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Mathematics
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Probability, Statistics and Random Processes
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Probability, Statistics and Random Processes

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

Deadline for manuscript submissions: 31 August 2024 | Viewed by 6291

Special Issue Editor

Prof. Dr. Velizar Pavlov

E-Mail Website
Guest Editor
Department of Applied Mathematics and Statistics, University of Ruse, 7017 Ruse, Bulgaria
Interests: applied mathematics; applied statistics

Special Issue Information

Dear Colleagues,

The probability, statistics and modelling of random processes are well-established yet actively developing domains in science. They feature various applications in diverse areas as finance and economics, engineering, physics, biology and many others. The governing laws of a great deal of the adequate models describing natural and anthropogenic phenomena are stochastic in nature. Furthermore, in modern computation, probability and statistics, both frequentist and Bayesian, play fundamental roles as they are used to solve a multitude of problems in hypothesis testing, regression modelling, sensitivity estimation, etc. 

Some applications of statistics and probability in economics include regression, classification, AR(I)MA and (G)ARCH modelling, and the Wiener process, Brownian motion and Levy processes are where stochastics meets finance. Estimating and modelling new distributions, possibly heavy-tailed, and other theoretical results are attractive aspects of the research as well. 

This Special Issue aims to collect high-quality and interesting papers considering recent advances in theoretical and applied statistics, probability and stochastic processes. Manuscripts providing pioneering results and achievements, solving complex problems or suggesting novel and advanced methods and techniques are warmly welcome for submission.

Prof. Dr. Velizar Pavlov
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

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

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • mathematics
  • statistics
  • probability
  • algorithms
  • computation
  • stochastics
  • simulation
  • random variables
  • random processes
  • networks
  • methodology

Published Papers (7 papers)

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Research

22 pages, 2318 KiB  
Article
Asymptotic Form of the Covariance Matrix of Likelihood-Based Estimator in Multidimensional Linear System Model for the Case of Infinity Number of Nuisance Parameters
by Alexander Varypaev
Mathematics 2024, 12(3), 473; https://doi.org/10.3390/math12030473 - 01 Feb 2024
Viewed by 455
Abstract
This article is devoted to the synthesis and analysis of the quality of the statistical estimate of parameters of a multidimensional linear system (MLS) with one input and m outputs. A nontrivial case is investigated when the one-dimensional input signal of MLS is [...] Read more.
This article is devoted to the synthesis and analysis of the quality of the statistical estimate of parameters of a multidimensional linear system (MLS) with one input and m outputs. A nontrivial case is investigated when the one-dimensional input signal of MLS is a deterministic process, the values of which are unknown nuisance parameters. The estimate is based only on observations of MLS output signals distorted by random Gaussian stationary m-dimensional noise with a known spectrum. It is assumed that the likelihood function of observations of the output signals of MLS satisfies the conditions of local asymptotic normality. The n-consistency of the estimate is established. Under the assumption of asymptotic normality of an objective function, the limiting covariance matrix of the estimate is calculated for case where the number of observations tends to infinity. Full article
(This article belongs to the Special Issue Probability, Statistics and Random Processes)
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25 pages, 400 KiB  
Article
Modified Block Bootstrap Testing for Persistence Change in Infinite Variance Observations
by Si Zhang, Hao Jin and Menglin Su
Mathematics 2024, 12(2), 258; https://doi.org/10.3390/math12020258 - 12 Jan 2024
Viewed by 580
Abstract
This paper investigates the properties of the change in persistence detection for observations with infinite variance. The innovations are assumed to be in the domain of attraction of a stable law with index κ∈(0,2]. We provide [...] Read more.
This paper investigates the properties of the change in persistence detection for observations with infinite variance. The innovations are assumed to be in the domain of attraction of a stable law with index κ∈(0,2]. We provide a new test statistic and show that its asymptotic distribution under the null hypothesis of non-stationary I(1) series is a functional of a stable process. When the change point in persistence is not known, the consistency is always given under the alternative, either from stationary I(0) to non-stationary I(1) or vice versa. The proposed tests can be used to identify the direction of change and do not over-reject against constant I(0) series, even in relatively small samples. Furthermore, we also consider the change point estimator which is consistent and the asymptotic behavior of the test statistic in the case of near-integrated time series. A block bootstrap method is suggested to determine critical values because the null asymptotic distribution contains the unknown tail index, which results in critical values depending on it. The simulation study demonstrates that the block bootstrap-based test is robust against change in persistence for heavy-tailed series with infinite variance. Finally, we apply our methods to the two series of the US inflation rate and USD/CNY exchange rate, and find significant evidence for persistence changes, respectively, from I(0) to I(1) and from I(1) to I(0). Full article
(This article belongs to the Special Issue Probability, Statistics and Random Processes)
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18 pages, 1046 KiB  
Article
Validation of Stock Price Prediction Models in the Conditions of Financial Crisis
by Vesela Mihova, Ivan Georgiev, Elitsa Raeva, Slavi Georgiev and Velizar Pavlov
Mathematics 2024, 12(1), 33; https://doi.org/10.3390/math12010033 - 22 Dec 2023
Viewed by 601
Abstract
The distribution laws of various natural and anthropogenic processes in the world around us are stochastic in nature. The development of mathematics and, in particular, of stochastic modeling allows us to study regularities in such processes. In practice, stochastic modeling finds a huge [...] Read more.
The distribution laws of various natural and anthropogenic processes in the world around us are stochastic in nature. The development of mathematics and, in particular, of stochastic modeling allows us to study regularities in such processes. In practice, stochastic modeling finds a huge number of applications in various fields, including finance and economics. In this work, some particular applications of stochastic processes in finance are examined in the conditions of financial crisis, aiming to provide a solid approach for stock price forecasting. More specifically, autoregressive integrated moving average (ARIMA) models and modified ordinary differential equation (ODE) models, previously developed by some of the authors to predict the asset prices of four Bulgarian companies, are validated against a time period during the crisis. Estimated rates of return are calculated from the models for one period ahead. The errors are estimated and the models are compared. The return values predicted with each of the two approaches are used to derive optimal risk portfolios based on the Markowitz model, which is the second major aim of this study. The third aim is to compare the resulting portfolios in terms of distribution (i.e., weights of the stocks), risk, and rate of return. Full article
(This article belongs to the Special Issue Probability, Statistics and Random Processes)
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10 pages, 302 KiB  
Article
New Ways to Calculate the Probability in the Bertrand Problem
by Javier Rodrigo, Mariló López and Sagrario Lantarón
Mathematics 2024, 12(1), 3; https://doi.org/10.3390/math12010003 - 19 Dec 2023
Viewed by 528
Abstract
We give two new ways of calculating the probability of a chord of circumference randomly selected being larger than the side of an equilateral triangle inscribed in the circumference (this problem is known as the Bertrand paradox). The first one employs an immersion [...] Read more.
We give two new ways of calculating the probability of a chord of circumference randomly selected being larger than the side of an equilateral triangle inscribed in the circumference (this problem is known as the Bertrand paradox). The first one employs an immersion in R4, and the second one uses a direct probability measure over the set of chords. Full article
(This article belongs to the Special Issue Probability, Statistics and Random Processes)
14 pages, 472 KiB  
Article
Estimation of Pianka Overlapping Coefficient for Two Exponential Distributions
by Suad Alhihi and Maalee Almheidat
Mathematics 2023, 11(19), 4152; https://doi.org/10.3390/math11194152 - 02 Oct 2023
Cited by 1 | Viewed by 1009
Abstract
Overlapping coefficients (OVL) are commonly used to estimate the similarity between populations in terms of their density functions. In this paper, we consider Pianka’s overlap coefficient for two exponential populations. The methods for statistical inference of Pianka’s coefficient are presented. The bias and [...] Read more.
Overlapping coefficients (OVL) are commonly used to estimate the similarity between populations in terms of their density functions. In this paper, we consider Pianka’s overlap coefficient for two exponential populations. The methods for statistical inference of Pianka’s coefficient are presented. The bias and mean square error (MSE) of the maximum likelihood estimator (MLE) and the Bayes estimator of Pianka’s overlap coefficient are investigated by simulation. Confidence intervals for Pianka’s overlap measure are constructed. Full article
(This article belongs to the Special Issue Probability, Statistics and Random Processes)
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21 pages, 543 KiB  
Article
Bertrand’s Paradox Resolution and Its Implications for the Bing–Fisher Problem
by Richard A. Chechile
Mathematics 2023, 11(15), 3282; https://doi.org/10.3390/math11153282 - 26 Jul 2023
Cited by 1 | Viewed by 1841
Abstract
Bertrand’s paradox is a problem in geometric probability that has resisted resolution for more than one hundred years. Bertrand provided three seemingly reasonable solutions to his problem — hence the paradox. Bertrand’s paradox has also been influential in philosophical debates about frequentist versus [...] Read more.
Bertrand’s paradox is a problem in geometric probability that has resisted resolution for more than one hundred years. Bertrand provided three seemingly reasonable solutions to his problem — hence the paradox. Bertrand’s paradox has also been influential in philosophical debates about frequentist versus Bayesian approaches to statistical inference. In this paper, the paradox is resolved (1) by the clarification of the primary variate upon which the principle of maximum entropy is employed and (2) by imposing constraints, based on a mathematical analysis, on the random process for any subsequent nonlinear transformation to a secondary variable. These steps result in a unique solution to Bertrand’s problem, and this solution differs from the classic answers that Bertrand proposed. It is shown that the solutions proposed by Bertrand and others reflected sampling processes that are not purely random. It is also shown that the same two steps result in the resolution of the Bing–Fisher problem, which has to do with the selection of a consistent prior for Bayesian inference. The resolution of Bertrand’s paradox and the Bing–Fisher problem rebuts philosophical arguments against the Bayesian approach to statistical inference, which were based on those two ostensible problems. Full article
(This article belongs to the Special Issue Probability, Statistics and Random Processes)
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8 pages, 242 KiB  
Article
Consistency of Decision in Finite and Numerable Multinomial Models
by Isaac Akoto and João T. Mexia
Mathematics 2023, 11(11), 2434; https://doi.org/10.3390/math11112434 - 24 May 2023
Viewed by 566
Abstract
The multinomial distribution is often used in modeling categorical data because it describes the probability of a random observation being assigned to one of several mutually exclusive categories. Given a finite or numerable multinomial model M|n,p whose decision is [...] Read more.
The multinomial distribution is often used in modeling categorical data because it describes the probability of a random observation being assigned to one of several mutually exclusive categories. Given a finite or numerable multinomial model M|n,p whose decision is indexed by a parameter θ and having a cost cθ,p depending on θ and on p, we show that, under general conditions, the probability of taking the least cost decision tends to 1 when n tends to ∞, i.e., we showed that the cost decision is consistent, representing a Statistical Decision Theory approach to the concept of consistency, which is not much considered in the literature. Thus, under these conditions, we have consistency in the decision making. The key result is that the estimator p˜n with components p˜n,i=nin,i=1,⋯, where ni is the number of times we obtain the ith result when we have a sample of size n, is a consistent estimator of p. This result holds both for finite and numerable models. By this result, we were able to incorporate a more general form for consistency for the cost function of a multinomial model. Full article
(This article belongs to the Special Issue Probability, Statistics and Random Processes)
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