<abstract xmlns="http://www.w3.org/1999/xhtml">

<p>In this paper, according to the concept of multivariate composite function, the limit and continuity conditions of multivariate composite function are given, and the continuity relationship between the inner function and the outer function of the composite function at the same point is investigated. The relationship between the limit and continuity of a composite function composed of two functions at the same point is investigated by examples, while some limit results are explained from the perspective of composite functions, and the limit calculation law of multivariate composite functions is proposed. Using the consistent central limit theorem in the limit calculation law, the absolute value and limit of the interval of the multivariate composite function are obtained, and the iterative limit of the variational reduction approximation method is confirmed by combining the absolute value and limit of the interval of the multivariate composite function. On the basis of the application of limit operation of a multivariate composite function, the limit distribution of high-dimensional likelihood ratio statistics of multivariate composite function is analyzed by using data simulation analysis. The results show that both limit distributions have better simulation results when <italic>p</italic> are small, such as <italic>p</italic> = 5 or <italic>p</italic> = 10, but the traditional limit is better than the normal limit, indicating that the results of multivariate composite function limit calculations have good accuracy. A method for solving the limits of multivariate composite functions is proposed in this study, which is of great theoretical and practical significance in the study of higher mathematics.</p>
</abstract>

In this paper, according to the concept of multivariate composite function, the limit and continuity conditions of multivariate composite function are given, and the continuity relationship between the inner function and the outer function of the composite function at the same point is investigated. The relationship between the limit and continuity of a composite function composed of two functions at the same point is investigated by examples, while some limit results are explained from the perspective of composite functions, and the limit calculation law of multivariate composite functions is proposed. Using the consistent central limit theorem in the limit calculation law, the absolute value and limit of the interval of the multivariate composite function are obtained, and the iterative limit of the variational reduction approximation method is confirmed by combining the absolute value and limit of the interval of the multivariate composite function. On the basis of the application of limit operation of a multivariate composite function, the limit distribution of high-dimensional likelihood ratio statistics of multivariate composite function is analyzed by using data simulation analysis. The results show that both limit distributions have better simulation results when p are small, such as p = 5 or p = 10, but the traditional limit is better than the normal limit, indicating that the results of multivariate composite function limit calculations have good accuracy. A method for solving the limits of multivariate composite functions is proposed in this study, which is of great theoretical and practical significance in the study of higher mathematics.

A method for solving the limit of multivariate composite functions

Applied Mathematics and Nonlinear Sciences

This work is licensed under the Creative Commons Attribution 4.0 International License.

{"article-title":"A method for solving the limit of multivariate composite functions"}

In this paper, according to the concept of multivariate composite function, the limit and continuity conditions of multivariate composite function are given,...

A method for solving the limit of multivariate composite functions

Published Online: Jan 31, 2024

Page range: -

Received: Dec 15, 2023

Accepted: Dec 24, 2023

DOI: https://doi.org/10.2478/amns-2024-0096

Keywords
Multivariate composite function, Continuity, Likelihood ratio statistic, Limit, Variance reduction approximation method

© 2024 Yue Liang et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

A method for solving the limit of multivariate composite functions

Published Online: Jan 31, 2024

Page range: -

Received: Dec 15, 2023

Accepted: Dec 24, 2023

DOI: https://doi.org/10.2478/amns-2024-0096

KeywordsMultivariate composite function, Continuity, Likelihood ratio statistic, Limit, Variance reduction approximation method

© 2024 Yue Liang et al., published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Keywords
Multivariate composite function, Continuity, Likelihood ratio statistic, Limit, Variance reduction approximation method