Probability, Statistics and Their Applications 2021

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

Deadline for manuscript submissions: closed (31 January 2023) | Viewed by 76558

Special Issue Editor


E-Mail Website1 Website2
Guest Editor
1. "Gheorghe Mihoc-Caius Iacob" Institute of Mathematical Statistics and Applied Mathematics, Calea 13 Septembrie 13, 050711 Bucharest, Romania
2. "Costin C. Kiriţescu" National Institute of Economic Research, Calea 13 Septembrie 13, 050711 Bucharest, Romania 3. Faculty of Mathematics and Computer Science, University of Bucharest, Str. Academiei 14, 010014 Bucharest, Romania
Interests: statistics; decision theory; operational research; variational inequalities; equilibrium theory; generalized convexity; information theory; biostatistics; actuarial statistics; functional analysis; approximation theory
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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

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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

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Published Papers (26 papers)

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Research

29 pages, 4100 KiB  
Article
Analytically Computing the Moments of a Conic Combination of Independent Noncentral Chi-Square Random Variables and Its Application for the Extended Cox–Ingersoll–Ross Process with Time-Varying Dimension
by Sanae Rujivan, Athinan Sutchada, Kittisak Chumpong and Napat Rujeerapaiboon
Mathematics 2023, 11(5), 1276; https://doi.org/10.3390/math11051276 - 6 Mar 2023
Cited by 5 | Viewed by 2113
Abstract
This paper focuses mainly on the problem of computing the γth, γ>0, moment of a random variable Yn:=i=1nαiXi in which the αi’s are positive [...] Read more.
This paper focuses mainly on the problem of computing the γth, γ>0, moment of a random variable Yn:=i=1nαiXi in which the αi’s are positive real numbers and the Xi’s are independent and distributed according to noncentral chi-square distributions. Finding an analytical approach for solving such a problem has remained a challenge due to the lack of understanding of the probability distribution of Yn, especially when not all αi’s are equal. We analytically solve this problem by showing that the γth moment of Yn can be expressed in terms of generalized hypergeometric functions. Additionally, we extend our result to computing the γth moment of Yn when Xi is a combination of statistically independent Zi2 and Gi in which the Zi’s are distributed according to normal or Maxwell–Boltzmann distributions and the Gi’s are distributed according to gamma, Erlang, or exponential distributions. Our paper has an immediate application in interest rate modeling, where we can explicitly provide the exact transition probability density function of the extended Cox–Ingersoll–Ross (ECIR) process with time-varying dimension as well as the corresponding γth conditional moment. Finally, we conduct Monte Carlo simulations to demonstrate the accuracy and efficiency of our explicit formulas through several numerical tests. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
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16 pages, 8765 KiB  
Article
Comparison of Statistical Production Models for a Solar and a Wind Power Plant
by Irina Meghea
Mathematics 2023, 11(5), 1115; https://doi.org/10.3390/math11051115 - 23 Feb 2023
Viewed by 1751
Abstract
Mathematical models to characterize and forecast the power production of photovoltaic and eolian plants are justified by the benefits of these sustainable energies, the increased usage in recent years, and the necessity to be integrated into the general energy system. In this paper, [...] Read more.
Mathematical models to characterize and forecast the power production of photovoltaic and eolian plants are justified by the benefits of these sustainable energies, the increased usage in recent years, and the necessity to be integrated into the general energy system. In this paper, starting from two collections of data representing the power production hourly measured at a solar plant and a wind farm, adequate time series methods have been used to draw appropriate statistical models for their productions. The data are smoothed in both cases using moving average and continuous time series have been obtained leading to some models in good agreement with experimental data. For the solar power plant, the developed models can predict the specific power of the next day, next week, and next month, with the most accurate being the monthly model, while for wind power only a monthly model could be validated. Using the CUSUM (cumulative sum control chart) method, the analyzed data formed stationary time series with seasonality. The similar methods used for both sets of data (from the solar plant and wind farm) were analyzed and compared. When compare with other studies which propose production models starting from different measurements involving meteorological data and/or machinery characteristics, an innovative element of this paper consists in the data set on which it is based, this being the production itself. The novelty and the importance of this research reside in the simplicity and the possibility to be reproduced for other related conditions even though every new set of data (provided from other power plants) requires further investigation. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
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24 pages, 5283 KiB  
Article
Statistical Features and Estimation Methods for Half-Logistic Unit-Gompertz Type-I Model
by Anum Shafiq, Tabassum Naz Sindhu, Sanku Dey, Showkat Ahmad Lone and Tahani A. Abushal
Mathematics 2023, 11(4), 1007; https://doi.org/10.3390/math11041007 - 16 Feb 2023
Cited by 9 | Viewed by 1726
Abstract
In this study, we propose a new three-parameter lifetime model based on the type-I half-logistic G family and the unit-Gompertz model, which we named the half-logistic unit Gompertz type-I distribution. The key feature of such a novel model is that it adds a [...] Read more.
In this study, we propose a new three-parameter lifetime model based on the type-I half-logistic G family and the unit-Gompertz model, which we named the half-logistic unit Gompertz type-I distribution. The key feature of such a novel model is that it adds a new tuning parameter to the unit-Gompertz model using the type-I half-logistic family in order to make the unit-Gompertz model more flexible. Diagrams and numerical results are used to look at the new model’s mathematical and statistical aspects. The efficiency of estimating the distribution parameters is measured using a variety of well-known classical methodologies, including Anderson–Darling, maximum likelihood, least squares, weighted least squares, right tail Anderson–Darling, and Cramer–von Mises estimation. Finally, using the maximum likelihood estimation method, the flexibility and ability of the proposed model are illustrated by means of re-analyzing two real datasets, and comparisons are provided with the fit accomplished by the unit-Gompertz, Kumaraswamy, unit-Weibull, and Kumaraswamy beta distributions for illustrative purposes. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
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28 pages, 734 KiB  
Article
Polynomial Distributions and Transformations
by Yue Yu and Pavel Loskot
Mathematics 2023, 11(4), 985; https://doi.org/10.3390/math11040985 - 15 Feb 2023
Cited by 2 | Viewed by 5806
Abstract
Polynomials are common algebraic structures, which are often used to approximate functions, such as probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of systems rather than to assume polynomials for only approximating known or empirically [...] Read more.
Polynomials are common algebraic structures, which are often used to approximate functions, such as probability distributions. This paper proposes to directly define polynomial distributions in order to describe stochastic properties of systems rather than to assume polynomials for only approximating known or empirically estimated distributions. Polynomial distributions offer great modeling flexibility and mathematical tractability. However, unlike canonical distributions, polynomial functions may have non-negative values in the intervals of support for some parameter values; their parameter numbers are usually much larger than for canonical distributions, and the interval of support must be finite. Hence, polynomial distributions are defined here assuming three forms of a polynomial function. Transformations and approximations of distributions and histograms by polynomial distributions are also considered. The key properties of the polynomial distributions are derived in closed form. A piecewise polynomial distribution construction is devised to ensure that it is non-negative over the support interval. A goodness-of-fit measure is proposed to determine the best order of the approximating polynomial. Numerical examples include the estimation of parameters of the polynomial distributions and generating polynomially distributed samples. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
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18 pages, 6664 KiB  
Article
An Analysis of the New Reliability Model Based on Bathtub-Shaped Failure Rate Distribution with Application to Failure Data
by Tabassum Naz Sindhu, Sadia Anwar, Marwa K. H. Hassan, Showkat Ahmad Lone, Tahani A. Abushal and Anum Shafiq
Mathematics 2023, 11(4), 842; https://doi.org/10.3390/math11040842 - 7 Feb 2023
Cited by 14 | Viewed by 1973
Abstract
The reliability of software has a tremendous influence on the reliability of systems. Software dependability models are frequently utilized to statistically analyze the reliability of software. Numerous reliability models are based on the nonhomogeneous Poisson method (NHPP). In this respect, in the current [...] Read more.
The reliability of software has a tremendous influence on the reliability of systems. Software dependability models are frequently utilized to statistically analyze the reliability of software. Numerous reliability models are based on the nonhomogeneous Poisson method (NHPP). In this respect, in the current study, a novel NHPP model established on the basis of the new power function distribution is suggested. The mathematical formulas for its reliability measurements were found and are visually illustrated. The parameters of the suggested model are assessed utilizing the weighted nonlinear least-squares, maximum-likelihood, and nonlinear least-squares estimation techniques. The model is subsequently verified using a variety of reliability datasets. Four separate criteria were used to assess and compare the estimating techniques. Additionally, the effectiveness of the novel model is assessed and evaluated with two foundation models both objectively and subjectively. The implementation results reveal that our novel model performed well in the failure data that we examined. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
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22 pages, 874 KiB  
Article
Change Point Analysis for Kumaraswamy Distribution
by Weizhong Tian, Liyuan Pang, Chengliang Tian and Wei Ning
Mathematics 2023, 11(3), 553; https://doi.org/10.3390/math11030553 - 20 Jan 2023
Cited by 4 | Viewed by 2068
Abstract
The Kumaraswamy distribution is a common type of bounded distribution, which is widely used in agriculture, hydrology, and other fields. In this paper, we use the methods of the likelihood ratio test, modified information criterion, and Schwarz information criterion to analyze the change [...] Read more.
The Kumaraswamy distribution is a common type of bounded distribution, which is widely used in agriculture, hydrology, and other fields. In this paper, we use the methods of the likelihood ratio test, modified information criterion, and Schwarz information criterion to analyze the change point of the Kumaraswamy distribution. Simulation experiments give the performance of the three methods. The application section illustrates the feasibility of the proposed method by applying it to a real dataset. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
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14 pages, 295 KiB  
Article
A New Stochastic Order of Multivariate Distributions: Application in the Study of Reliability of Bridges Affected by Earthquakes
by Luigi-Ionut Catana and Vasile Preda
Mathematics 2023, 11(1), 102; https://doi.org/10.3390/math11010102 - 26 Dec 2022
Viewed by 1414
Abstract
In this article, we introduce and study a new stochastic order of multivariate distributions, namely, the conditional likelihood ratio order. The proposed order and other stochastic orders are analyzed in the case of a bivariate exponential distributions family. The theoretical results obtained are [...] Read more.
In this article, we introduce and study a new stochastic order of multivariate distributions, namely, the conditional likelihood ratio order. The proposed order and other stochastic orders are analyzed in the case of a bivariate exponential distributions family. The theoretical results obtained are applied for studying the reliability of bridges affected by earthquakes. The conditional likelihood ratio order involves the multivariate stochastic ordering; it resembles the likelihood ratio order in the univariate case but is much easier to verify than the likelihood ratio order in the multivariate case. Additionally, the likelihood ratio order in the multivariate case implies this ordering. However, the conditional likelihood ratio order does not imply the weak hard rate order, and it is not an order relation on the multivariate distributions set. The new conditional likelihood ratio order, together with the likelihood ratio order and the weak hazard rate order, were studied in the case of the bivariate Marshall–Olkin exponential distributions family, which has a lack of memory type property. At the end of the paper, we also presented an application of the analyzed orderings for this bivariate distributions family to the study of the effects of earthquakes on bridges. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
29 pages, 937 KiB  
Article
On the Élö–Runyan–Poisson–Pearson Method to Forecast Football Matches
by José Daniel López-Barrientos, Damián Alejandro Zayat-Niño, Eric Xavier Hernández-Prado and Yolanda Estudillo-Bravo
Mathematics 2022, 10(23), 4587; https://doi.org/10.3390/math10234587 - 3 Dec 2022
Viewed by 2857
Abstract
This is a work about football. In it, we depart from two well-known approaches to forecast the outcome of a football match (or even a full tournament) and take advantage of their strengths to develop a new method of prediction. We illustrate the [...] Read more.
This is a work about football. In it, we depart from two well-known approaches to forecast the outcome of a football match (or even a full tournament) and take advantage of their strengths to develop a new method of prediction. We illustrate the Élö–Runyan rating system and the Poisson technique in the English Premier League and we analyze their accuracies with respect to the actual results. We obtained an accuracy of 84.37% for the former, and 79.99% for the latter in this first exercise. Then, we present a criticism of these methods and use it to complement the aforementioned procedures, and hence, introduce the so-called Élö–Runyan–Poisson–Pearson method, which consists of adopting the distribution that best fits the historical distribution of goals to simulate the score of each match. Finally, we obtain a Monte Carlo-based forecast of the result. We test our mechanism to backcast the World Cup of Russia 2018, obtaining an accuracy of 87.09%; and forecast the results of the World Cup of Qatar 2022. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
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18 pages, 466 KiB  
Article
The Leader Property in Quasi Unidimensional Cases
by Anișoara Maria Răducan and Gheorghiță Zbăganu
Mathematics 2022, 10(22), 4199; https://doi.org/10.3390/math10224199 - 9 Nov 2022
Viewed by 1128
Abstract
The following problem was studied: let Zjj1 be a sequence of i.i.d. d-dimensional random vectors. Let F be their probability distribution and for every n1 consider the sample [...] Read more.
The following problem was studied: let Zjj1 be a sequence of i.i.d. d-dimensional random vectors. Let F be their probability distribution and for every n1 consider the sample Sn={Z1,Z2,,Zn}. Then Zj was called a “leader” in the sample Sn if ZjZk,k{1,2,,n} and Zj was an “anti-leader” if ZjZk,k{1,2,,n}. The comparison of two vectors was the usual one: if Zj=Zj1,Zj2,,Zjd,j1, then ZjZk means ZjiZki, while ZjZk means ZjiZki,1id,j,k1. Let an be the probability that Sn has a leader, bn be the probability that Sn has an anti-leader and cn be the probability that Sn has both a leader and an anti-leader. Sometimes these probabilities can be computed or estimated, for instance in the case when F is discrete or absolutely continuous. The limits a=liminfan,b=liminfbn,c=liminfcn were considered. If a>0 it was said that F has the leader property, if b>0 they said that F has the anti-leader property and if c>0 then F has the order property. In this paper we study an in-between case: here the vector Z has the form Z=fX where f=f1,,fd:0,1Rd and X is a random variable. The aim is to find conditions for f in order that a>0,b>0 or c>0. The most examples will focus on a more particular case Z=X,f2X,,fdX with X uniformly distributed on the interval [0,1]. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
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20 pages, 2248 KiB  
Article
Modeling for the Relationship between Monetary Policy and GDP in the USA Using Statistical Methods
by Andre Amaral, Taysir E. Dyhoum, Hussein A. Abdou and Hassan M. Aljohani
Mathematics 2022, 10(21), 4137; https://doi.org/10.3390/math10214137 - 5 Nov 2022
Cited by 3 | Viewed by 7876
Abstract
The Federal Reserve has played an arguably important role in financial crises in the United States since its creation in 1913 through monetary policy tools. Thus, this paper aims to analyze the impact of monetary policy on the United States’ economic growth in [...] Read more.
The Federal Reserve has played an arguably important role in financial crises in the United States since its creation in 1913 through monetary policy tools. Thus, this paper aims to analyze the impact of monetary policy on the United States’ economic growth in the short and long run, measured by Gross Domestic Product (GDP). The Vector Autoregressive (VAR) method explores the relationship among the variables, and the Granger causality test assesses the predictability of the variables. Moreover, the Impulse Response Function (IRF) examines the behavior of one variable after a change in another, utilizing the time-series dataset from the first quarter of 1959 to the second quarter of 2022. This work demonstrates that expansionary monetary policy does have a positive impact on economic growth in the short term though it does not last long. However, in the long term, inflation, measured by the Consumer Price Index (CPI), is affected by expansionary monetary policy. Therefore, if the Federal Reserve wants to cease the expansionary monetary policy in the short run, this should be done appropriately, with the fiscal surplus, to preserve its credibility and trust in the US dollar as a global store of value asset. Also, the paper’s findings suggest that continuous expansion of the Money Supply will lead to a long-term inflationary problem. The purpose of this research is to bring the spotlight to the side effects of expansionary monetary policy on the US economy, but also allow other researchers to test this model in different economies with different dynamics. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
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22 pages, 445 KiB  
Article
Lie Symmetries of the Nonlinear Fokker-Planck Equation Based on Weighted Kaniadakis Entropy
by Iulia-Elena Hirica, Cristina-Liliana Pripoae, Gabriel-Teodor Pripoae and Vasile Preda
Mathematics 2022, 10(15), 2776; https://doi.org/10.3390/math10152776 - 4 Aug 2022
Cited by 7 | Viewed by 1812
Abstract
The paper studies the Lie symmetries of the nonlinear Fokker-Planck equation in one dimension, which are associated to the weighted Kaniadakis entropy. In particular, the Lie symmetries of the nonlinear diffusive equation, associated to the weighted Kaniadakis entropy, are found. The MaxEnt problem [...] Read more.
The paper studies the Lie symmetries of the nonlinear Fokker-Planck equation in one dimension, which are associated to the weighted Kaniadakis entropy. In particular, the Lie symmetries of the nonlinear diffusive equation, associated to the weighted Kaniadakis entropy, are found. The MaxEnt problem associated to the weighted Kaniadakis entropy is given a complete solution, together with the thermodynamic relations which extend the known ones from the non-weighted case. Several different, but related, arguments point out a subtle dichotomous behavior of the Kaniadakis constant k, distinguishing between the cases k(1,1) and k=±1. By comparison, the Lie symmetries of the NFPEs based on Tsallis q-entropies point out six “exceptional” cases, for: q=12, q=32, q=43, q=73, q=2 and q=3. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
12 pages, 294 KiB  
Article
A Continuous-Time Semi-Markov System Governed by Stepwise Transitions
by Vlad Stefan Barbu, Guglielmo D’Amico and Andreas Makrides
Mathematics 2022, 10(15), 2745; https://doi.org/10.3390/math10152745 - 3 Aug 2022
Cited by 2 | Viewed by 1891
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 an additional insight to a classical semi-Markov process: the transition between two states is accomplished through two or several steps. [...] Read more.
In this paper, we introduce a class of stochastic processes in continuous time, called step semi-Markov processes. The main idea comes from bringing an additional insight to a classical semi-Markov process: the transition between two states is accomplished through two or several steps. This is an extension of a previous work on discrete-time step 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. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
16 pages, 2389 KiB  
Article
Quantile-Zone Based Approach to Normality Testing
by Atif Avdović and Vesna Jevremović
Mathematics 2022, 10(11), 1828; https://doi.org/10.3390/math10111828 - 26 May 2022
Cited by 5 | Viewed by 2042
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 [...] Read more.
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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17 pages, 7945 KiB  
Article
A Generalization of the Bivariate Gamma Distribution Based on Generalized Hypergeometric Functions
by Christian Caamaño-Carrillo and Javier E. Contreras-Reyes
Mathematics 2022, 10(9), 1502; https://doi.org/10.3390/math10091502 - 1 May 2022
Cited by 3 | Viewed by 2307
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 [...] Read more.
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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12 pages, 266 KiB  
Article
Non-Markovian Inverse Hawkes Processes
by Youngsoo Seol
Mathematics 2022, 10(9), 1413; https://doi.org/10.3390/math10091413 - 22 Apr 2022
Cited by 2 | Viewed by 1576
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, [...] Read more.
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)
15 pages, 306 KiB  
Article
Conformal Control Tools for Statistical Manifolds and for γ-Manifolds
by 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
Cited by 2 | Viewed by 1995
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)
31 pages, 833 KiB  
Article
Categorical Functional Data Analysis. The cfda R Package
by Cristian Preda, Quentin Grimonprez and Vincent Vandewalle
Mathematics 2021, 9(23), 3074; https://doi.org/10.3390/math9233074 - 29 Nov 2021
Cited by 2 | Viewed by 3539
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 [...] Read more.
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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29 pages, 1202 KiB  
Article
A New Extended Cosine—G Distributions for Lifetime Studies
by Mustapha Muhammad, Rashad A. R. Bantan, Lixia Liu, Christophe Chesneau, Muhammad H. Tahir, Farrukh Jamal and Mohammed Elgarhy
Mathematics 2021, 9(21), 2758; https://doi.org/10.3390/math9212758 - 30 Oct 2021
Cited by 40 | Viewed by 2519
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, [...] Read more.
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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15 pages, 395 KiB  
Article
Causality Distance Measures for Multivariate Time Series with Applications
by Achilleas Anastasiou, Peter Hatzopoulos, Alex Karagrigoriou and George Mavridoglou
Mathematics 2021, 9(21), 2708; https://doi.org/10.3390/math9212708 - 25 Oct 2021
Cited by 1 | Viewed by 1768
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 [...] Read more.
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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22 pages, 3297 KiB  
Article
Mastering the Body and Tail Shape of a Distribution
by Matthias Wagener, Andriette Bekker and Mohammad Arashi
Mathematics 2021, 9(21), 2648; https://doi.org/10.3390/math9212648 - 20 Oct 2021
Cited by 1 | Viewed by 2866
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 [...] Read more.
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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16 pages, 384 KiB  
Article
Conditions for the Existence of Absolutely Optimal Portfolios
by Marius Rădulescu, Constanta Zoie Rădulescu and Gheorghiță Zbăganu
Mathematics 2021, 9(17), 2032; https://doi.org/10.3390/math9172032 - 24 Aug 2021
Cited by 2 | Viewed by 1301
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 [...] Read more.
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 -absolutely optimal portfolio if EuξTx*EuξTx for every x Δn and uU. In this paper, we investigate the following problem: For what random vectors, ξ, do U-absolutely optimal portfolios 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-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 ξi, given the return of portfolio x0, are the same. We prove that if ξ is bounded below then CARA-absolutely optimal portfolios are also U2-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 U2-absolutely optimal portfolios. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
17 pages, 572 KiB  
Article
A New Flexible Family of Continuous Distributions: The Additive Odd-G Family
by Emrah Altun, Mustafa Ç. Korkmaz, Mahmoud El-Morshedy and Mohamed S. Eliwa
Mathematics 2021, 9(16), 1837; https://doi.org/10.3390/math9161837 - 4 Aug 2021
Cited by 15 | Viewed by 2232
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 [...] Read more.
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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17 pages, 1131 KiB  
Article
Reliability Properties of the NDL Family of Discrete Distributions with Its Inference
by 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 5 | Viewed by 2084
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 [...] Read more.
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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20 pages, 2097 KiB  
Article
An Exhaustive Power Comparison of Normality Tests
by Jurgita Arnastauskaitė, Tomas Ruzgas and Mindaugas Bražėnas
Mathematics 2021, 9(7), 788; https://doi.org/10.3390/math9070788 - 6 Apr 2021
Cited by 29 | Viewed by 6960
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 [...] Read more.
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 n118. Full article
(This article belongs to the Special Issue Probability, Statistics and Their Applications 2021)
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21 pages, 347 KiB  
Article
On the Omega Distribution: Some Properties and Estimation
by Abdelaziz Alsubie, Zuber Akhter, Haseeb Athar, Mahfooz Alam, 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 4 | Viewed by 3462
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 [...] Read more.
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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16 pages, 311 KiB  
Article
Robust Estimation and Tests for Parameters of Some Nonlinear Regression Models
by Pengfei Liu, Mengchen Zhang, Ru Zhang and Qin Zhou
Mathematics 2021, 9(6), 599; https://doi.org/10.3390/math9060599 - 11 Mar 2021
Cited by 2 | Viewed by 2203
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) [...] Read more.
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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