UDC 330.43
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.
statistical methods of economics, mathematical statistics, estimation, maximum likelihood method, statistics of non-numerical data, limit theorems
1. Borovkov A.A. Matematicheskaya statistika. Izd. 5-e, stereotipnoe. - Sankt- Peterburg : Lan', 2021. - 704 s.
2. Orlov A.I. Ocenivanie parametrov: odnoshagovye ocenki predpochtitel'nee ocenok maksimal'nogo pravdopodobiya // Nauchnyy zhurnal KubGAU. 2015. №109. S. 208 – 237.
3. Orlov A.I. Prikladnoy statisticheskiy analiz. — M.: Ay Pi Ar Media, 2022. — 812 c.
4. Orlov A.I. Iskusstvennyy intellekt: nechislovaya statistika : uchebnik. — M.: Ay Pi Ar Media, 2022. — 446 c.
5. Tyurin Yu.N. Ob ocenivanii funkcii raspredeleniya // Teoriya veroyatnostey i ee primeneniya. 1970. T. 15. № 3. S. 567-568.
6. Orlov A.I. O proverke odnorodnosti svyazannyh vyborok // Politematicheskiy setevoy elektronnyy nauchnyy zhurnal Kubanskogo gosudarstvennogo agrarnogo universiteta. 2016. № 123. S. 708–726.
7. Astaf'ev, R. U. Metodika formirovaniya bazy znaniy dlya sistemy upravleniya kachestvom programmnogo obespecheniya / R. U. Astaf'ev // Nauchno-tehnologicheskoe razvitie 2025: sbornik statey Mezhdunarodnoy nauchno-prakticheskoy konferencii, Petrozavodsk, 26 iyunya 2025 goda. – Petrozavodsk: Mezhdunarodnyy centr nauchnogo partnerstva «Novaya Nauka» (IP Ivanovskaya I.I.), 2025. – S. 126-130. – EDN WHXWFG.
8. Astaf'ev, R. U. Rol' imitacionnyh modeley v sistemah podderzhki prinyatiya resheniy v oblasti razrabotki programmnyh produktov / R. U. Astaf'ev // Opticheskie tehnologii, materialy i sistemy (Optoteh - 2024): Mezhdunarodnaya nauchno-tehnicheskaya konferenciya, Moskva, 02–08 dekabrya 2024 goda. – Moskva: MIREA - Rossiyskiy tehnologicheskiy universitet, 2024. – S. 789-790. – EDN JTFOGS.
9. Sidorov, A. A. Dokazatel'stvo svoystv srednih stepennyh / A. A. Sidorov // Innovacionnye tehnologii v elektronike i priborostroenii: sbornik dokladov Rossiyskoy nauchno-tehnicheskoy konferencii s mezhdunarodnym uchastiem Fiziko-tehnologicheskogo instituta RTU MIREA, Moskva, 16–17 aprelya 2020 goda. Tom 1. – Moskva: MIREA - Rossiyskiy tehnologicheskiy universitet, 2020. – S. 287-293. – EDN ELMJXA.
10. Ob odnom aspekte v voprose opredeleniya analitichnostifunkcii kompleksnogo peremennogo / O. Yu. Kozlova, T. A. Manaenkova, A. I. Novikova [i dr.] // Perspektivnye materialy i tehnologii (PMT-2024) : Sbornik dokladov Mezhdunarodnoy nauchno-tehnicheskoy konferencii, Moskva, 12–16 aprelya 2024 goda. – Moskva: MIREA - Rossiyskiy tehnologicheskiy universitet, 2024. – S. 422-425. – EDN EMGWJP.
11. SIDOROV Andrei, 2024, THE IMPACT OF ANNOUNCEMENTS ON CRYPTOCURRENCY PRICES, Revista Economică, Lucian Blaga University of Sibiu, Faculty of Economic Sciences, vol.76(4), pages 69-94, December. DOI: https://doi.org/10.56043/reveco-2024-0035
