 2021, published December, 2, 2021. Publisher Copyright: © 2021 Chebunin M.G., Kovalevskii A.P.

PY - 2021

Y1 - 2021

N2 - We prove theorems about the Gaussian asymptotics of anempirical bridge built from residuals of a linear model under multipleregressor orderings. We study the testing of the hypothesis of a linearmodel for the components of a random vector: one of the componentsis a linear combination of the others up to an error that does notdepend on the other components of the random vector. The independentcopies of the random vector are sequentially ordered in ascending orderof several of its components. The result is a sequence of vectors ofhigher dimension, consisting of induced order statistics (concomitants)corresponding to different orderings. For this sequence of vectors, withoutthe assumption of a linear model for the components, we prove alemma of weak convergence of the distributions of an appropriatelycentered and normalized process to a centered Gaussian process withalmost surely continuous trajectories. Assuming a linear relationship ofthe components, standard least squares estimates are used to computeregression residuals, that is, the differences between response values andthe predicted ones by the linear model. We prove a theorem of weakconvergence of the process of sums of of regression residuals under thenecessary normalization to a centered Gaussian process.

AB - We prove theorems about the Gaussian asymptotics of anempirical bridge built from residuals of a linear model under multipleregressor orderings. We study the testing of the hypothesis of a linearmodel for the components of a random vector: one of the componentsis a linear combination of the others up to an error that does notdepend on the other components of the random vector. The independentcopies of the random vector are sequentially ordered in ascending orderof several of its components. The result is a sequence of vectors ofhigher dimension, consisting of induced order statistics (concomitants)corresponding to different orderings. For this sequence of vectors, withoutthe assumption of a linear model for the components, we prove alemma of weak convergence of the distributions of an appropriatelycentered and normalized process to a centered Gaussian process withalmost surely continuous trajectories. Assuming a linear relationship ofthe components, standard least squares estimates are used to computeregression residuals, that is, the differences between response values andthe predicted ones by the linear model. We prove a theorem of weakconvergence of the process of sums of of regression residuals under thenecessary normalization to a centered Gaussian process.

KW - Concomitants

KW - Copula

KW - Regression residuals

KW - Weak convergence

UR - http://www.scopus.com/inward/record.url?scp=85123521151&partnerID=8YFLogxK

UR - https://elibrary.ru/item.asp?id=47669588

U2 - 10.33048/semi.2021.18.111

DO - 10.33048/semi.2021.18.111

M3 - Article

AN - SCOPUS:85123521151

VL - 18

SP - 1482

EP - 1492

JO - Сибирские электронные математические известия

JF - Сибирские электронные математические известия

SN - 1813-3304

IS - 2

M1 - 48

ER -