Symmetry

4 pages, 198 KiB

Open AccessEditorial

Special Issue Editorial “Symmetric Distributions, Moments and Applications”

by Zivorad Tomovski

Symmetry 2022, 14(9), 1863; https://doi.org/10.3390/sym14091863 - 7 Sep 2022

Viewed by 1219

In 1933, Kolmogorov published his book, Foundations of the Theory of Probability, laying the modern axiomatic foundations of probability theory and establishing his reputation as the world’s leading expert in this field [...] Full article

(This article belongs to the Special Issue Symmetric Distributions, Moments and Applications)

15 pages, 292 KiB

Open AccessReview

Convolutions for Bernoulli and Euler–Genocchi Polynomials of Order (r,m) and Their Probabilistic Interpretation

by Robert Frontczak and Živorad Tomovski

Symmetry 2022, 14(6), 1220; https://doi.org/10.3390/sym14061220 - 13 Jun 2022

Cited by 2 | Viewed by 1558

Abstract

The main purpose of this article is to derive several convolutions for generalized Bernoulli and Euler–Genocchi polynomials of order

(r, m)

,

B_{n}^{(r, m)} (x)

and [...] Read more.

The main purpose of this article is to derive several convolutions for generalized Bernoulli and Euler–Genocchi polynomials of order

(r, m)

,

B_{n}^{(r, m)} (x)

and

A_{n}^{(r, m)} (x)

, respectively. These polynomials have been introduced recently and contain the generalized Bernoulli, Euler and Genocchi polynomials as special members. Some of our results extend the results of M. Merca and others concerning Bernoulli numbers and polynomials. Probabilistic interpretations of the presented results are also given. Full article

(This article belongs to the Special Issue Symmetric Distributions, Moments and Applications)

17 pages, 4297 KiB

Open AccessArticle

A Flexible Extension to an Extreme Distribution

by Mohamed S. Eliwa, Fahad Sameer Alshammari, Khadijah M. Abualnaja and Mahmoud El-Morshedy

Symmetry 2021, 13(5), 745; https://doi.org/10.3390/sym13050745 - 23 Apr 2021

Cited by 7 | Viewed by 1630

Abstract

The aim of this paper is not only to propose a new extreme distribution, but also to show that the new extreme model can be used as an alternative to well-known distributions in the literature to model various kinds of datasets in different [...] Read more.

The aim of this paper is not only to propose a new extreme distribution, but also to show that the new extreme model can be used as an alternative to well-known distributions in the literature to model various kinds of datasets in different fields. Several of its statistical properties are explored. It is found that the new extreme model can be utilized for modeling both asymmetric and symmetric datasets, which suffer from over- and under-dispersed phenomena. Moreover, the hazard rate function can be constant, increasing, increasing–constant, or unimodal shaped. The maximum likelihood method is used to estimate the model parameters based on complete and censored samples. Finally, a significant amount of simulations was conducted along with real data applications to illustrate the use of the new extreme distribution. Full article

(This article belongs to the Special Issue Symmetric Distributions, Moments and Applications)

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13 pages, 384 KiB

Open AccessArticle

Taming Tail Risk: Regularized Multiple β Worst-Case CVaR Portfolio

by Kei Nakagawa and Katsuya Ito

Symmetry 2021, 13(6), 922; https://doi.org/10.3390/sym13060922 - 21 May 2021

Cited by 5 | Viewed by 2521

Abstract

The importance of proper tail risk management is a crucial component of the investment process and conditional Value at Risk (CVaR) is often used as a tail risk measure. CVaR is the asymmetric risk measure that controls and manages the downside risk of [...] Read more.

The importance of proper tail risk management is a crucial component of the investment process and conditional Value at Risk (CVaR) is often used as a tail risk measure. CVaR is the asymmetric risk measure that controls and manages the downside risk of a portfolio while symmetric risk measures such as variance consider both upside and downside risk. In fact, minimum CVaR portfolio is a promising alternative to traditional mean-variance optimization. However, there are three major challenges in the minimum CVaR portfolio. Firstly, when using CVaR as a risk measure, we need to determine the distribution of asset returns, but it is difficult to actually grasp the distribution; therefore, we need to invest in a situation where the distribution is uncertain. Secondly, the minimum CVaR portfolio is formulated with a single β and may output significantly different portfolios depending on the β. Finally, most portfolio allocation strategies do not account for transaction costs incurred by each rebalancing of the portfolio. In order to improve these challenges, we propose a Regularized Multiple β Worst-case CVaR (RM-WCVaR) portfolio. The characteristics of this portfolio are as follows: it makes CVaR robust with worst-case CVaR which is still an asymmetric risk measure, it is stable among multiple β, and against changes in weights over time. We perform experiments on well-known benchmarks to evaluate the proposed portfolio.RM-WCVaR demonstrates superior performance of having both higher risk-adjusted returns and lower maximum drawdown. Full article

(This article belongs to the Special Issue Symmetric Distributions, Moments and Applications)

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26 pages, 417 KiB

Open AccessArticle

Inventory Models for Non-Instantaneous Deteriorating Items with Expiration Dates and Imperfect Quality under Hybrid Payment Policy in the Three-Level Supply Chain

by Jui-Jung Liao, Hari Mohan Srivastava, Kun-Jen Chung, Shih-Fang Lee, Kuo-Nan Huang and Shy-Der Lin

Symmetry 2021, 13(9), 1695; https://doi.org/10.3390/sym13091695 - 14 Sep 2021

Cited by 9 | Viewed by 3107

Abstract

This article considers an inventory model for non-instantaneous deteriorating items with expiration dates, such as seasonal items, first-hand vegetables, and fruits. Interestingly, an inspection will be performed to manage the quality of the items during the state of no deterioration because it is [...] Read more.

This article considers an inventory model for non-instantaneous deteriorating items with expiration dates, such as seasonal items, first-hand vegetables, and fruits. Interestingly, an inspection will be performed to manage the quality of the items during the state of no deterioration because it is difficult to purchase items with 100% perfection. Additionally, we assume that the upstream member has the power of controlling or influencing downstream members’ decisions. That is, the supplier asks the retailer for a partial advance payment to avoid cancellation of orders and offers them a credit payment to stimulate sales; in turn, the customer must pay some cash when placing an order and pay the remainder in credit for the retailer. The goal of this article is to determine an optimal replenishment cycle and the total annual cost function, so we explore the functional properties of the total annual cost function and show that the total annual cost function is convex. Theoretical analysis of the optimal properties shows the existence and uniqueness of the optimal solution. Then, we obtain simple and easy solution procedures for the inventory system. Moreover, numerical analysis of the inventory model is conducted, and the corresponding examples are considered with a view to illustrating the application of the supply chain model that we have investigated in this article. Finally, in the concluding section, we have not only provided the motivation and the need for our usages of mathematical analytic solution procedures based upon the convexity, monotonicity (increasing and decreasing) and differentiability properties of the object function (that is, the total annual cost function), which involve some symmetry aspects of the object function, but we have also indicated the limitations and shortcomings in our investigation, which will naturally lead to some potential directions for further research on the supply chain model, which we have considered and mathematically analyzed in this article. Full article

(This article belongs to the Special Issue Symmetric Distributions, Moments and Applications)

26 pages, 906 KiB

Open AccessArticle

An Alternate Generalized Odd Generalized Exponential Family with Applications to Premium Data

by Sadaf Khan, Oluwafemi Samson Balogun, Muhammad Hussain Tahir, Waleed Almutiry and Amani Abdullah Alahmadi

Symmetry 2021, 13(11), 2064; https://doi.org/10.3390/sym13112064 - 1 Nov 2021

Cited by 9 | Viewed by 1919

Abstract

In this article, we use Lehmann alternative-II to extend the odd generalized exponential family. The uniqueness of this family lies in the fact that this transformation has resulted in a multitude of inverted distribution families with important applications in actuarial field. We can [...] Read more.

In this article, we use Lehmann alternative-II to extend the odd generalized exponential family. The uniqueness of this family lies in the fact that this transformation has resulted in a multitude of inverted distribution families with important applications in actuarial field. We can characterize the density of the new family as a linear combination of generalised exponential distributions, which is useful for studying some of the family’s properties. Among the structural characteristics of this family that are being identified are explicit expressions for numerous types of moments, the quantile function, stress-strength reliability, generating function, Rényi entropy, stochastic ordering, and order statistics. The maximum likelihood methodology is often used to compute the new family’s parameters. To confirm that our results are converging with reduced mean square error and biases, we perform a simulation analysis of one of the special model, namely OGE2-Fréchet. Furthermore, its application using two actuarial data sets is achieved, favoring its superiority over other competitive models, especially in risk theory. Full article

(This article belongs to the Special Issue Symmetric Distributions, Moments and Applications)

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20 pages, 1791 KiB

Open AccessArticle

Optimal Plan of Multi-Stress–Strength Reliability Bayesian and Non-Bayesian Methods for the Alpha Power Exponential Model Using Progressive First Failure

by Ehab M. Almetwally, Refah Alotaibi, Aned Al Mutairi, Chanseok Park and Hoda Rezk

Symmetry 2022, 14(7), 1306; https://doi.org/10.3390/sym14071306 - 23 Jun 2022

Cited by 13 | Viewed by 1996

Abstract

It is extremely frequent for systems to fail in their demanding operating environments in many real-world contexts. When systems reach their lowest, highest, or both extreme operating conditions, they usually fail to perform their intended functions, which is something that researchers pay little [...] Read more.

It is extremely frequent for systems to fail in their demanding operating environments in many real-world contexts. When systems reach their lowest, highest, or both extreme operating conditions, they usually fail to perform their intended functions, which is something that researchers pay little attention to. The goal of this paper is to develop inference for multi-reliability using unit alpha power exponential distributions for stress–strength variables based on the progressive first failure. As a result, the problem of estimating the stress–strength function R, where X, Y, and Z come from three separate alpha power exponential distributions, is addressed in this paper. The conventional methods, such as maximum likelihood for point estimation, Bayesian and asymptotic confidence, boot-p, and boot-t methods for interval estimation, are also examined. Various confidence intervals have been obtained. Monte Carlo simulations and real-world application examples are used to evaluate and compare the performance of the various proposed estimators. Full article

(This article belongs to the Special Issue Symmetric Distributions, Moments and Applications)

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16 pages, 864 KiB

Open AccessArticle

Simple Closed-Form Formulas for Conditional Moments of Inhomogeneous Nonlinear Drift Constant Elasticity of Variance Process

by Kittisak Chumpong, Raywat Tanadkithirun and Chanon Tantiwattanapaibul

Symmetry 2022, 14(7), 1345; https://doi.org/10.3390/sym14071345 - 29 Jun 2022

Cited by 4 | Viewed by 2014

Abstract

The stochastic differential equation (SDE) has been used to model various phenomena and investigate their properties. Conditional moments of stochastic processes can be used to price financial derivatives whose payoffs depend on conditional moments of underlying assets. In general, the transition probability density [...] Read more.

The stochastic differential equation (SDE) has been used to model various phenomena and investigate their properties. Conditional moments of stochastic processes can be used to price financial derivatives whose payoffs depend on conditional moments of underlying assets. In general, the transition probability density function (PDF) of a stochastic process is often unavailable in closed form. Thus, the conditional moments, which can be directly computed by applying the transition PDFs, may be unavailable in closed form. In this work, we studied an inhomogeneous nonlinear drift constant elasticity of variance (IND-CEV) process, which is a class of diffusions that have time-dependent parameter functions; therefore, their sample paths are asymmetric. The closed-form formulas for conditional moments of the IND-CEV process were derived without having a condition on eigenfunctions or the transition PDF. The analytical results were examined through Monte Carlo simulations. Full article

(This article belongs to the Special Issue Symmetric Distributions, Moments and Applications)

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