KEY MATHEMATICAL PROPERTIES OF THE GENERALIZED MAXIMUM LIKELIHOOD METHOD

Tat'yana P. Krasnoslobodceva; Nataliya A. Mihaylova; Tat'yana V. Timchenko

doi:10.55186/2413046X_2025_10_9_209

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KEY MATHEMATICAL PROPERTIES OF THE GENERALIZED MAXIMUM LIKELIHOOD METHOD

Published in MOSCOW ECONOMIC JOURNAL · Volume 10, Issue 9 · Pages 70–84 · Rubric: Economic theory

DOI: https://doi.org/10.55186/2413046X_2025_10_9_209

Received: 14.09.2025 Accepted: 02.10.2025 Published: 02.10.2025

Authors

Tat'yana P. Krasnoslobodceva

Nataliya A. Mihaylova

Tat'yana V. Timchenko

Tat'yana P. Krasnoslobodceva

Academic degree: candidate of physical and mathematical sciences

Science rang: docent

Federal State Budgetary Educational Institution of Higher Education "MIREA – Russian Technological University" Associate Professor · Department of Higher and Applied Mathematics · Russian Federation · Moscow

Nataliya A. Mihaylova

MIREA - Russian Technological University Senior Lecturer · Department of Higher and Applied Mathematics · Russian Federation · Moscow

Tat'yana V. Timchenko

Federal State Budgetary Educational Institution of Higher Education «MIREA – Russian Technological University» Senior Lecturer · Department of Higher and Applied Mathematics · Russian Federation · Moscow

ABSTRACT

A variety of statistical methods are widely used in modern economic research and management practice, as well as in various industries and fields of activity. Applied statistics is a scientific discipline that develops methods for processing empirical data. Within the framework of this discipline, there are traditionally three main areas: descriptive data analysis, parameter estimation theory, and statistical hypothesis testing. One of the fundamental approaches in estimation theory is the maximum likelihood method, based on optimizing the corresponding likelihood function. In this paper, this method is investigated in its most general formulation and its extension is proposed for the case when the standard likelihood function optimization problem has no solution. For the first time, a necessary and sufficient condition for the consistency of the maximum likelihood estimate in the general formulation has been obtained. To achieve this result, it was necessary to apply the apparatus of mathematical statistics in spaces of arbitrary nature, which refers to the central sections of statistics of non-numerical data. Next, we study the situation when the problem of maximizing the likelihood function has no solution. In this case, it is proposed to use the generalized maximum likelihood estimate introduced in the paper to estimate the distribution function from the corresponding set. This approach has certain conceptual similarities with A. N. Tikhonov's regularization method, developed to solve incorrectly posed operator equations. The paper provides examples of calculating generalized maximum likelihood estimates. It is shown that such estimates include, in particular, the empirical distribution function and its symmetrized version obtained under the assumption of symmetry of the estimated distribution relative to zero. The symmetrized distribution function finds application, in particular, in testing hypotheses about the homogeneity of related samples. A number of unsolved problems related to the development of the generalized maximum likelihood method are formulated. It is planned to devote further scientific research to solving these problems.

KEYWORDS

statistical methods of economics, mathematical statistics, estimation, maximum likelihood method, statistics of non-numerical data, limit theorems

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Aleksandr Fomin

e-science@list.ru

Publishing editor

Ekaterina Syamina

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Krasnoslobodceva T. P., Mihaylova N. A., Timchenko T. V. KEY MATHEMATICAL PROPERTIES OF THE GENERALIZED MAXIMUM LIKELIHOOD METHOD // MOSCOW ECONOMIC JOURNAL. 2025. no. 9. pp. 70-84. DOI: https://doi.org/10.55186/2413046X_2025_10_9_209. (Date of access 10.10.2026).

MLA

Tat'yana P. Krasnoslobodceva, Mihaylova N. A., Timchenko T. V.. "KEY MATHEMATICAL PROPERTIES OF THE GENERALIZED MAXIMUM LIKELIHOOD METHOD." MOSCOW ECONOMIC JOURNAL 10.9 (2025): 70-84. DOI: https://doi.org/10.55186/2413046X_2025_10_9_209.

APA

Krasnoslobodceva T. P., Mihaylova N. A., Timchenko T. V. (2025). KEY MATHEMATICAL PROPERTIES OF THE GENERALIZED MAXIMUM LIKELIHOOD METHOD. MOSCOW ECONOMIC JOURNAL, 10(9), 70-84. https://doi.org/10.55186/2413046X_2025_10_9_209

Chicago

Tat'yana P. Krasnoslobodceva, Mihaylova N. A., Timchenko T. V.. "KEY MATHEMATICAL PROPERTIES OF THE GENERALIZED MAXIMUM LIKELIHOOD METHOD." MOSCOW ECONOMIC JOURNAL 10, no. 9 (2025): 70-84. https://doi.org/10.55186/2413046X_2025_10_9_209.

WIKI

{{cite journal | author = Krasnoslobodceva T. P., Mihaylova N. A., Timchenko T. V. | title = KEY MATHEMATICAL PROPERTIES OF THE GENERALIZED MAXIMUM LIKELIHOOD METHOD | journal = MOSCOW ECONOMIC JOURNAL | year = 2025 | volume = 10 | issue = 9 | pages = 70-84 | doi = 10.55186/2413046X_2025_10_9_209 }}

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