Yule Distribution Research Articles

Doubly Generalized Yule Distribution of Order k and Its Applications

Doubly Generalized Yule Distribution of Order k and Its Applications

Segmental estimation and testing method for power-law distributions and some extensions

For discrete segmental power-law distributions, the probability ratio ft/ft+1 is a linear function of the exponent parameter. Based on this property, the estimation of the exponent parameter and a goodness-of-fit testing method are provided. The proposed testing method is parameter-independent and the testing statistic is proved to asymptotically follow a chi-square distribution. In the region where power-law properties exist, the testing method and the estimation method can be applied by segments, so they can also be used to determine interval endpoints. These methods can also be extended to other discrete distributions such as Yule distribution, Poisson distribution, Geometry distribution and so on. Some simulation results of synthetic truncated power-law distributions provide support for the effectiveness of the proposed methods. To demonstrate the applicability of the method, two empirical examples, word frequencies of the novel Moby Dick and US casuality numbers in the American Indian War, are analyzed.

Conservatism and representativeness heuristic in peer reviews: Evidence from the finance literature 1946–2020

This paper explores whether journal referees take into account the author's scholarly credentials in their decisions using the stochastic process underlying the Yule-Simon distribution as a descriptive model of the peer-review process. We provide evidence that referees consider the author's publication record valuable information in their decision-making, and such referee behavior helps improve the efficacy of the peer-review process. We show that the probability that a newly accepted paper in a given journal is written by authors with no prior publications is lower for top journals, specialty journals, journals with higher Eigenfactor or Article Influence Score, and journals with more publications.

On the MLE of the Waring distribution

The two-parameter Waring is an important heavy-tailed discrete distribution, which extends the famous Yule-Simon distribution and provides more flexibility when modelling the data. The commonly used EFF (Expectation-First Frequency) for parameter estimation can only be applied when the first moment exists, and it only uses the information of the expectation and the first frequency, which is not as efficient as the maximum likelihood estimator (MLE). However, the MLE may not exist for some sample data. We apply the profile method to the log-likelihood function and derive the necessary and sufficient conditions for the existence of the MLE of the Waring parameters. We use extensive simulation studies to compare the MLE and EFF methods, and the goodness-of-fit comparison with the Yule-Simon distribution. We also apply the Waring distribution to fit an insurance data.

Forward–backward multiplicity distribution with the Chou–Yang model for pp collisions at s = 0.9, 7 and 8 TeV from the CMS experiment

In this paper, a Chou–Yang-type multiplicity distribution comprising a total multiplicity component and a binomial asymmetry component is used to describe charged hadron multiplicity data at [Formula: see text] = 0.9, 7 and 8 TeV from the CMS experiment at CERN. The data was obtained and processed from the CERN Open Data Portal. For the total multiplicity component, it was found that a weighted superposition of a Negative Binomial Distribution (NBD) and a Furry–Yule Distribution (FYD) is able to describe the shoulder-like structure characteristic of Koba–Nielsen–Olesen (KNO) scaling violation well. The mean cluster size produced from collisions was also found to increase with collision energy. A prediction is given for pp collisions at [Formula: see text] = 14 TeV.

Bivariate generalized Yule distribution: properties and applications

A bivariate generalized Yule distribution is introduced here as the distribution of the random sum of certain types of independent and identically distributed bivariate Bernoulli random variables. We study some important properties of the distribution through deriving explicit expressions for its probability mass function, factorial moments, and the probability generating function (p.g.f.) of its conditional distributions. Certain recurrence relations for its probabilities, raw moments and factorial moments are also developed. The method of maximum likelihood is employed for estimating the parameters of the distribution and applied to real life data sets for illustrating the relevance of the distribution.

Bivariate Extended Yule Distribution

Here we introduce a bivariate version of the extended Yule distribution (EYD) of Martinez-Rodriguez et al (Computational Statistics and Data Analysis, 2011). The EYD has been found suitable for heavy tailed distributions. For emphasizing the practical relevance of the model, it is shown that the proposed distribution can be viewed as the distribution of the random sum of independently and identically distributed bivariate Bernoulli random variables. It is also observed that the proposed bivariate distribution is a very flexible distribution and it includes the bivariate geometric distribution as its special case. A genesis of the distribution and explicit closed form expressions for its probability mass function, factorial moments and the probability generating function of its conditional distributions are derived. Certain recurrence relations for probabilities, raw moments and factorial moments of the BEYD are also obtained. The method of maximum likelihood estimation is employed for estimating the parameters of the distribution. Some examples of application using data from different fields are included in order to illustrate all these procedures.

Superstars and “The Voice”

ABSTRACT The previous literature has explored theories and models of the economics of superstars in the context of the music industry and in general finds mixed evidence for superstar effects and the data being close to the Yule distribution. The present article extends that literature by examining if the same patterns are present for near superstars, specifically contestants on the reality television show ‘The Voice’. Our results support the presence of the same features in the data for these near superstars.

On Some Aspects of Generalized Extended Yule Distribution: Properties and Applications

Martinez-Rodriguez (Comp. Statist. Dat. Anal., 2011) studied an extended version of the Yule distribution, namely “the extended Yule distribution (EYD)” which they obtained as a mixture of geometric distribution and generalized beta distribution. Through the present paper, we propose a generalized version of the EYD and named it “the generalized extended Yule distribution (GEYD)”. Several statistical properties of the distribution are obtained, including probability generating function (p.g.f), moments, recursion formulae etc. The maximum likelihood estimation of the parameters of the GEYD is discussed and fitted to two real-life data sets for illustrating its usefulness compared to the existing models. Further, the generalized likelihood ratio test procedure is considered for testing the significance of the parameters of the GEYD.

On some aspects of a general class of Yule distribution and its applications

A wide class of Yule distribution is introduced here as a generalization of the extended Yule distribution of Martinez-Rodríguez et al. and the generalized Yule distribution of Mishra and investigate some of its properties. Various methods of estimation are employed for estimating the parameters of the distribution and generalized likelihood ratio test procedures are suggested for testing the significance of the parameters of the distribution. All these procedures are illustrated with the help of real data sets.

Bivariate Version of Yule Distribution and Some of its Properties

Bivariate Version of Yule Distribution and Some of its Properties

On a Modified Yule Distribution

A modified version of Yule distribution is introduced here and discuss some of its properties by deriving expressions for its probability generating function, raw moments, factorial moments etc. Certain recursion formulae for its probabilities, raw moments and factorial moments are also developed. Various methods of estimation are employed for estimating the parameters of the distribution and certain test procedures are suggested for testing the significance of the additional parameters of the distribution. The distribution has been fitted to certain real-life data sets for illustrating its usefulness, compared with certain existing models available in the literature. Further, a simulation study is conducted for assessing the performance of the maximum likelihood estimators.

Population Dynamic Models Leading to Logarithmic and Yule Distribution

Population Dynamic Models Leading to Logarithmic and Yule Distribution

Parametric Point Estimation of the Geeta Distribution

Geeta distribution is a new discrete random variable distribution defined over all the positive integers with two parameters. This distribution belongs to the family of Location-parameter (LDPD) system and is of the form L – shaped model. Pareto and Yule distributions belong to the same family but these distributions have a disadvantage of having a single parameter which makes them not versatile to meet the needs of modern complex data sets. Geeta distribution is found to be very versatile and flexible to fit observed count data sets and can be used efficiently to model different types of sets. This paper investigates the characteristics of Geeta distribution, such as the existence of the mean, variance, moment generating function, probability generating function and that the sum of probabilities for all values of X for Geeta Distribution model is unity. It is well known that the sample mean is the estimator of a population mean from a given population of interest as a point estimator which assume a single number that is obtained by taking a random sample of a specified size from the entire population, depending on whether the population mean and variance is known or unknown These point estimators were obtained by employing the method of Moments, Maximum Likelihood (MLE) and Bayesian estimator. Further the estimators were subjected to the conditions like unbiasedness, efficiency, sufficiency and completeness which are properties of a good estimator. For the first aspect, the results of the mean, variance, moments and generating functions were achieved that proves the distribution is a probability density function (pdf). The methods of moments and the maximum likelihood and their properties were applied and yielded the desired and expected results for any given probability distribution. The best estimator obtained is best linear unbiased estimator (BLUE).

Intriguing yet simple skewness: kurtosis relation in economic and demographic data distributions, pointing to preferential attachment processes

ABSTRACTIn this paper, we propose that relations between high-order moments of data distributions, for example, between the skewness (S) and kurtosis (K), allow to point to theoretical models with understandable structural parameters. The illustrative data concern two cases: (i) the distribution of income taxes and (ii) that of inhabitants, after aggregation over each city in each province of Italy in 2011. Moreover, from the rank-size relationship, for either S or K, in both cases, it is shown that one obtains the parameters of the underlying (hypothetical) modeling distribution: in the present cases, the 2-parameter Beta function, itself related to the Yule–Simon distribution function, whence suggesting a growth model based on the preferential attachment process.

Objective bayesian analysis of the Yule\u2013Simon distribution with applications

The Yule–Simon distribution is usually employed in the analysis of frequency data. As the Bayesian literature, so far, has ignored this distribution, here we show the derivation of two objective priors for the parameter of the Yule–Simon distribution. In particular, we discuss the Jeffreys prior and a loss-based prior, which has recently appeared in the literature. We illustrate the performance of the derived priors through a simulation study and the analysis of real datasets.

A Note on the Posterior Inference for the Yule-Simon Distribution

The Yule-Simon distribution has been out of the radar of the Bayesian community, so far. In this note, we propose an explicit Gibbs sampling scheme when a Gamma prior is chosen for the shape parameter. The performance of the algorithm is illustrated with simulation studies, including count data regression, and a real data application to text analysis. We compare our proposal to the frequentist counterparts showing better performance of our algorithm when a small sample size is considered.

A note on the posterior inference for the Yule–Simon distribution

ABSTRACTThe Yule–Simon distribution has been out of the radar of the Bayesian community, so far. In this note, we propose an explicit Gibbs sampling scheme when a Gamma prior is chosen for the shape parameter. The performance of the algorithm is illustrated with simulation studies, including count data regression, and a real data application to text analysis. We compare our proposal to the frequentist counterparts showing better performance of our algorithm when a small sample size is considered.

A Universal Rank-Size Law.

A mere hyperbolic law, like the Zipf’s law power function, is often inadequate to describe rank-size relationships. An alternative theoretical distribution is proposed based on theoretical physics arguments starting from the Yule-Simon distribution. A modeling is proposed leading to a universal form. A theoretical suggestion for the “best (or optimal) distribution”, is provided through an entropy argument. The ranking of areas through the number of cities in various countries and some sport competition ranking serves for the present illustrations.

Phase transition of social learning collectives and the echo chamber.

We study a simple model for social learning agents in a restless multiarmed bandit. There are N agents, and the bandit has M good arms that change to bad with the probability q_{c}/N. If the agents do not know a good arm, they look for it by a random search (with the success probability q_{I}) or copy the information of other agents' good arms (with the success probability q_{O}) with probabilities 1-p or p, respectively. The distribution of the agents in M good arms obeys the Yule distribution with the power-law exponent 1+γ in the limit N,M→∞, and γ=1+(1-p)q_{I}/pq_{O}. The system shows a phase transition at p_{c}=q_{I}/q_{I}+q_{o}. For p<p_{c}(>p_{c}), the variance of N_{1} per agent is finite (diverges as ∝N^{2-γ} with N). There is a threshold value N_{s} for the system size that scales as lnN_{s}∝1/(γ-1). The expected value of the number of the agents with a good arm N_{1} increases with p for N>N_{s}. For p>p_{c} and N<N_{s}, all agents tend to share only one good arm. If the shared arm changes to be bad, it takes a long time for the agents to find another good one. E(N_{1}) decreases to zero as p→1, which is referred to as the "echo chamber."