Conducted jointly by UL, this research explores generalized inverse Gaussian and Kummer distributions through Stein's method.
The Essentials: Conducted jointly by UL, this research explores generalized inverse Gaussian and Kummer distributions through Stein's method.
Essomanda Konzou, researcher at IECL (Université de Lorraine).
Thesis defended in 2020 at the doctoral school IAEM Lorraine Doctoral School - Computer Science, Automation, Electronics - Electrical Engineering, Mathematics of Lorraine.
This research is the product of an international joint supervision (cotutelle) between several partner institutions.
Context and Research Question
Generalized inverse Gaussian distributions and Kummer distributions are important objects of study in probability theory. These distributions have unique characteristics that make them attractive candidates for applications in various fields, notably statistics and the modeling of random phenomena. Stein's method, a powerful tool for analyzing the convergence of probability distributions, is applied here to examine these specific distributions. The central interest of this research lies in establishing bounds for the rate of convergence of these distributions toward relevant limiting distributions, such as the gamma distribution and the generalized inverse Gaussian distribution.
The study of inverse Gaussian distributions is particularly relevant in contexts where modeling risk and returns is necessary. Practical applications of these distributions in fields such as finance and biostatistics highlight the importance of a thorough understanding of their properties. By focusing on convergence bounds, this research aims to provide mathematical tools that can facilitate better use of these distributions in complex statistical models.
Methodology
Essomanda Konzou's research centers on applying Stein's method within the framework of generalized inverse Gaussian distributions and Kummer distributions. The author begins by establishing the Stein operator for each of these distributions, solving the associated differential equations. This resolution is necessary to apply Stein's method, which relies on the relationship between the distribution of the random variable under study and the limiting distribution.
The techniques used to obtain bounds on the solution and its derivatives are based on the fact that the densities of these distributions satisfy a particular differential equation, where ( g ) represents the density, and ( s ) and ( \tau ) are polynomial functions. Applying this method yields bounds for the rate of convergence of the generalized inverse Gaussian and Kummer distributions toward the gamma distribution, as well as toward other relevant distributions.
The iterative approach used for bounding successive derivatives is a central aspect of the methodology. Although some of the bounds are not explicit, they are derived from a rigorous mathematical framework that guarantees their validity. This rigor is essential to ensure the robustness of the results obtained.
Key Findings
One of the major contributions of this thesis is establishing a bound for the rate of convergence of the distributions studied. The results demonstrate that generalized inverse Gaussian distributions converge toward the gamma distribution, while the generalized hyperbolic distribution approaches the generalized inverse Gaussian distribution. Furthermore, a sequence of random variables associated with random-resistance contexts also converges toward the reciprocal inverse Gaussian distribution.
The Stein operators for the inverse Gaussian and Kummer distributions were explicitly characterized, and the corresponding differential equations were successfully solved. The bounds obtained for the solution and its successive derivatives provide valuable information on the asymptotic behavior of the distributions considered.
The research also highlighted the convolution relationship between the distribution of the sequence of random variables and the limiting distribution, a key element for estimating convergence rates. These results pave the way for practical applications in fields such as finance, where statistical models can benefit from a better understanding of the probability distributions involved.
Discussion and Outlook
The results of this thesis provide significant advances in the analysis of generalized inverse Gaussian and Kummer distributions. The mathematical tools developed, along with the bounds established for the rate of convergence, offer interesting prospects for practical applications in various fields. It is worth considering how these results can be integrated into statistical models used by researchers and practitioners.
Integrating the theories developed in this research into practical contexts, such as finance and health, could improve the accuracy of predictive models. Furthermore, exploring the generalization of these results to other families of probability distributions could enrich the field of research and offer new avenues of investigation.
The implications of this research are far-reaching, and it would be fruitful to continue studies on inverse Gaussian distributions and their applications. Future work could focus on extending the results to other families of probability distributions, as well as exploring new convergence methods. This could not only reinforce the validity of statistical models but also make significant contributions to probability theory.
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Sources and Access
Essomanda Konzou. Lois gaussiennes inverses (généralisées), lois de Kummer et méthode de Stein. Mathématiques [math]. Université de Lorraine; Université de Lomé (Togo), 2020. Français. ⟨NNT : 2020LORR0147⟩. ⟨tel-03127767⟩