Study on geometric stable laws

Abstract

This thesis delves into the profound area of geometric stable (GS) laws, which emerge as a critical class within the framework of random summation schemes when the number of terms follows geometric distribution. These distributions naturally find applications in a multitude of fields, particularly excelling in the modeling of heavy -tailed kind of data distribution. The study presented herein not only advances our understanding of GS laws but also extends there applicability through generilizations and parameter estimation techniques.Furthermore, this research extends its purview to encompass circular data which arises in diverse domains such as earth sciences, meteorology, biology, and image analysis.The results emanating from this study can be summarized as follows : • New generilizations of geometric table distributions (GGS) are introduced, broadening there utility in diverse practical scenarios. These extensions facilitate more accurate representations of heavy-tailed phenomena in various fields. • Robust parametric estimation procedures are developed for both GS and GGS distributions. There procedures empower researchers and practitioners to estimate distribution parameters more effectively, enhancing the reliability of these models in statistical applications. • The research systematically explores the inherent properties and mathematical representations of the newly proposed models. This comprehensive analysis deepens our understanding of GS laws, shedding light on their behaviour under varying conditions. • A significant milestone in the research is the introduction of Generilized normal GS models (GNGS). These models constitute a notable expansion of GS laws offering versatile tools for modeling complex heavy-tailed phenomena in diverse fields. • The research presents geometric extensions of the new distributions and derives autoregressive process based on these models, expanding their application to dynamic real- world phenomena. • The study extends its scope to encompass circular data, which plays a pivotal role in fields such as earth sciences, meteorology, biology, and image analysis. Circular versions of the newly proposed models are derived and their properties are examined. • Multivariate extensions of the new models are developed, allowing for the modeling of complex interdependencies and interactions within data sets. • The thesis substantiates the validity of the developed models by applying them to real-life data, demonstrating their efficacy in capturing and explaining complex phenomena encountered in practical applications.