Parametric Distributions for Survival and Reliability Analyses, a Review and Historical Sketch

Computational Journal of Mathematical and Statistical Sciences

2023

The copula approach provides an option for capturing the structure of dependence between two quantitative variables. This approach is based on special bivariate functions called copulas. In brief, a copula can be presented as a "functional print" of a structure of dependence. Among the numerous types of copulas, the strict Archimedean copulas are the most famous. However, some of them appear to be unfairly neglected and understudied. We attempt to rehabilitate two of them, namely the Nelsen strict Archimedean copulas numbers 10 and 17 by extending their mathematical functionality. For each of them, the extension is made by the thorough addition of a parameter, which reveals to have a significant impact on several important aspects. As the main results, we determine the range of admissible values for the involved parameters. Then, the basic properties of the proposed extended copulas are studied, including symmetry, tail dependence, bounds, and correlation. A focus is put on the tau of Kendall as the main correlation measure. For one of the extended copulas, it is shown that there are some connections with the Frank copula and that the associated tau of Kendall almost reached the perfect interval of [−1, 1], demonstrating the significance of our findings. Graphics, short statistical works, and numerical tables are given to support the theoretical results.

Section: 4)mentioning

confidence: 99%

Extensions of Two Bivariate Strict Archimedean Copulas

Computational Journal of Mathematical and Statistical Sciences

2023

“…There are countless new two-dimensional distributions that could be created based on this function. The options for motivated lifetime cumulative distribution functions are discussed in [27], among others.…”

Section: Name Formula Valuementioning

confidence: 99%

Theoretical Advancements on a Few New Dependence Models Based on Copulas with an Original Ratio Form

2023

Modelling

Self Cite

Copulas are well-known tools for describing the relationship between two or more quantitative variables. They have recently received a lot of attention, owing to the variable dependence complexity that appears in heterogeneous modern problems. In this paper, we offer five new copulas based on a common original ratio form. All of them are defined with a single tuning parameter, and all reduce to the independence copula when this parameter is equal to zero. Wide admissible domains for this parameter are established, and the mathematical developments primarily rely on non-trivial limits, two-dimensional differentiations, suitable factorizations, and mathematical inequalities. The corresponding functions and characteristics of the proposed copulas are looked at in some important details. In particular, as common features, it is shown that they are diagonally symmetric, but not Archimedean, not radially symmetric, and without tail dependence. The theory is illustrated with numerical tables and graphics. A final part discusses the multi-dimensional variation of our original ratio form. The contributions are primarily theoretical, but they provide the framework for cutting-edge dependence models that have potential applications across a wide range of fields. Some established two-dimensional inequalities may be of interest beyond the purposes of this paper.

“…Proposition 3 can be the starting point for various two-dimensional data analyses. For a panel of choices of lifetime baseline CDFs, we may suggest the overview in [26] and the references therein.…”

Section: Proof For (Iii)mentioning

confidence: 99%

“…In a two-dimensional lifetime data analysis scenario, one can consider lifetime baseline distributions. On this topic, we again refer to the survey in [26], and the references therein.…”

Section: Fundamental Propertiesmentioning

confidence: 99%

Theoretical Validation of New Two-Dimensional One-Variable-Power Copulas

2023

Axioms

Self Cite

One of the most effective ways to illustrate the relationship between two quantitative variables is to describe the corresponding two-dimensional copula. This approach is acknowledged as practical, nonredundant, and computationally manageable in the context of data analysis. Modern data, however, contain a wide variety of dependent structures, and the copulas now in use may not provide the best model for all of them. As a result, researchers seek to innovate by building novel copulas with appealing properties that are also based on original methodologies. The foundations are theoretical; for a copula to be validated, it must meet specific requirements, which frequently dictate the constraints that must be placed on the relevant parameters. In this article, we make a contribution to the understudied field of one-variable-power copulas. We first identify the specific assumptions that, in theory, validate copulas of such nature. Some other general copulas and inequalities are discussed. Our general results are illustrated with numerous examples depending on two or three parameters. We also prove that strong connections exist between our assumptions and well-established distributions. To highlight the importance of our findings, we emphasize a particular two-parameter, one-variable-power copula that unifies the definition of some other copulas. We reveal its versatile shapes, related functions, various symmetry, Archimedean nature, geometric invariance, copula ordering, quadrant dependence, tail dependence, correlations, and distribution generation. Numerical tables and graphics are produced to support some of these properties. The estimation of the parameters based on data is discussed. As a complementary contribution, two new, intriguing one-variable-power copulas beyond the considered general form are finally presented and studied.