Eggenberger Distribution Research Articles

Parametric Bernstein operators based on contagion distribution

This paper proposes the extended form of the ‐Bernstein operators with the Pólya–Eggenberger distribution, also known as contagion distribution, using the integral operators for . We will refer to these sequence of operators as the “Parametric‐Bernstein operators based on contagion distribution” or simply the modified form of ‐Bernstein operators. These operators are defined with the help of two parameters, namely, and . We observe the significance of both these parameters. Subsequently, we calculate their moments and on the basis of Korovkin's approximation theorem we assert that our defined operators converge for a real valued continuous function . We give some basic properties, study the Voronovskaya type asymptotic result and discuss the ‐statistical approximation of our operators. We also use the modulus of continuity to study the rate of convergence. Alongside our primary work, we have also defined the King‐type modification that preserves the operators at and . Further, using modulus of continuity, we compare the properties and behavior of both the operators.

Bézier-Summation-Integral-Type Operators That Include Pólya–Eggenberger Distribution

We define the summation-integral-type operators involving the ideas of Pólya–Eggenberger distribution and Bézier basis functions, and study some of their basic approximation properties. In addition, by means of the Ditzian–Totik modulus of smoothness, we study a direct theorem as well as a quantitative Voronovskaja-type theorem for our newly constructed operators. Moreover, we investigate the approximation of functions with derivatives of bounded variation (DBV) of the aforesaid operators.

Generalized Durrmeyer operators based on inverse Pólya–Eggenberger distribution

Generalized Durrmeyer operators based on inverse Pólya–Eggenberger distribution

Higher order Kantorovich operators based on inverse Pólya–Eggenberger distribution

Higher order Kantorovich operators based on inverse Pólya–Eggenberger distribution

Stancu Type Baskakov—Durrmeyer Operators and Approximation Properties

In this article, we introduce Stancu type generalization of Baskakov–Durrmeyer operators by using inverse Pólya–Eggenberger distribution. We discuss some basic results and approximation properties. Moreover, we study the statistical convergence for these operators.

New modified Baskakov operators based on the inverse Pólya-Eggenberger distribution

In the present article we introduce some modifications of the Baskakov operators in sense of the Lupa? operators based on the inverse P?lya-Eggenberger distribution. For these new modifications we derive some direct results concerning the uniform convergence and the asymptotic formula, as well as some quantitative type theorems.

Jain–Durrmeyer Operators Involving Inverse Pólya–Eggenberger Distribution

Stancu generalized Baskakov operators using inverse Polya–Eggenberger distribution for a real valued bounded function on $$[0,\infty )$$ and a non-negative real number $$\alpha $$ (which may depend on n). Dhamija and Deo (Appl Math Comput 286:15–22, 2016) introduced a Durrmeyer type modification of these generalized operators and studied the uniform convergence, the rate of convergence by means of the moduli of continuity and the Peetre’s K-functional. The purpose of the present paper is to continue to study the approximation properties of these Durrmeyer type operators. Local and global direct approximation theorems and a Voronovskaya type asymptotic theorem are established. The quantitative Voronovskaya and Gruss Voronovskaya type theorems are investigated by calculating the indispensible sixth order moment. The weighted approximation properties and the approximation of functions with derivatives of bounded variation are also studied.

Generalized Positive Linear Operators Based on PED and IPED

The paper deals with generalized positive linear operators based on Polya–Eggenberger distribution (PED) as well as inverse Polya–Eggenberger distribution (IPED). Initially, we give the moments using Stirling numbers of second kind and then establish direct results for proposed operators.

Approximation by Stancu-Durrmeyer type operators based on Pólya-Eggenberger distribution

In the present paper we introduce the Durrmeyer type modification of Stancu operators based on P?lya-Eggenberger distribution. For these new operators some indispensable auxiliary results are established in the second section. Our further study focuses on a Voronovskaja type asymptotic formula and some estimates of the rate of approximation involving modulus of smoothness, respectively Ditzian-Totik modulus of smoothness. The rate of convergence for differential functions whose derivatives are of bounded variation is also obtained.

New sufficient conditions for strong unimodality of multivariate discrete distributions

New sufficient conditions that ensure the strong unimodality of multivariate discrete distributions are obtained by the use of a special simplicial subdivision of multidimensional space. Strong unimodality of multivariate Pólya–Eggenberger distribution is shown.

Approximation degree of a Kantorovich variant of Stancu operators based on Polya–Eggenberger distribution

This paper is a continuation of the work done by Deo et al. (Appl. Math. Comput. 273, 281–289, 2016), in which the authors have established some approximation properties of the Stancu–Kantorovich operators based on Polya–Eggenberger distribution. We obtain some direct results for these operators by means of the Lipschitz class function, the modulus of continuity and the weighted space. Also, we study an approximation theorem with the aid of the unified Ditzian–Totik modulus of smoothness $$\omega _{\phi ^{\tau }}(f;t),\,\,\,0\le \tau \le 1$$ and the rate of convergence of the operators for the functions having a derivative which is locally of bounded variation on $$[0,\infty )$$ .

Modified Stancu operators based on inverse Polya Eggenberger distribution

In this paper, we construct a sequence of modified Stancu-Baskakov operators for a real valued function bounded on [0,infty), based on a function tau(x). This function tau(x) is infinite times continuously differentiable on [0,infty) and satisfy the conditions tau (0)=0,~tau ^{prime}(x)>0 and tau^{primeprime}(x) is bounded for all xin {}[0,infty). We study the degree of approximation of these operators by means of the Peetre K-functional and the Ditzian-Totik modulus of smoothness. The quantitative Voronovskaja-type theorems are also established in terms of the first order Ditzian-Totik modulus of smoothness.

Baskakov–Szász-type operators based on inverse Pólya–Eggenberger distribution

The present article deals with the modified forms of the Baskakov and Szász basis functions. We introduce a Durrmeyer-type operator having the basis functions in summation and integration due to Stancu (1970) and Pǎltǎnea (2008). We obtain some approximation results, which include the Voronovskaja-type asymptotic formula, local approximation, error estimation in terms of the modulus of continuity, and weighted approximation. Also, the rate of convergence for functions with derivatives of bounded variation is established. Furthermore, the convergence of these operators to certain functions is shown by illustrative graphics using MAPLE algorithms.

Approximation by generalized positive linear Kantorovich operators

In the present article, we introduce generalized positive linear-Kantorovich operators depending on P?lya-Eggenberger distribution (PED) as well as inverse P?lya-Eggenberger distribution (IPED) and for these operators, we study some approximation properties like local approximation theorem, weighted approximation and estimation of rate of convergence for absolutely continuous functions having derivatives of bounded variation.

Approximation of Baskakov type Pólya–Durrmeyer operators

In the present paper we propose the Durrmeyer type modification of Baskakov operators based on inverse Pólya–Eggenberger distribution. First we estimate a recurrence relation by using hypergeometric series. We give a global approximation theorem in terms of second order modulus of continuity, a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness and a Voronovskaja type theorem. Some approximation results in weighted space are obtained. Also, we show the rate of convergence of these operators to certain functions by illustrative graphics using the Maple algorithms.

Jain–Durrmeyer operators associated with the inverse Pólya–Eggenberger distribution

The present paper deals with the Jain–Durrmeyer operators based on inverse Pólya–Eggenberger distribution. First, we give the moments with the help of Vandermonde convolution formula and then study approximation properties of these operators which include uniform convergence and degree of approximation.

Stancu–Kantorovich operators based on inverse Pólya–Eggenberger distribution

The purpose of this paper is to investigate approximation properties of Stancu–Kantorovich operators based on inverse Pólya–Eggenberger distribution. For these new operators we establish some approximation properties including uniform convergence, asymptotic formula and degree of approximation.

Application of a truncated distribution for tsumugi yarn width

Hand-spun, silk yarn made from floss silk is referred to as tsumugi yarn. It gives a unique appearance on the products, also called tsumugi, which are considered to be an important type of fabric for their aesthetics. The essential factor of this visual effect is originated by the thickness variation of tsumugi yarn; however, the detail of the variation is unknown. In this paper, as a fundamental study of tsumugi yarn, its width distribution is discussed. To model the width distribution, the Polya–Eggenberger distribution is introduced. Moreover some modifications are applied for the distribution to describe the yarn's situation. That is to say, it is impossible to be thinner than a level of thickness for the yarn to maintain a shape of a thread; on the other hand, tsumugi yarn is not limited to only containing thicker parts. Additionally, it is unavoidable to involve errors and fluctuations in the actual data. Hence, the distribution is truncated and convoluted. By using an image scanner, the width of tsumugi yarn is measured. With the measured data, the parameters of the width distribution are estimated using the maximum likelihood method. The results show that the width distribution of tsumugi yarn is asymmetric and well summarized by the distribution.

Hierarchical and spatially aggregated plant cover data

Most plant species are spatially aggregated and here the importance of taking the spatial variation into account when analyzing plant cover data is demonstrated in a general stochastic model where both the within-site and the among-site spatial variation of species cover data are parameterized. Using a generalised binomial distribution (or Pólya–Eggenberger distribution), where the among-site variation in mean cover is modeled by a zero-inflated beta distribution, it is possible to adequately analyze hierarchical plant cover data and link the estimates to the underlying ecological processes. The model is demonstrated in a case-study of pin-point cover data of Erica tetralix from 1148 wet heathland plots at 84 Danish sites, and it is shown that both parameter estimates and the conclusions of hypotheses testing critically depend on the correct modeling of the observed spatial variation. Finally, statistical power simulations of plant cover measurements are presented, which will be useful for planning ecological experiments and monitoring programs.

Probability generating function of GPED2

The Probability generating function of a random variable which has Generalized Polya Eggenberger Distribution of the second kind (GPED 2) is obtained. The probability density function of the range R, in random sampling from a uniform distribution on (k, l) and exponential distribution with parameter λ is obtained, when the sample size is a random variable from GPED 2. The results of Bazargan-Lari (2004) follow as special cases.