Mathematics

Research

29 pages, 4100 KiB

Open AccessArticle

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 2468

Abstract

This paper focuses mainly on the problem of computing the

γ^{th}

,

γ > 0

, moment of a random variable

Y_{n} : = \sum_{i = 1}^{n} α_{i} X_{i}

in which the

α_{i}

’s are positive [...] Read more.

This paper focuses mainly on the problem of computing the

γ^{th}

,

γ > 0

, moment of a random variable

Y_{n} : = \sum_{i = 1}^{n} α_{i} X_{i}

in which the

α_{i}

’s are positive real numbers and the

X_{i}

’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

Y_{n}

, especially when not all

α_{i}

’s are equal. We analytically solve this problem by showing that the

γ^{th}

moment of

Y_{n}

can be expressed in terms of generalized hypergeometric functions. Additionally, we extend our result to computing the

γ^{th}

moment of

Y_{n}

when

X_{i}

is a combination of statistically independent

Z_{i}^{2}

and

G_{i}

in which the

Z_{i}

’s are distributed according to normal or Maxwell–Boltzmann distributions and the

G_{i}

’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

γ^{th}

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

Open AccessArticle

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

Cited by 1 | Viewed by 2106

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

Open AccessArticle

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 11 | Viewed by 2035

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

Open AccessArticle

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 3 | Viewed by 7116

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

Open AccessArticle

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 15 | Viewed by 2292

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

Open AccessArticle

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 5 | Viewed by 2386

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

Open AccessArticle

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 1579

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

Open AccessArticle

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

Cited by 2 | Viewed by 3688

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

Open AccessFeature PaperArticle

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 1242

Abstract

The following problem was studied: let

{(Z_{j})}_{j \geq 1}

be a sequence of i.i.d. d-dimensional random vectors. Let F be their probability distribution and for every

n \geq 1

consider the sample [...] Read more.

The following problem was studied: let

{(Z_{j})}_{j \geq 1}

be a sequence of i.i.d. d-dimensional random vectors. Let F be their probability distribution and for every

n \geq 1

consider the sample

S_{n} = {Z_{1}, Z_{2}, \dots, Z_{n}} .

Then

Z_{j}

was called a “leader” in the sample

S_{n}

if

Z_{j} \geq Z_{k},

\forall k \in {1, 2, \dots, n}

and

Z_{j}

was an “anti-leader” if

Z_{j} \leq Z_{k},

\forall k \in {1, 2, \dots, n}

. The comparison of two vectors was the usual one: if

Z_{j} = (Z_{j}^{(1)}, Z_{j}^{(2)}, \dots, Z_{j}^{(d)}),

j \geq 1,

then

Z_{j} \geq Z_{k}

means

Z_{j}^{(i)} \geq Z_{k}^{(i)},

while

Z_{j} \leq Z_{k}

means

Z_{j}^{(i)} \leq Z_{k}^{(i)},

\forall 1 \leq i \leq d,

\forall j, k \geq 1 .

Let

a_{n}

be the probability that

S_{n}

has a leader,

b_{n}

be the probability that

S_{n}

has an anti-leader and

c_{n}

be the probability that

S_{n}

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 = lim inf a_{n},

b = lim inf b_{n},

c = lim inf c_{n}

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 = f (X)

where

f = (f_{1}, \dots, f_{d}) : [0, 1] \to R^{d}

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, f_{2} (X), \dots, f_{d} (X))

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

Open AccessArticle

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 9 | Viewed by 9566

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

Open AccessArticle

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 8 | Viewed by 2234

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 \in (- 1, 1)

and

k = \pm 1

. By comparison, the Lie symmetries of the NFPEs based on Tsallis q-entropies point out six “exceptional” cases, for:

q = \frac{1}{2}

,

q = \frac{3}{2}

,

q = \frac{4}{3}

,

q = \frac{7}{3}

,

q = 2

and

q = 3

. Full article

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

12 pages, 294 KiB

Open AccessEditor’s ChoiceArticle

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 3 | Viewed by 2194

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

Open AccessArticle

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 7 | Viewed by 2438

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

Open AccessArticle

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 2627

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

Open AccessArticle

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 1786

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

Open AccessFeature PaperArticle

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 2180

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

Open AccessFeature PaperArticle

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 4131

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

Open AccessArticle

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 46 | Viewed by 2915

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

Open AccessArticle

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 2033

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

Open AccessArticle

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 2 | Viewed by 3316

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

Open AccessArticle

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 1464

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)

17 pages, 572 KiB

Open AccessArticle

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 16 | Viewed by 2515

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

Open AccessArticle

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 2292

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

Open AccessEditor’s ChoiceArticle

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 35 | Viewed by 8337

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

n \geq 118

. Full article

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

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21 pages, 347 KiB

Open AccessArticle

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 3789

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

Open AccessArticle

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 2530

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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