TY - JOUR

T1 - Asymptotics of an empirical bridge of regression on induced order statistics

AU - Kovalevskii, Artyom Pavlovich

N1 - Funding Information: Kovalevskii, A.P., Asymptotics of an empirical bridge of regression on induced order statistics. © 2020 Kovalevskii A.P. The research is supported by Mathematical Center in Akademgorodok under agreement No. Funding Information: The proof is complete. Acknowledgements 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. The author would like to thank an anonimous referee for helpful and constructive comments and suggestions. Publisher Copyright: © 2020 Kovalevskii A.P. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020

Y1 - 2020

N2 - We develop a class of statistical tests for analysis of multivariate data. These statistical tests verify the hypothesis of a linear regression model. To solve the question of the applicability of the regression model, one needs a statistical test to determine whether the actual multivariate data corresponds to this model. If the data does not correspond to the model, then the latter should be corrected. The developed statistical tests are based on an ordering of data array by some null variable. With this ordering, all observed variables become concomitants (induced order statistics). Statistical tests are based on functionals of the process of sequential (under the introduced ordering) sums of regression residuals. We prove a theorem on weak convergence of this process to a centered Gaussian process with continuous trajectories. This theorem is the basis of an algorithm for analysis of multivariate data for matching a linear regression model. The proposed statistical tests have several advantages compared to the commonly used statistical tests based on recursive regression residuals. So, unlike the latter, the statistics of the new tests are invariant to a change in ordering from direct to reverse. The proof of the theorem is based on the Central Limit Theorem for induced order statistics by Davydov and Egorov (2000).

AB - We develop a class of statistical tests for analysis of multivariate data. These statistical tests verify the hypothesis of a linear regression model. To solve the question of the applicability of the regression model, one needs a statistical test to determine whether the actual multivariate data corresponds to this model. If the data does not correspond to the model, then the latter should be corrected. The developed statistical tests are based on an ordering of data array by some null variable. With this ordering, all observed variables become concomitants (induced order statistics). Statistical tests are based on functionals of the process of sequential (under the introduced ordering) sums of regression residuals. We prove a theorem on weak convergence of this process to a centered Gaussian process with continuous trajectories. This theorem is the basis of an algorithm for analysis of multivariate data for matching a linear regression model. The proposed statistical tests have several advantages compared to the commonly used statistical tests based on recursive regression residuals. So, unlike the latter, the statistics of the new tests are invariant to a change in ordering from direct to reverse. The proof of the theorem is based on the Central Limit Theorem for induced order statistics by Davydov and Egorov (2000).

KW - Concomitants

KW - Empirical bridge

KW - Regression residuals

KW - Weak convergence

UR - http://www.scopus.com/inward/record.url?scp=85092396393&partnerID=8YFLogxK

UR - https://www.elibrary.ru/item.asp?id=44726581

U2 - 10.33048/semi.2020.17.070

DO - 10.33048/semi.2020.17.070

M3 - Article

AN - SCOPUS:85092396393

VL - 17

SP - 954

EP - 963

JO - Сибирские электронные математические известия

JF - Сибирские электронные математические известия

SN - 1813-3304

M1 - 68

ER -