# Guaranteed root-mean-square estimates of the forecast of matrix observations under conditions of statistical uncertainty

## DOI:

https://doi.org/10.20535/SRIT.2308-8893.2023.2.07## Keywords:

matrix observations, linear estimations, guaranteed RMS estimates, guaranteed RMS estimate errors, quasi-minimax guaranteed vector estimates, difference equation, small parameter method, matrix perturbation## Abstract

We investigate the problem of linear estimation of unknown mathematical expectations based on observations of realizations of random matrix sequences. Constructive mathematical methods have been developed for finding linear guaranteed RMS estimates of unknown non-stationary parameters of average values based on observations of realizations of random matrix sequences. It is shown that such guaranteed estimates are obtained either as solutions to boundary value problems for systems of linear differential equations or as solutions to the corresponding Cauchy problems. We establish the form and look for errors for the guaranteed RMS quasi-minimax estimates of the special forecast vector and parameters of unknown average values. In the presence of small perturbations of known matrices in the model of matrix observations, quasi-minimax RMS estimates are found, and their guaranteed RMS errors are obtained in the first approximation of the small parameter method. Two test examples for calculating the guaranteed root mean square estimates and their errors are given.

## References

Sheng Yue, Application of random matrix theory in Statistics and machine learning. Publicly Accessible Penn Dissertations, 4146, 2021, 240 p. Available: https://repository.upenn.edu/edissertations/4146

Yuan Ke, Stanislav Minsker, Zhao Ren, Qiang Sun, and Wen-Xin Zhou, “Uzer friendly covariance estimation for heavy-tailed distributions,” Statistical Science, 34 (3), pp. 454–471, 2019.

Stanislav Minsker, “Sub-gaussian estimators of mean of a random matrix with heavy-tailed entries,” The Annals of Statistics, 46 (6A), pp. 2871–2903, 2018.

Jun Tong, Rui Hu, Jiangtao Xi, Zhitao Xiao, Qinghua Guo, and Yu. Yanguang, “Linear shrinkage estimation of covariance matrices using complexity cross-validation,” Signal Processing, 148, pp. 223–233, 2018.

Roberto Cabal Lopes, Robust estimation of the mean a random matrix: a non-asymptotic study. Centro de Investigacion en Matematicas, A.C., 2020, 187 p. Available: https://cimat.repositorioinstitucional.mx/jspui/bitstream/1008/1082/1/TE%20785.pdf

H. Battey, J. Fan, J. Lu, and Z. Zhu, “Distributed testing and estimation under sparse high dimensional models,” The Annals of Statistics, 46 (3), pp. 1352–1382, 2018.

T.T. Cai and H. Wei, “Distributed Gaussian mean estimation under communication constraints: Optimal rates and communication-efficient algorithms,” arXiv preprint, arXiv: 2001.08877, 2020.

T. Ke, Y. Ma, and X. Lin, “Estimation of the number of spiked eigenvalues in a covariance matrix by bulk eigenvalue matching analysis,” arXiv preprint, arXiv: 2006.00436, 2020.

C. McKennan, “Factor analysis in high dimensional biological data with dependent observations,” arXiv preprint, arXiv: 2009.11134, 2020.

Sourav Chatterjee, “Matrix Estimation by Universal Singular Value Thresholding,” The Annals of Statistics, vol. 43, no. 1, 2015, pp. 177–214.

O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, and T.P. Zinko, “Perturbation Method in Problems of Linear Matrix Regression,” Problems of Control and Informatics, no. 1, pp. 38–47, 2020.

O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, and T.P. Zinko, “Approximate guaranteed estimates of matrices in linear regression problems with a small parameter,” System Research & Information Technologies, no. 4, pp. 88–102, 2020.

O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, and T.P. Zinko, “Guaranteed Root-Mean-Square Estimates of Linear Transformations of Matrices under Statistical Uncertainty,” Problems of Control and Informatics, no. 2, pp. 24–37, 2021.

O.G. Nakonechnyi, G.I. Kudin, P.M. Zinko, and T.P. Zinko, “Minimax Root-Mean-Square Estimates of Matrix Parameters in Linear Regression Problems under Uncertainty,” Problems of Control and Informatics, no. 4, pp. 28–37, 2021.

K. Ostrem, Introduction to Stochastic Control Theory. M.: Mir, 1973, 324 p.