scholarly journals Some New Results of Residual and Past Entropy Measures

2021 ◽  
pp. 21-33
Author(s):  
Abdul Basit ◽  
Zafar Iqbal ◽  
En-Bing Lin
Keyword(s):  
Residual Entropy ◽  
Exponential Distributions ◽  
Generalized Entropy ◽  
Special Cases ◽  
Entropy Measures ◽  
Different Types ◽  
Past Entropy ◽  
Parameter Values ◽  
Generalized Entropies ◽  
Cumulative Entropy

In this paper, two new generalized entropies have been introduced with their respective properties. The results of these entropies have been verified for the exponential and weighted exponential distributions. These two entropies produce the results in the form of simple entropy, generalized entropy, residual entropy, cumulative entropy and mixtures of all these entropies. Some characteristics of residual & past entropy have been derived and special cases have also been obtained. These cases indicate that new generalized entropies are more comprehensive and useful. The main advantage of this study is to derive different types of generalization of entropies using the different parameter values of α and β.

scholarly journals Generalized Entropies, Variance and Applications

Entropy ◽  
2020 ◽  
Vol 22 (6) ◽  
pp. 709 ◽  
Author(s):  
Abdolsaeed Toomaj ◽  
Antonio Di Crescenzo
Keyword(s):  
Residual Entropy ◽  
Dispersion Measure ◽  
Homogeneous Poisson Process ◽  
Cumulative Residual Entropy ◽  
Excess Wealth Order ◽  
Dynamic Version ◽  
Cumulative Residual ◽  
Non Homogeneous Poisson Process ◽  
Generalized Entropies ◽  
Cumulative Entropy

The generalized cumulative residual entropy is a recently defined dispersion measure. In this paper, we obtain some further results for such a measure, in relation to the generalized cumulative residual entropy and the variance of random lifetimes. We show that it has an intimate connection with the non-homogeneous Poisson process. We also get new expressions, bounds and stochastic comparisons involving such measures. Moreover, the dynamic version of the mentioned notions is studied through the residual lifetimes and suitable aging notions. In this framework we achieve some findings of interest in reliability theory, such as a characterization for the exponential distribution, various results on k-out-of-n systems, and a connection to the excess wealth order. We also obtain similar results for the generalized cumulative entropy, which is a dual measure to the generalized cumulative residual entropy.

Univariate and bivariate extensions of the generalized exponential distributions

2021 ◽  
Vol 71 (6) ◽  
pp. 1581-1598
Author(s):  
Vahid Nekoukhou ◽  
Ashkan Khalifeh ◽  
Hamid Bidram
Keyword(s):  
Em Algorithm ◽  
Exponential Distribution ◽  
Bivariate Distribution ◽  
Generalized Exponential Distribution ◽  
Exponential Distributions ◽  
Unknown Parameters ◽  
Data Set ◽  
New Class ◽  
Special Cases ◽  
Univariate Distribution

Abstract The main aim of this paper is to introduce a new class of continuous generalized exponential distributions, both for the univariate and bivariate cases. This new class of distributions contains some newly developed distributions as special cases, such as the univariate and also bivariate geometric generalized exponential distribution and the exponential-discrete generalized exponential distribution. Several properties of the proposed univariate and bivariate distributions, and their physical interpretations, are investigated. The univariate distribution has four parameters, whereas the bivariate distribution has five parameters. We propose to use an EM algorithm to estimate the unknown parameters. According to extensive simulation studies, we see that the effectiveness of the proposed algorithm, and the performance is quite satisfactory. A bivariate data set is analyzed and it is observed that the proposed models and the EM algorithm work quite well in practice.

Circles on the Complex Plane

2021 ◽  
pp. 3-12
Author(s):  
A. Girsh
Keyword(s):  
Euclidean Space ◽  
Complex Plane ◽  
Euclidean Plane ◽  
Complex Space ◽  
Radical Center ◽  
Special Cases ◽  
Different Types ◽  
Different Dimensions ◽  
The Given ◽  
Made In

The Euclidean plane and Euclidean space themselves do not contain imaginary elements by definition, but are inextricably linked with them through special cases, and this leads to the need to propagate geometry into the area of imaginary values. Such propagation, that is adding a plane or space, a field of imaginary coordinates to the field of real coordinates leads to various variants of spaces of different dimensions, depending on the given axiomatics. Earlier, in a number of papers, were shown examples for solving some urgent problems of geometry using imaginary geometric images [2, 9, 11, 13, 15]. In this paper are considered constructions of orthogonal and diametrical positions of circles on a complex plane. A generalization has been made of the proposition about a circle on the complex plane orthogonally intersecting three given spheres on the proposition about a sphere in the complex space orthogonally intersecting four given spheres. Studies have shown that the diametrical position of circles on the Euclidean E-plane is an attribute of the orthogonal position of the circles’ imaginary components on the pseudo-Euclidean M-plane. Real, imaginary and degenerated to a point circles have been involved in structures and considered, have been demonstrated these circles’ forms, properties and attributes of their orthogonal position. Has been presented the construction of radical axes and a radical center for circles of the same and different types. A propagation of 2D mutual orthogonal position of circles on 3D spheres has been made. In figures, dashed lines indicate imaginary elements.

A Novel Approach for Analyzing Single Buffer Queueing Systems with State-Dependent Vacation and Correlated Input Process under Four Different Service Disciplines

2015 ◽  
Vol 6 (3) ◽  
pp. 19-59 ◽  
Author(s):  
Thomas Yew Sing Lee
Keyword(s):  
Performance Measures ◽  
Probability Generating Function ◽  
Performance Measure ◽  
Idle Time ◽  
Separate Analysis ◽  
Input Process ◽  
State Dependent ◽  
Special Cases ◽  
Service Disciplines ◽  
Different Types

The author presents performance analysis of a single buffer multiple-queue system. Four different types of service disciplines (i.e., non-preemptive, pre-emptive repeat different, state dependent random polling and globally gated) are analyzed. His model includes correlated input process and three different types of non-productive time (i.e., switchover, vacation and idle time). Special cases of the model includes server with mixed multiple and single vacations, stopping server with delayed vacation and stopping server with alternating vacation and idle time. For each of the four service disciplines the key performance measures such as average customer waiting time, loss probability, and throughput are computed. The results permit a detailed discussion of how these performance measures depends on the customer arrival rate, the customer service time, the switchover time, the vacation time, and the idle time. Moreover, extensive numerical results are presented and the four service disciplines are compared with respect to the performance measure. Previous studies of the single buffer multiple-queue systems tend to provide separate analysis for the two cases of zero and nonzero switchover time. The author is able to provide a unified analysis for the two cases. His results generalize and improve a number of known results on single buffer multiple-queue systems. Furthermore, this method does not require differentiation while it is needed if one uses the probability generating function approach. Lastly, the author's approach works for all single buffer multiple-queue systems in which the next queue to be served is determines solely on the basis of the occupancy states at the end of the cycle time.

scholarly journals Generalising Exponential Distributions Using an Extended Marshall–Olkin Procedure

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 464
Author(s):  
Victoriano García ◽  
María Martel-Escobar ◽  
F.J. Vázquez-Polo
Keyword(s):  
Failure Rate ◽  
Real Data ◽  
Rate Function ◽  
Exponential Distributions ◽  
Symmetric Distributions ◽  
Failure Rate Function ◽  
Numerical Examples ◽  
Special Cases ◽  
The Common ◽  
Family Of Distributions

This paper presents a three-parameter family of distributions which includes the common exponential and the Marshall–Olkin exponential as special cases. This distribution exhibits a monotone failure rate function, which makes it appealing for practitioners interested in reliability, and means it can be included in the catalogue of appropriate non-symmetric distributions to model these issues, such as the gamma and Weibull three-parameter families. Given the lack of symmetry of this kind of distribution, various statistical and reliability properties of this model are examined. Numerical examples based on real data reflect the suitable behaviour of this distribution for modelling purposes.

Hypernetwork models based on random hypergraphs

2019 ◽  
Vol 30 (08) ◽  
pp. 1950052
Author(s):  
Feng Hu ◽  
Jin-Li Guo ◽  
Fa-Xu Li ◽  
Hai-Xing Zhao
Keyword(s):  
Field Theory ◽  
Mean Field Theory ◽  
Mean Field ◽  
World Systems ◽  
Uniform Hypergraphs ◽  
Different Types ◽  
Powerful Means ◽  
Simulation Results ◽  
Parameter Values ◽  
Random Hypergraphs

Hypernetworks are ubiquitous in real-world systems. They provide a powerful means of accurately depicting networks of different types of entity and will attract more attention from researchers in the future. Most previous hypernetwork research has been focused on the application and modeling of uniform hypernetworks, which are based on uniform hypergraphs. However, random hypernetworks are generally more common, therefore, it is useful to investigate the evolution mechanisms of random hypernetworks. In this paper, we construct three dynamic evolutional models of hypernetworks, namely the equal-probability random hypernetwork model, the Poisson-probability random hypernetwork model and the certain-probability random hypernetwork model. Furthermore, we analyze the hyperdegree distributions of the three models with mean-field theory, and we simulate each model numerically with different parameter values. The simulation results agree well with the results of our theoretical analysis, and the findings indicate that our models could help understand the structure and evolution mechanisms of real systems.

scholarly journals Hyers-Ulam stability of first-order homogeneous linear dynamic equations on time scales

2018 ◽  
Vol 51 (1) ◽  
pp. 198-210 ◽  
Author(s):  
Douglas R. Anderson ◽  
Masakazu Onitsuka
Keyword(s):  
Time Scales ◽  
Dynamic Equations ◽  
Ulam Stability ◽  
Step Size ◽  
First Order ◽  
Linear Dynamic ◽  
Discrete Step ◽  
Special Cases ◽  
Parameter Values ◽  
Homogeneous Linear

Abstract We establish theHyers-Ulam stability (HUS) of certain first-order linear constant coefficient dynamic equations on time scales, which include the continuous (step size zero) and the discrete (step size constant and nonzero) dynamic equations as important special cases. In particular, for certain parameter values in relation to the graininess of the time scale, we find the minimum HUS constants. A few nontrivial examples are provided. Moreover, an application to a perturbed linear dynamic equation is also included.

Individual Components of Three Inequality Measures for Analyzing Shapes of Inequality

2019 ◽  
pp. 004912411987596
Author(s):  
Tim Futing Liao
Keyword(s):  
Sociological Research ◽  
Aggregate Level ◽  
Current Population Survey Data ◽  
Single Measure ◽  
Generalized Entropy ◽  
Individual Level ◽  
Inequality Measures ◽  
Entropy Measures ◽  
Individual Contributions ◽  
The Individual

In common sociological research, income inequality is measured only at the aggregate level. The main purpose of this article is to demonstrate that there is more than meets the eye when inequality is indicated by a single measure. In this article, I introduce an alternative method that evaluates individuals’ contributions to inequality as well as the between-group and within-group components of these individual contributions. I first highlight three common inequality measures, the Gini index and two generalized entropy measures—Theil’s T and Theil’s L indices—by presenting their individual components as a method for evaluating inequality. Five artificial data examples illustrate the use of these individual components first. An empirical analysis of the 2007 and 2017 Current Population Survey data then focuses on the differences in inequality revealed by such individual inequality components between the 2007 and 2017. The individual-level inequality measures can reveal patterns of inequality concealed by single measures at the aggregate level. In particular, the Gini individual measures differentiate cases better than the generalized entropy measures and tend to have smaller standard errors in a regression analysis.

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