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The Opone family of distributions : beyond the power continuous Bernoulli distribution

Opone Festus C., Chesneau Christophe

Operations Research and Decisions

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2024

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Vol. 34, No. 4

103--124

EN

Recent developments in applied statistics have given rise to the continuous Bernoulli distribution, a one-parameter distribution with support of [0, 1]. In this paper, we use it for a more general purpose: the creation of a family of distributions. We thus exploit the flexible functionalities of the continuous Bernoulli distribution to enhance the modeling properties of wellreferenced distributions. We first focus on the theory of this new family, including the quantiles, expansion of important functions, and moments. Then we exemplify it by considering a special baseline: the Topp–Leone distribution. Thanks to the functional structure of the continuous Bernoulli distribution, we create a new two-parameter distribution with support for [0, 1] that possesses versatile shape capacities. In particular, the corresponding probability density function has left-skewed, N-type and decreasing shapes, and the corresponding hazard rate function has increasing and bathtub shapes, beyond the possibilities of the corresponding functions of the Topp–Leone distribution. Its quantile and moment properties are also examined. We then use our modified Topp–Leone distribution from a statistical perspective. The two parameters are supposed to be unknown and then estimated from proportional-type data with the maximum likelihood method. Two different data sets are considered, and reveal that the modified Topp–Leone distribution can fit them better than popular rival distributions, including the unitWeibull, unit-Gompertz, and log-weighted exponential distributions. It also outperforms the Topp–Leone and continuous Bernoulli distributions.

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Diverse copulas through Durante’s method : exploring parametric functions

Chesneau Christophe

Operations Research and Decisions

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2024

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Vol. 34, No. 3

61--86

EN

This article unveils the often underestimated potential of a copula methodology introduced by Durante in 2009. It highlights the remarkable ability of the method to generate a broad spectrum of copulas by exploiting various parametric functions. We determine a collection of power-like, exponential-like, trigonometric-like, logarithmic-like, hyperbolic-like and error-like functions, each dependent on one, two, or three parameters, effectively satisfying the necessary assumptions of Durante’s method. The proofs provided rely on suitable differentiation, comprehensive factorizations, and judicious application of mathematical inequalities. In the vast repertoire of copulas derived from this methodology, we present three distinct series of eight new copulas, supported by a graphical analysis of their respective densities. This theoretical study expands the understanding of copula generation and also introduces a new perspective on their construction in various contexts.

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A unit Weibull loss distribution with quantile regression and practical applications to actuarial science

Abubakari Abdul Ghaniyyu, Nasiru Suleman, Chesneau Christophe

Operations Research and Decisions

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2024

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Vol. 34, No. 4

1--39

EN

A new bounded distribution called the unit Weibull loss distribution has been studied. The corresponding probability density function plots reveal that it is suitable to analyze data that exhibit right skewness, left skewness, and approximately symmetric and decreasing shapes. Furthermore, the corresponding hazard rate function plots indicate that it is adequate to fit data that have J, bathtub, and modified bathtub hazard rate shapes. This makes the new distribution suitable for modeling data with complex characteristics. Statistical properties such as the quantile, moments, and moment-generating function are determined. Risk measures, including the value-at-risk, tail value-at-risk, and tail variance are also calculated. Furthermore, different principles are derived for the computation of insurance premiums. The parameters of the distribution are estimated using different methods, and their performance is assessed via Monte Carlo simulations. The accuracy of the estimates is thus empirically demonstrated. A quantile regression model with responses following the unit distribution is developed. Applications of the proposed distribution and its corresponding regression model to three insurance data sets are carried out, with their performance compared with other models. The results show that they outperform the competitors. Thus, the new methodology can serve as an alternative option to analyze insurance data.