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Special Issue "Probability, Statistics and Their Applications 2021"

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A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Probability and Statistics".

Deadline for manuscript submissions: 30 June 2022 | Viewed by 9712

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Special Issue Editor

Prof. Dr. Vasile Preda
E-Mail Website1 Website2

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Guest Editor

1. “Gheorghe Mihoc-Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics, 050711 Bucharest, Romania
2. “Costin C. Kiritescu” National Institute of Economic Research, 050711 Bucharest, Romania
3. Faculty of Mathematics and Computer Science, University of Bucharest, 010014 Bucharest, Romania
Interests: statistics; decision theory; operational research; variational inequalities; equilibrium theory; generalized convexity; information theory; biostatistics; actuarial statistics
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Special Issue Information

Dear Colleagues,

Statistics and probability are important domains in the scientific world, having many applications in various fields, such as engineering, reliability, medicine, biology, economics, physics, and not only, probability laws providing an estimated image of the world we live in. This Special Volume deals targets some certain directions of the two domains as described below.

Some applications of statistics are clustering of random variables based on simulated and real data or scan statistics, the latter being introduced in 1963 by Joseph Naus. In reliability theory, some important statistical tools are hazard rate and survival functions, order statistics, and stochastic orders. In physics, the concept of entropy is at its core, while special statistics were introduced and developed, such as statistical mechanics and Tsallis statistics.

~In economics, statistics, mathematics, and economics formed a particular domain called econometrics. ARMA models, linear regressions, income analysis, and stochastic processes are discussed and analyzed in the context of real economic processes. Other important tools are Lorenz curves and broken stick models.

~Theoretical results such as modeling of discretization of random variables and estimation of parameters of new and old statistical models are welcome, some important probability laws being heavy-tailed distributions. In recent years, many distributions along with their properties have been introduced in order to better fit the growing data available.

The purpose of this Special Issue is to provide a collection of articles that reflect the importance of statistics and probability in applied scientific domains. Papers providing theoretical methodologies and applications in statistics are welcome.

Prof. Dr. Vasile Preda
Guest Editor

Manuscript Submission Information

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Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

Applied and theoretical statistics
New probability distributions and estimation methods
Broken stick models
Lorenz curve
Scan statistics
Discretization of random variables
Clustering of random variables

Published Papers (14 papers)

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Research

Open AccessArticle

Quantile-Zone Based Approach to Normality Testing

Atif Avdović

and

Vesna Jevremović

Mathematics 2022, 10(11), 1828; https://doi.org/10.3390/math10111828 - 26 May 2022

Viewed by 308

Abstract

Normality testing remains an important issue for researchers, despite many solutions that have been published and in use for a long time. There is a need for testing normality in many areas of research and application, among them in Quality control, or more precisely, in the investigation of Shewhart-type control charts. We modified some of our previous results concerning control charts by using the empirical distribution function, proper choice of quantiles and a zone function that quantifies the discrepancy from a normal distribution. That was our approach in constructing a new normality test that we present in this paper. Our results show that our test is more powerful than any other known normality test, even in the case of alternatives with small departures from normality and for small sample sizes. Additionally, many test statistics are sensitive to outliers when testing normality, but that is not the case with our test statistic. We provide a detailed distribution of the test statistic for the presented test and comparable power analysis with highly illustrative graphics. The discussion covers both the cases for known and for estimated parameters. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

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Open AccessArticle

A Generalization of the Bivariate Gamma Distribution Based on Generalized Hypergeometric Functions

Christian Caamaño-Carrillo

and

Javier E. Contreras-Reyes

Mathematics 2022, 10(9), 1502; https://doi.org/10.3390/math10091502 - 01 May 2022

Viewed by 436

Abstract

In this paper, we provide a new bivariate distribution obtained from a Kibble-type bivariate gamma distribution. The stochastic representation was obtained by the sum of a Kibble-type bivariate random vector and a bivariate random vector builded by two independent gamma random variables. In addition, the resulting bivariate density considers an infinite series of products of two confluent hypergeometric functions. In particular, we derive the probability and cumulative distribution functions, the moment generation and characteristic functions, the Hazard, Bonferroni and Lorenz functions, and an approximation for the differential entropy and mutual information index. Numerical examples showed the behavior of exact and approximated expressions. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

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Open AccessArticle

Non-Markovian Inverse Hawkes Processes

Youngsoo Seol

Mathematics 2022, 10(9), 1413; https://doi.org/10.3390/math10091413 - 22 Apr 2022

Viewed by 303

Abstract

Hawkes processes are a class of self-exciting point processes with a clustering effect whose jump rate is determined by its past history. They are generally regarded as continuous-time processes and have been widely applied in a number of fields, such as insurance, finance, queueing, and statistics. The Hawkes model is generally non-Markovian because its future development depends on the timing of past events. However, it can be Markovian under certain circumstances. If the exciting function is an exponential function or a sum of exponential functions, the model can be Markovian with a generator of the model. In contrast to the general Hawkes processes, the inverse Hawkes process has some specific features and self-excitation indicates severity. Inverse Markovian Hawkes processes were introduced by Seol, who studied some asymptotic behaviors. An extended version of inverse Markovian Hawkes processes was also studied by Seol. With this paper, we propose a non-Markovian inverse Hawkes process, which is a more general inverse Hawkes process that features several existing models of self-exciting processes. In particular, we established both the law of large numbers (LLN) and Central limit theorems (CLT) for a newly considered non-Markovian inverse Hawkes process. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

Open AccessFeature PaperArticle

Conformal Control Tools for Statistical Manifolds and for γ-Manifolds

Iulia-Elena Hirica

Cristina-Liliana Pripoae

Gabriel-Teodor Pripoae

and

Vasile Preda

Mathematics 2022, 10(7), 1061; https://doi.org/10.3390/math10071061 - 25 Mar 2022

Viewed by 518

Abstract

The theory of statistical manifolds w.r.t. a conformal structure is reviewed in a creative manner and developed. By analogy, the

γ

-manifolds are introduced. New conformal invariant tools are defined. A necessary condition for the f-conformal equivalence of

γ

-manifolds is found, [...] Read more.

The theory of statistical manifolds w.r.t. a conformal structure is reviewed in a creative manner and developed. By analogy, the

γ

-manifolds are introduced. New conformal invariant tools are defined. A necessary condition for the f-conformal equivalence of

γ

-manifolds is found, extending that for the

α

-conformal equivalence for statistical manifolds. Certain examples of these new defined geometrical objects are given in the theory of Iinformation. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

Open AccessFeature PaperArticle

Categorical Functional Data Analysis. The cfda R Package

Cristian Preda

Quentin Grimonprez

and

Vincent Vandewalle

Mathematics 2021, 9(23), 3074; https://doi.org/10.3390/math9233074 - 29 Nov 2021

Viewed by 628

Abstract

Categorical functional data represented by paths of a stochastic jump process with continuous time and a finite set of states are considered. As an extension of the multiple correspondence analysis to an infinite set of variables, optimal encodings of states over time are approximated using an arbitrary finite basis of functions. This allows dimension reduction, optimal representation, and visualisation of data in lower dimensional spaces. The methodology is implemented in the cfda R package and is illustrated using a real data set in the clustering framework. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

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Open AccessArticle

A New Extended Cosine—G Distributions for Lifetime Studies

and

Mathematics 2021, 9(21), 2758; https://doi.org/10.3390/math9212758 - 30 Oct 2021

Cited by 1 | Viewed by 462

Abstract

In this article, we introduce a new extended cosine family of distributions. Some important mathematical and statistical properties are studied, including asymptotic results, a quantile function, series representation of the cumulative distribution and probability density functions, moments, moments of residual life, reliability parameter, and order statistics. Three special members of the family are proposed and discussed, namely, the extended cosine Weibull, extended cosine power, and extended cosine generalized half-logistic distributions. Maximum likelihood, least-square, percentile, and Bayes methods are considered for parameter estimation. Simulation studies are used to assess these methods and show their satisfactory performance. The stress–strength reliability underlying the extended cosine Weibull distribution is discussed. In particular, the stress–strength reliability parameter is estimated via a Bayes method using gamma prior under the square error loss, absolute error loss, maximum a posteriori, general entropy loss, and linear exponential loss functions. In the end, three real applications of the findings are provided for illustration; one of them concerns stress–strength data analyzed by the extended cosine Weibull distribution. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

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Open AccessArticle

Causality Distance Measures for Multivariate Time Series with Applications

and

Mathematics 2021, 9(21), 2708; https://doi.org/10.3390/math9212708 - 25 Oct 2021

Viewed by 408

Abstract

In this work, we focus on the development of new distance measure algorithms, namely, the Causality Within Groups (CAWG), the Generalized Causality Within Groups (GCAWG) and the Causality Between Groups (CABG), all of which are based on the well-known Granger causality. The proposed distances together with the associated algorithms are suitable for multivariate statistical data analysis including unsupervised classification (clustering) purposes for the analysis of multivariate time series data with emphasis on financial and economic data where causal relationships are frequently present. For exploring the appropriateness of the proposed methodology, we implement, for illustrative purposes, the proposed algorithms to hierarchical clustering for the classification of 19 EU countries based on seven variables related to health resources in healthcare systems. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

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Open AccessArticle

Mastering the Body and Tail Shape of a Distribution

Matthias Wagener

Andriette Bekker

and

Mohammad Arashi

Mathematics 2021, 9(21), 2648; https://doi.org/10.3390/math9212648 - 20 Oct 2021

Viewed by 447

Abstract

The normal distribution and its perturbation have left an immense mark on the statistical literature. Several generalized forms exist to model different skewness, kurtosis, and body shapes. Although they provide better fitting capabilities, these generalizations do not have parameters and formulae with a clear meaning to the practitioner on how the distribution is being modeled. We propose a neat integration approach generalization which intuitively gives direct control of the body and tail shape, the body-tail generalized normal (BTGN). The BTGN provides the basis for a flexible distribution, emphasizing parameter interpretation, estimation properties, and tractability. Basic statistical measures are derived, such as the density function, cumulative density function, moments, moment generating function. Regarding estimation, the equations for maximum likelihood estimation and maximum product spacing estimation are provided. Finally, real-life situations data, such as log-returns, time series, and finite mixture modeling, are modeled using the BTGN. Our results show that it is possible to have more desirable traits in a flexible distribution while still providing a superior fit to industry-standard distributions, such as the generalized hyperbolic, generalized normal, tail-inflated normal, and t distributions. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

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Open AccessArticle

Conditions for the Existence of Absolutely Optimal Portfolios

Marius Rădulescu

Constanta Zoie Rădulescu

and

Gheorghiță Zbăganu

Mathematics 2021, 9(17), 2032; https://doi.org/10.3390/math9172032 - 24 Aug 2021

Viewed by 385

Abstract

Let Δ_n be the n-dimensional simplex, ξ = (ξ₁, ξ₂,…, ξ_n) be an n-dimensional random vector, and

U

be a set of utility functions. A vector x^*

\in

Δ_n is a

U

[...] Read more.

Let Δ_n be the n-dimensional simplex, ξ = (ξ₁, ξ₂,…, ξ_n) be an n-dimensional random vector, and

U

be a set of utility functions. A vector x^*

\in

Δ_n is a

U

-absolutely optimal portfolio if

E (u (ξ^{T} x^{*})) \geq E (u (ξ^{T} x))

for every x

\in

Δ_n and u ∈

U

. In this paper, we investigate the following problem: For what random vectors, ξ, do

U

-absolutely optimal portfolios exist? If

U_{2}

is the set of concave utility functions, we find necessary and sufficient conditions on the distribution of the random vector, ξ, in order that it admits a

U_{2}

-absolutely optimal portfolio. The main result is the following: If x⁰ is a portfolio having all its entries positive, then x⁰ is an absolutely optimal portfolio if and only if all the conditional expectations of

ξ_{i}

, given the return of portfolio x⁰, are the same. We prove that if ξ is bounded below then CARA-absolutely optimal portfolios are also

U_{2}

-absolutely optimal portfolios. The classical case when the random vector ξ is normal is analyzed. We make a complete investigation of the simplest case of a bi-dimensional random vector ξ = (ξ₁, ξ₂). We give a complete characterization and we build two dimensional distributions that are absolutely continuous and admit

U_{2}

-absolutely optimal portfolios. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

Open AccessArticle

A New Flexible Family of Continuous Distributions: The Additive Odd-G Family

and

Mathematics 2021, 9(16), 1837; https://doi.org/10.3390/math9161837 - 04 Aug 2021

Cited by 3 | Viewed by 443

Abstract

This paper introduces a new family of distributions based on the additive model structure. Three submodels of the proposed family are studied in detail. Two simulation studies were performed to discuss the maximum likelihood estimators of the model parameters. The log location-scale regression model based on a new generalization of the Weibull distribution is introduced. Three datasets were used to show the importance of the proposed family. Based on the empirical results, we concluded that the proposed family is quite competitive compared to other models. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

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Open AccessArticle

Reliability Properties of the NDL Family of Discrete Distributions with Its Inference

Mohammed Mohammed Ahmed Almazah

Badr Alnssyan

Abdul Hadi N. Ahmed

and

Ahmed Z. Afify

Mathematics 2021, 9(10), 1139; https://doi.org/10.3390/math9101139 - 18 May 2021

Cited by 2 | Viewed by 583

Abstract

The natural discrete Lindley (NDL) distribution is an intuitive idea that uses discrete analogs to well-known continuous distributions rather than using any of the published discretization techniques. The NDL is a flexible extension of both the geometric and the negative binomial distributions. In the present article, we further investigate new results of value in the areas of both theoretical and applied reliability. To be specific, several closure properties of the NDL are proved. Among the results, sufficient conditions that maintain the preservation properties under useful partial orderings, convolution, and random sum of random variables are introduced. Eight different methods of estimation, including the maximum likelihood, least squares, weighted least squares, Cramér–von Mises, the maximum product of spacing, Anderson–Darling, right-tail Anderson–Darling, and percentiles, have been used to estimate the parameter of interest. The performance of these estimators has been evaluated through extensive simulation. We have also demonstrated two applications of NDL in modeling real-life problems, including count data. It is worth noting that almost all the methods have resulted in very satisfactory estimates on both simulated and real-world data. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

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Open AccessEditor’s ChoiceArticle

An Exhaustive Power Comparison of Normality Tests

Jurgita Arnastauskaitė

Tomas Ruzgas

and

Mindaugas Bražėnas

Mathematics 2021, 9(7), 788; https://doi.org/10.3390/math9070788 - 06 Apr 2021

Cited by 7 | Viewed by 1182

Abstract

A goodness-of-fit test is a frequently used modern statistics tool. However, it is still unclear what the most reliable approach is to check assumptions about data set normality. A particular data set (especially with a small number of observations) only partly describes the process, which leaves many options for the interpretation of its true distribution. As a consequence, many goodness-of-fit statistical tests have been developed, the power of which depends on particular circumstances (i.e., sample size, outlets, etc.). With the aim of developing a more universal goodness-of-fit test, we propose an approach based on an N-metric with our chosen kernel function. To compare the power of 40 normality tests, the goodness-of-fit hypothesis was tested for 15 data distributions with 6 different sample sizes. Based on exhaustive comparative research results, we recommend the use of our test for samples of size

n \geq 118

. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

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Open AccessArticle

On the Omega Distribution: Some Properties and Estimation

Abd EL-Baset A. Ahmad

Gauss M. Cordeiro

and

Ahmed Z. Afify

Mathematics 2021, 9(6), 656; https://doi.org/10.3390/math9060656 - 19 Mar 2021

Cited by 2 | Viewed by 1142

Abstract

We obtain explicit expressions for single and product moments of the order statistics of an omega distribution. We also discuss seven methods to estimate the omega parameters. Various simulation results are performed to compare the performance of the proposed estimators. Furthermore, the maximum likelihood method is adopted to estimate the omega parameters under the type II censoring scheme. The usefulness of the omega distribution is proven using a real data set. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

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Open AccessArticle

Robust Estimation and Tests for Parameters of Some Nonlinear Regression Models

and

Mathematics 2021, 9(6), 599; https://doi.org/10.3390/math9060599 - 11 Mar 2021

Viewed by 602

Abstract

This paper uses the median-of-means (MOM) method to estimate the parameters of the nonlinear regression models and proves the consistency and asymptotic normality of the MOM estimator. Especially when there are outliers, the MOM estimator is more robust than nonlinear least squares (NLS) estimator and empirical likelihood (EL) estimator. On this basis, we propose hypothesis testing Statistics for the parameters of the nonlinear regression models using empirical likelihood method, and the simulation performance shows the superiority of MOM estimator. We apply the MOM method to analyze the top 50 data of GDP of China in 2019. The result shows that MOM method is more feasible than NLS estimator and EL estimator. Full article

(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)

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Planned Papers

The below list represents only planned manuscripts. Some of these manuscripts have not been received by the Editorial Office yet. Papers submitted to MDPI journals are subject to peer-review.

Title: On Time Series Clustering Metrics
Authors: Achilleas ANASTASIOU 1; Petros HATZOPOULOS 1; Alexandros KARAGRIGORIOU 1*; George MAVRIDOGLOU 2
Affiliation: 1University of the Aegean, Greece, [email protected]; 2University of Peloponnese, Greece *Correspondence: [email protected]
Abstract: In this work our purpose is to present and discuss various Multivariate Time Series Clustering Techniques for classification purposes and see how similarity measures affect the statistical procedures. One of the main contributions of this work is the development of two new distance measure algorithms, called Causality Within Groups (CAWG) and Causality Between Groups (CABG) both of which are based on the well-known Granger Causality. The proposed distance algorithms are suitable for classification purposes for the analysis of multivariate time series data with emphasis on financial and economic data where causal relationships are frequently present.

Title: Continuous-Time Step semi-Markov Models and Applications
Authors: Vlad Stefan BARBU1*, Guglielmo D'AMICO 2 and Andreas MAKRIDES 3
Affiliation: 1Laboratory of Mathematics Raphael Salem, University of Rouen Normandy, France; [email protected] 2Department of Pharmacy, University G. d'Annunzio of Chieti-Pescara, Italy; [email protected] 3University of Uclan, Cyprus & University of the Aegean, Greece; [email protected] *Correspondence: [email protected]
Abstract: In this paper we introduce a class of stochastic processes in continuous time, called step semi-Markov processes. The main idea comes from bringing and additional insight to a classical semi-Markov process: the transition between two states is done through two or several steps. This is an extension of a previous work on discrete-time strep semi-Markov processes. After defining the models and the main characteristics of interest, we derive the recursive evolution equations for two-step semi-Markov processes.

Title: A non-extensive joint representation for minimal entropy martingale measures with applications to semi-markov regime switching interest rate modeling
Authors: Silvia DEDU1*, Mihaita DRAGAN2, Muhammad SHERAZ3
Affiliation: 1Department of Applied Mathematics, Bucharest University of Economic Studies, Romania, [email protected] 2Faculty of Mathematics and Computer Science, University of Bucharest, Romania, [email protected] 3Department of Mathematical Sciences, Institute of Business Administration Karachi, Pakistan, [email protected] *Correspondence: [email protected]
Abstract: In this paper a joint representation of extensive and non-extensive entropy measures is developed. The underlying theoretical properties are investigated and applied to solve the minimal entropy martingale measure problem for deriving risk-neutral densities and interest rate modelling. The Lambert function and a new type of approach are used to obtain results without depending on stochastic calculus techniques.

Title: Comparison of Poisson probabilistic models using Chi square statistics as criterion. Application in exact scores of football matches
Authors: Miltiadis CHALIKIAS1*, Dimitrios KALIVOKAS2, Panagiota LALOU1
Affiliation: 1Department of Accounting and Finance, University of West Attica, Greece, [email protected]; [email protected] 2Department of Business Administration, University of Peloponnese, Greece, [email protected] *Correspondence: [email protected]
Abstract: The purpose of this paper is to suggest and analyze a new method to predict exact scores in football matches and to exhibit its results. It is well known that the Poisson distribution is used in this field. Wanting to give a better approach, we thought to use not only the goals, but the final efforts’ average too, so as to create parameters more beneficial to us. For the comparison of the Poisson probabilistic models Chi square goodness of it statistics was used. Greek Football league of 2014-2015 was used as an application. The model and its results could become useful in the prediction of exact scores.

Title: On the existence of absolute portfolios
Authors: Marius RĂDULESCU1*, Constanta Zoie RĂDULESCU2, Gheorghiţă ZBĂGANU1
Affiliation: 1“Gheorghe Mihoc – Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, [email protected]; [email protected] Casa Academiei Române, Calea 13 Septembrie no. 13, 050711 Bucharest, Romania. 2National Institute for Research and Development in Informatics, 8-10 Averescu Avenue, 011455, Bucharest 1, Romania, [email protected] *Correspondence: [email protected]
Abstract: Let Δn be the n-dimensional simplex, ξ=( ξ1, ξ2,…, ξn) be an n-dimensional random vector and U be a set of utility functions. A vector x*∈ Δn is a U -absolute portfolio if E(u(ξ^T x^* ))≥E(u(ξ^T x)) for every x∈Δn and u∈U. In this paper we investigate the following problem: For what random vectors ξ, U-absolute portfolios do exist? If U2 is the set of concave utility functions we find necessary and sufficient conditions on the distribution of the random vector ξ in order that it admits a U2-absolute portfolio. We prove that if ξ is bounded below then CARA-absolute portfolios are U2-absolute portfolios too. The classical case when the random vector ξ is normal is analyzed. We make a complete investigation of the simplest case of a bi-dimensional random vector ξ = (ξ1, ξ2). We give a complete characterization and we build two dimensional distributions that are absolutely continuous and admit U2-absolute portfolios.

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