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Open AccessArticle

A Novel Generalization of Zero-Truncated Binomial Distribution by Lagrangian Approach with Applications for the COVID-19 Pandemic

by

Muhammed Rasheed Irshad

,

Christophe Chesneau

,

Damodaran Santhamani Shibu

,

Mohanan Monisha

and

Radhakumari Maya

Stats 2022, 5(4), 1004-1028; https://doi.org/10.3390/stats5040060 - 30 Oct 2022

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Abstract

The importance of Lagrangian distributions and their applicability in real-world events have been highlighted in several studies. In light of this, we create a new zero-truncated Lagrangian distribution. It is presented as a generalization of the zero-truncated binomial distribution (ZTBD) and hence named [...] Read more.

The importance of Lagrangian distributions and their applicability in real-world events have been highlighted in several studies. In light of this, we create a new zero-truncated Lagrangian distribution. It is presented as a generalization of the zero-truncated binomial distribution (ZTBD) and hence named the Lagrangian zero-truncated binomial distribution (LZTBD). The moments, probability generating function, factorial moments, as well as skewness and kurtosis measures of the LZTBD are discussed. We also show that the new model’s finite mixture is identifiable. The unknown parameters of the LZTBD are estimated using the maximum likelihood method. A broad simulation study is executed as an evaluation of the well-established performance of the maximum likelihood estimates. The likelihood ratio test is used to assess the effectiveness of the third parameter in the new model. Six COVID-19 datasets are used to demonstrate the LZTBD’s applicability, and we conclude that the LZTBD is very competitive on the fitting objective. Full article

Open AccessArticle

Comparison of Positivity in Two Epidemic Waves of COVID-19 in Colombia with FDA

by

Cristhian Leonardo Urbano-Leon

and

Manuel Escabias

Stats 2022, 5(4), 993-1003; https://doi.org/10.3390/stats5040059 - 28 Oct 2022

Viewed by 39

Abstract

We use the functional data methodology to examine whether there are significant differences between two waves of contagion by COVID-19 in Colombia between 7 July 2020 and 20 July 2021. A pointwise functional t-test is initially used, then an alternative statistical test [...] Read more.

We use the functional data methodology to examine whether there are significant differences between two waves of contagion by COVID-19 in Colombia between 7 July 2020 and 20 July 2021. A pointwise functional t-test is initially used, then an alternative statistical test proposal for paired samples is presented, which has a theoretical distribution and performs well in small samples. Our statistical test generates a scalar p-value, which provides a global idea about the significance of the positivity curves, complementing the existing punctual tests, as an advantage. Full article

(This article belongs to the Section Applied Stochastic Models)

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Open AccessArticle

Snooker Statistics and Zipf’s Law

by

Wim Hordijk

Stats 2022, 5(4), 985-992; https://doi.org/10.3390/stats5040058 - 21 Oct 2022

Viewed by 226

Abstract

Zipf’s law is well known in linguistics: the frequency of a word is inversely proportional to its rank. This is a special case of a more general power law, a common phenomenon in many kinds of real-world statistical data. Here, it is shown [...] Read more.

Zipf’s law is well known in linguistics: the frequency of a word is inversely proportional to its rank. This is a special case of a more general power law, a common phenomenon in many kinds of real-world statistical data. Here, it is shown that snooker statistics also follow such a mathematical pattern, but with varying parameter values. Two types of rankings (prize money earned and centuries scored), and three different time frames (all-time, decade, and year) are considered. The results indicate that the power law parameter values depend on the type of ranking used, as well as the time frame considered. Furthermore, in some cases, the resulting parameter values vary significantly over time, for which a plausible explanation is provided. Finally, it is shown how individual rankings can be described somewhat more accurately using a log-normal distribution, but that the overall conclusions derived from the power law analysis remain valid. Full article

(This article belongs to the Section Data Science)

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Open AccessArticle

Extreme Tail Ratios and Overrepresentation among Subpopulations with Normal Distributions

by

Theodore P. Hill

and

Ronald F. Fox

Stats 2022, 5(4), 977-984; https://doi.org/10.3390/stats5040057 - 20 Oct 2022

Viewed by 283

Abstract

Given several different populations, the relative proportions of each in the high (or low) end of the distribution of a given characteristic are often more important than the overall average values or standard deviations. In the case of two different normally-distributed random variables, [...] Read more.

Given several different populations, the relative proportions of each in the high (or low) end of the distribution of a given characteristic are often more important than the overall average values or standard deviations. In the case of two different normally-distributed random variables, as is shown here, one of the (right) tail ratios will not only eventually be greater than 1 from some point on, but will even become infinitely large. More generally, in every finite mixture of different normal distributions, there will always be exactly one of those distributions that is not only overrepresented in the right tail of the mixture but even completely overwhelms all other subpopulations in the rightmost tails. This property (and the analogous result for the left tails), although not unique to normal distributions, is not shared by other common continuous centrally symmetric unimodal distributions, such as Laplace, nor even by other bell-shaped distributions, such as Cauchy (Lorentz) distributions. Full article

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Open AccessCommunication

Ordinal Cochran-Mantel-Haenszel Testing and Nonparametric Analysis of Variance: Competing Methodologies

by

J. C. W. Rayner

and

G. C. Livingston, Jr.

Stats 2022, 5(4), 970-976; https://doi.org/10.3390/stats5040056 - 17 Oct 2022

Viewed by 176

Abstract

The Cochran-Mantel-Haenszel (CMH) and nonparametric analysis of variance (NP ANOVA) methodologies are both sets of tests for categorical response data. The latter are competitor tests for the ordinal CMH tests in which the response variable is necessarily ordinal; the treatment variable may be [...] Read more.

The Cochran-Mantel-Haenszel (CMH) and nonparametric analysis of variance (NP ANOVA) methodologies are both sets of tests for categorical response data. The latter are competitor tests for the ordinal CMH tests in which the response variable is necessarily ordinal; the treatment variable may be either ordinal or nominal. The CMH mean score test seeks to detect mean treatment differences, while the CMH correlation test assesses ordinary or (1, 1) generalized correlation. Since the corresponding nonparametric ANOVA tests assess arbitrary univariate and bivariate moments, the ordinal CMH tests have been extended to enable a fuller comparison. The CMH tests are conditional tests, assuming that certain marginal totals in the data table are known. They have been extended to have unconditional analogues. The NP ANOVA tests are unconditional. Here, we give a brief overview of both methodologies to address the question “which methodology is preferable?”. Full article

(This article belongs to the Section Statistical Methods)

Open AccessArticle

On the Bivariate Composite Gumbel–Pareto Distribution

by

Alexandra Badea

,

Catalina Bolancé

and

Raluca Vernic

Stats 2022, 5(4), 948-969; https://doi.org/10.3390/stats5040055 - 16 Oct 2022

Viewed by 156

Abstract

In this paper, we propose a bivariate extension of univariate composite (two-spliced) distributions defined by a bivariate Pareto distribution for values larger than some thresholds and by a bivariate Gumbel distribution on the complementary domain. The purpose of this distribution is to capture [...] Read more.

In this paper, we propose a bivariate extension of univariate composite (two-spliced) distributions defined by a bivariate Pareto distribution for values larger than some thresholds and by a bivariate Gumbel distribution on the complementary domain. The purpose of this distribution is to capture the behavior of bivariate data consisting of mainly small and medium values but also of some extreme values. Some properties of the proposed distribution are presented. Further, two estimation procedures are discussed and illustrated on simulated data and on a real data set consisting of a bivariate sample of claims from an auto insurance portfolio. In addition, the risk of loss in this insurance portfolio is estimated by Monte Carlo simulation. Full article

Open AccessArticle

Benford Networks

by

Roeland de Kok

and

Giulia Rotundo

Stats 2022, 5(4), 934-947; https://doi.org/10.3390/stats5040054 - 30 Sep 2022

Viewed by 238

Abstract

The Benford law applied within complex networks is an interesting area of research. This paper proposes a new algorithm for the generation of a Benford network based on priority rank, and further specifies the formal definition. The condition to be taken into account [...] Read more.

The Benford law applied within complex networks is an interesting area of research. This paper proposes a new algorithm for the generation of a Benford network based on priority rank, and further specifies the formal definition. The condition to be taken into account is the probability density of the node degree. In addition to this first algorithm, an iterative algorithm is proposed based on rewiring. Its development requires the introduction of an ad hoc measure for understanding how far an arbitrary network is from a Benford network. The definition is a semi-distance and does not lead to a distance in mathematical terms, instead serving to identify the Benford network as a class. The semi-distance is a function of the network; it is computationally less expensive than the degree of conformity and serves to set a descent condition for the rewiring. The algorithm stops when it meets the condition that either the network is Benford or the maximum number of iterations is reached. The second condition is needed because only a limited set of densities allow for a Benford network. Another important topic is assortativity and the extremes which can be achieved by constraining the network topology; for this reason, we ran simulations on artificial networks and explored further theoretical settings as preliminary work on models of preferential attachment. Based on our extensive analysis, the first proposed algorithm remains the best one from a computational point of view. Full article

(This article belongs to the Special Issue Benford's Law(s) and Applications)

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Stats, Volume 5, Issue 4 (December 2022) – 7 articles

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