scholarly journals Flexible models for overdispersed and underdispersed count data

2021 ◽  
Author(s):  
Dexter Cahoy ◽  
Elvira Di Nardo ◽  
Federico Polito
Keyword(s):  
Poisson Distribution ◽  
Count Data ◽  
Hypergeometric Functions ◽  
Natural Generalization ◽  
Model Parameters ◽  
Probability Models ◽  
Limiting Behavior ◽  
Poisson Models ◽  
Special Cases ◽  
Flexible Models

AbstractWithin the framework of probability models for overdispersed count data, we propose the generalized fractional Poisson distribution (gfPd), which is a natural generalization of the fractional Poisson distribution (fPd), and the standard Poisson distribution. We derive some properties of gfPd and more specifically we study moments, limiting behavior and other features of fPd. The skewness suggests that fPd can be left-skewed, right-skewed or symmetric; this makes the model flexible and appealing in practice. We apply the model to real big count data and estimate the model parameters using maximum likelihood. Then, we turn to the very general class of weighted Poisson distributions (WPD’s) to allow both overdispersion and underdispersion. Similarly to Kemp’s generalized hypergeometric probability distribution, which is based on hypergeometric functions, we analyze a class of WPD’s related to a generalization of Mittag–Leffler functions. The proposed class of distributions includes the well-known COM-Poisson and the hyper-Poisson models. We characterize conditions on the parameters allowing for overdispersion and underdispersion, and analyze two special cases of interest which have not yet appeared in the literature.

scholarly journals Analytical Investigation of Viscoelastic Stagnation-Point Flows with Regard to Their Singularity

2021 ◽  
Vol 11 (15) ◽  
pp. 6931
Author(s):  
Jie Liu ◽  
Martin Oberlack ◽  
Yongqi Wang
Keyword(s):  
Stress Distribution ◽  
Stress Field ◽  
Stagnation Point ◽  
Analytical Solutions ◽  
Wide Spectrum ◽  
Momentum Conservation ◽  
Stagnation Point Flow ◽  
Model Parameters ◽  
Viscoelastic Models ◽  
Special Cases

Singularities in the stress field of the stagnation-point flow of a viscoelastic fluid have been studied for various viscoelastic constitutive models. Analyzing the analytical solutions of these models is the most effective way to study this problem. In this paper, exact analytical solutions of two-dimensional steady wall-free stagnation-point flows for the generic Oldroyd 8-constant model are obtained for the stress field using different material parameter relations. For all solutions, compatibility with the conservation of momentum is considered in our analysis. The resulting solutions usually contain arbitrary functions, whose choice has a crucial effect on the stress distribution. The corresponding singularities are discussed in detail according to the choices of the arbitrary functions. The results can be used to analyze the stress distribution and singularity behavior of a wide spectrum of viscoelastic models derived from the Oldroyd 8-constant model. Many previous results obtained for simple viscoelastic models are reproduced as special cases. Some previous conclusions are amended and new conclusions are drawn. In particular, we find that all models have singularities near the stagnation point and most of them can be avoided by appropriately choosing the model parameters and free functions. In addition, the analytical solution for the stress tensor of a near-wall stagnation-point flow for the Oldroyd-B model is also obtained. Its compatibility with the momentum conservation is discussed and the parameters are identified, which allow for a non-singular solution.

scholarly journals New Modifications of Integral Inequalities via ℘-Convexity Pertaining to Fractional Calculus and Their Applications

Mathematics ◽  
2021 ◽  
Vol 9 (15) ◽  
pp. 1753
Author(s):  
Saima Rashid ◽  
Aasma Khalid ◽  
Omar Bazighifan ◽  
Georgia Irina Oros
Keyword(s):  
Integral Operator ◽  
Convex Functions ◽  
Hypergeometric Functions ◽  
Fractional Integral ◽  
Type Inequality ◽  
Fractional Integral Operator ◽  
Integral Inequalities ◽  
Convex Mappings ◽  
Special Cases ◽  
Harmonically Convex Functions

Integral inequalities for ℘-convex functions are established by using a generalised fractional integral operator based on Raina’s function. Hermite–Hadamard type inequality is presented for ℘-convex functions via generalised fractional integral operator. A novel parameterized auxiliary identity involving generalised fractional integral is proposed for differentiable mappings. By using auxiliary identity, we derive several Ostrowski type inequalities for functions whose absolute values are ℘-convex mappings. It is presented that the obtained outcomes exhibit classical convex and harmonically convex functions which have been verified using Riemann–Liouville fractional integral. Several generalisations and special cases are carried out to verify the robustness and efficiency of the suggested scheme in matrices and Fox–Wright generalised hypergeometric functions.

scholarly journals Agricultural drought periods analysis by using nonhomogeneous poisson models and regionalization of appropriate model parameters

2021 ◽  
Vol 73 (1) ◽  
pp. 1-15
Author(s):  
Asad Ellahi ◽  
Ijaz Hussain ◽  
Muhammad Zaffar Hashmi ◽  
Mohammed Mohammed Ahmed Almazah ◽  
Fuad S. Al-Duais
Keyword(s):  
Agricultural Drought ◽  
Model Parameters ◽  
Poisson Models

scholarly journals An Application Comparison of Two Poisson Models on Zero Count Data

2021 ◽  
Vol 1818 (1) ◽  
pp. 012165
Author(s):  
L. H. Hashim ◽  
K. H. Hashim ◽  
Mushtak A. K. Shiker
Keyword(s):  
Count Data ◽  
Poisson Models

scholarly journals Marginal regression models for clustered count data based on zero-inflated Conway-Maxwell-Poisson distribution with applications

Biometrics ◽  
2015 ◽  
Vol 72 (2) ◽  
pp. 606-618 ◽  
Author(s):  
Hyoyoung Choo-Wosoba ◽  
Steven M. Levy ◽  
Somnath Datta
Keyword(s):  
Poisson Distribution ◽  
Count Data ◽  
Regression Models ◽  
Marginal Regression

scholarly journals New Formulae for the High-Order Derivatives of Some Jacobi Polynomials: An Application to Some High-Order Boundary Value Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
W. M. Abd-Elhameed
Keyword(s):  
Galerkin Method ◽  
Boundary Value Problems ◽  
Chebyshev Polynomials ◽  
Hypergeometric Functions ◽  
Jacobi Polynomials ◽  
Boundary Value ◽  
High Order ◽  
Sixth Order ◽  
Special Cases ◽  
Derivatives Of

This paper is concerned with deriving some new formulae expressing explicitly the high-order derivatives of Jacobi polynomials whose parameters difference is one or two of any degree and of any order in terms of their corresponding Jacobi polynomials. The derivatives formulae for Chebyshev polynomials of third and fourth kinds of any degree and of any order in terms of their corresponding Chebyshev polynomials are deduced as special cases. Some new reduction formulae for summing some terminating hypergeometric functions of unit argument are also deduced. As an application, and with the aid of the new introduced derivatives formulae, an algorithm for solving special sixth-order boundary value problems are implemented with the aid of applying Galerkin method. A numerical example is presented hoping to ascertain the validity and the applicability of the proposed algorithms.

Unification of Software Reliability Models by Self-Exciting Point Processes

1997 ◽  
Vol 29 (2) ◽  
pp. 337-352 ◽  
Author(s):  
Yiping Chen ◽  
Nozer D. Singpurwalla
Keyword(s):  
Software Reliability ◽  
Point Processes ◽  
Computer Software ◽  
Probability Models ◽  
Reliability Models ◽  
Common Structure ◽  
Software Reliability Models ◽  
Special Cases ◽  
Active Debate ◽  
The Subject

Assessing the reliability of computer software has been an active area of research in computer science for the past twenty years. To date, well over a hundred probability models for software reliability have been proposed. These models have been motivated by seemingly unrelated arguments and have been the subject of active debate and discussion. In the meantime, the search for an ideal model continues to be pursued. The purpose of this paper is to point out that practically all the proposed models for software reliability are special cases of self-exciting point processes. This perspective unifies the very diverse approaches to modeling reliability growth and provides a common structure under which problems of software reliability can be discussed.

scholarly journals ON SOME NEW Pδ-TRANSFORMS OF KUMMER'S CONFLUENT HYPERGEOMETRIC FUNCTIONS

2019 ◽  
pp. 373
Author(s):  
Rakesh K. Parmar ◽  
Vivek Rohira ◽  
Arjun K. Rathie
Keyword(s):  
Hypergeometric Functions ◽  
Confluent Hypergeometric Functions ◽  
Special Cases ◽  
Summation Theorem

The aim of our paper is to present Pδ -transforms of the Kummer’s confluent hypergeometric functions by employing the generalized Gauss’s second summation the-orem, Bailey’s summation theorem and Kummer’s summation theorem obtained earlier by Lavoie, Grondin and Rathie [9]. Relevant connections of certain special cases of the main results presented here are also pointed out.

Bayesian Meta-analysis

2013 ◽  
Author(s):  
Christopher H. Schmid ◽  
Kerrie Mengersen
Keyword(s):  
Bayesian Approach ◽  
Meta Analysis ◽  
Model Parameters ◽  
Likelihood Methods ◽  
Rich Variety ◽  
Other Information ◽  
Maximum Likelihood Methods ◽  
Data Missing ◽  
Flexible Models ◽  
Parameters Of Interest

This chapter introduces a Bayesian approach to meta-analysis. It discusses the ways in which a Bayesian approach differs from the method of moments and maximum likelihood methods described in chapters 9 and 10, and summarizes the steps required for a Bayesian analysis. It shows that Bayesian methods provide the basis for a rich variety of very flexible models, explicit statements about uncertainty of model parameters, inclusion of other information relevant to an analysis, and direct probabilistic statements about parameters of interest. In a meta-analysis context, this allows for more straightforward accommodation of study-specific differences and similarities, nonnormality and other distributional features of the data, missing data, small studies, and so forth.

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