Copulas are probabilistic functions that are being used more and more frequently to describe, examine, and model the interdependence of continuous random variables. Among the numerous proposed copulas, renewed interest has recently been shown in the so-called Celebioglu–Cuadras copula. It is mainly because of its simplicity, exploitable dependence properties, and potential for applicability. In this article, we contribute to the development of this copula by proposing three generalized versions of it, each involving three tuning parameters. The main results are theoretical: they consist of determining wide and manageable intervals of admissible values for the involved parameters. The proofs are mainly based on limit, differentiation, and factorization techniques as well as mathematical inequalities. Some of the configuration parameters are new in the literature, and original phenomena are revealed. Subsequently, the basic properties of the proposed copulas are studied, such as symmetry, quadrant dependence, various expansions, concordance ordering, tail dependences, medial correlation, and Spearman correlation. Detailed examples, numerical tables, and graphics are used to support the theory.

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Theoretical Contributions to Three Generalized Versions of the Celebioglu–Cuadras Copula

In this article, a new two-parameter model called the truncated Cauchy power-inverted Topp–Leone (TCP-ITL) is constructed by merging the truncated Cauchy power -G (TCP-G) family with the inverted Topp–Leone (ITL) distribution. Some structural properties of the newly suggested model are obtained. Different types of entropies are proposed under the TCP-ITL distribution. Under the complete and hybrid censored data, the maximum likelihood (ML), maximum product of spacing (MPSP), and Bayesian estimate approaches are explored. A simulation study is developed to test the proposed distribution’s restricted sample attributes. In the majority of cases, the numerical data revealed that the Bayesian estimates provided more accurate outcomes than the equivalent alternative estimates. The adaptability of the proposed approach is proven using examples from dependability, medicine, and engineering. A real-world data set is utilized to demonstrate the potential of the TCP-ITL distribution in comparison to other well-known distributions. The results of the model selection revealed that the proposed distribution is the best choice for the data sets under consideration.

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Statistical Inference of Truncated Cauchy Power-Inverted Topp-Leone Distribution under Hybrid Censored Scheme with Applications

Copula analysis was created to explain the dependence of two or more quantitative variables. Due to the need for in-depth data analysis involving complex variable relationships, there is always a need for new copula models with original features. As a modern example, for the analysis of circular or periodic data types, trigonometric copulas are particularly attractive and recommended. This is, however, an underexploited topic. In this article, we propose a new collection of eight trigonometric and hyperbolic copulas, four based on the sine function and the others on the tangent function, all derived from the construction of the famous Farlie–Gumbel–Morgenstern copula. In addition to their original trigonometric and hyperbolic functionalities, the proposed copulas have the feature of depending on three parameters with complementary roles: one is a dependence parameter; one is a shape parameter; and the last can be viewed as an angle parameter. In our main findings, for each of the eight copulas, we determine a wide range of admissible values for these parameters. Subsequently, the capabilities, features, and functions of the new copulas are thoroughly examined. The shapes of the main functions of some copulas are illustrated graphically. Theoretically, symmetry in general, stochastic dominance, quadrant dependence, tail dependence, Archimedean nature, correlation measures, and inference on the parameters are investigated. Some copula shapes are illustrated with the help of figures. On the other hand, some two-dimensional inequalities are established and may be of separate interest.

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A Collection of New Trigonometric- and Hyperbolic-FGM-Type Copulas

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. 

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

In this paper, we propose a flexible and general family of distributions based on an original power-exponential transformation approach. We call it the modified generalized-G (MGG) family. The elegance and significance of this family lie in the ability to modify the standard distributions by changing their functional forms without adding new parameters, by compounding two distributions, or by adding one or two shape parameters. The aim of this modification is to provide flexible shapes for the corresponding probability functions. In particular, the distributions of the MGG family can possess increasing, constant, decreasing, “unimodal”, or “bathtub-shaped“ hazard rate functions, which are ideal for fitting several real data sets encountered in applied fields. Some members of the MGG family are proposed for special distributions. Following that, the uniform distribution is chosen as a baseline distribution to yield the modified uniform (MU) distribution with the goal of efficiently modeling measures with bounded values. Some useful key properties of the MU distribution are determined. The estimation of the unknown parameters of the MU model is discussed using seven methods, and then, a simulation study is carried out to explore the performance of the estimates. The flexibility of this model is illustrated by the analysis of two real-life data sets. When compared to fair and well-known competitor models in contemporary literature, better-fitting results are obtained for the new model. 

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Complete Study of an Original Power-Exponential Transformation Approach for Generalizing Probability Distributions

During its 330 years of history, parametric distributions have been useful for survival and reliability analyses. In this paper, we comprehensively review the historical backgrounds and statistical properties of a number of parametric distributions used in survival and reliability analyses. We provide encyclopedic coverage of the important parametric distributions, which is more extensive than the existing textbooks on survival and reliability analyses. We also explain how these distributions have been adopted in survival and reliability analyses with original and state-of-the-art references. We cover the exponential, Weibull, Rayleigh, lognormal, log-logistic, gamma, generalized gamma, Pareto (types I, II, and IV), Hjorth, Burr (types III and XII), Dagum, exponential power, Gompertz, Birnbaum-Saunders, exponential-logarithmic, piecewise exponential, generalized exponential, exponentiated Weibull, generalized modified Weibull, and spline distributions. We analyze a real dataset for illustration.

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Kazuki Yamamoto

Papers

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