Editorial

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Special Issue “Probability Theory and Stochastic Modeling with Applications”

by Francisco Germán Badía and María D. Berrade

Mathematics 2023, 11(14), 3196; https://doi.org/10.3390/math11143196 - 21 Jul 2023

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This Special Issue (SI), titled “Probability Theory and Stochastic Modeling with Applications”, is concerned with the theory and applications of stochastic models [...] Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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21 pages, 1145 KiB

Open AccessArticle

Spectral Analysis for Comparing Bitcoin to Currencies and Assets

by Maria Chiara Pocelli, Manuel L. Esquível and Nadezhda P. Krasii

Mathematics 2023, 11(8), 1775; https://doi.org/10.3390/math11081775 - 07 Apr 2023

Cited by 1 | Viewed by 1333

Abstract

We present an analysis on variability Bitcoin characteristics that help to quantitatively differentiate Bitcoin from the state-owned traditional currencies and the asset Gold. We provide a detailed study on returns of exchange rates—against the Swiss Franc—of several traditional currencies together with Bitcoin and [...] Read more.

We present an analysis on variability Bitcoin characteristics that help to quantitatively differentiate Bitcoin from the state-owned traditional currencies and the asset Gold. We provide a detailed study on returns of exchange rates—against the Swiss Franc—of several traditional currencies together with Bitcoin and Gold; for that purpose, we define a distance between currencies by means of the spectral densities of the ARMA models of the returns of the exchange rates, and we present the computed matrix of the distances between the chosen currencies. A statistical analysis of these matrix distances is further proposed, which shows that the distance between Bitcoin and any other currency or Gold is not comparable to any of the distances between currencies or between currencies and Gold and not involving Bitcoin. This result shows that Bitcoin is essentially different from the traditional currencies and from Gold, at least in what concerns the structure of its variance and auto-covariances. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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18 pages, 397 KiB

Open AccessArticle

A Wavelet-Based Computational Framework for a Block-Structured Markov Chain with a Continuous Phase Variable

by Shuxia Jiang, Nian Liu and Yuanyuan Liu

Mathematics 2023, 11(7), 1587; https://doi.org/10.3390/math11071587 - 24 Mar 2023

Cited by 1 | Viewed by 944

Abstract

We consider the computing issues of the steady probabilities for block-structured discrete-time Markov chains that are of upper-Hessenberg or lower-Hessenberg transition kernels with a continuous phase set. An effective computational framework is proposed based on the wavelet transform, which extends and modifies the [...] Read more.

We consider the computing issues of the steady probabilities for block-structured discrete-time Markov chains that are of upper-Hessenberg or lower-Hessenberg transition kernels with a continuous phase set. An effective computational framework is proposed based on the wavelet transform, which extends and modifies the arguments in the literature for quasi-birth-death (QBD) processes. A numerical procedure is developed for computing the steady probabilities based on the fast discrete wavelet transform, and several examples are presented to illustrate its effectiveness. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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30 pages, 811 KiB

Open AccessArticle

A New Probabilistic Approach: Estimation and Monte Carlo Simulation with Applications to Time-to-Event Data

by Huda M. Alshanbari, Zubair Ahmad, Hazem Al-Mofleh, Clement Boateng Ampadu and Saima K. Khosa

Mathematics 2023, 11(7), 1583; https://doi.org/10.3390/math11071583 - 24 Mar 2023

Cited by 2 | Viewed by 923

Abstract

In this paper, we propose a useful method without adding any extra parameters to obtain new probability distributions. The proposed family is a combination of the two existing families of distributions and is called a weighted sine-G family. A two-parameter special member [...] Read more.

In this paper, we propose a useful method without adding any extra parameters to obtain new probability distributions. The proposed family is a combination of the two existing families of distributions and is called a weighted sine-G family. A two-parameter special member of the weighted sine-G family, using the Weibull distribution as a baseline model, is considered and investigated in detail. Some distributional properties of the weighted sine-G family are derived. Different estimation methods are considered to estimate the parameters of the special model of the weighted sine-G family. Furthermore, simulation studies based on these different methods are also provided. Finally, the applicability and usefulness of the weighted sine-G family are demonstrated by analyzing two data sets taken from the engineering sector. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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12 pages, 272 KiB

Open AccessArticle

Finding the Best Dueler

by Zhengu Zhang and Sheldon M. Ross

Mathematics 2023, 11(7), 1568; https://doi.org/10.3390/math11071568 - 23 Mar 2023

Cited by 2 | Viewed by 821

Abstract

Consider a set of n players. We suppose that each game involves two players, that there is some unknown player who wins each game it plays with a probability greater than

1 / 2,

and that our objective is to determine this [...] Read more.

Consider a set of n players. We suppose that each game involves two players, that there is some unknown player who wins each game it plays with a probability greater than

1 / 2,

and that our objective is to determine this best player. Under the requirement that the policy employed guarantees a correct choice with a probability of at least some specified value, we look for a policy that has a relatively small expected number of games played before decision. We consider this problem both under the assumption that the best player wins each game with a probability of at least some specified value

p_{0} > 1 / 2,

and under a Bayesian assumption that the probability that player i wins a game against player j is

\frac{v_{i}}{v_{i} + v_{j}}

, where

v_{1}, \dots, v_{n}

are the unknown values of n independent and identically distributed exponential random variables. In the former case, we propose a policy where chosen pairs play a match that ends when one of them has had a specified number of wins more than the other; in the latter case, we propose a Thompson sampling type rule. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

20 pages, 1320 KiB

Open AccessArticle

Quantile Regression with a New Exponentiated Odd Log-Logistic Weibull Distribution

by Gabriela M. Rodrigues, Edwin M. M. Ortega, Gauss M. Cordeiro and Roberto Vila

Mathematics 2023, 11(6), 1518; https://doi.org/10.3390/math11061518 - 21 Mar 2023

Cited by 3 | Viewed by 1377

Abstract

We define a new quantile regression model based on a reparameterized exponentiated odd log-logistic Weibull distribution, and obtain some of its structural properties. It includes as sub-models some known regression models that can be utilized in many areas. The maximum likelihood method is [...] Read more.

We define a new quantile regression model based on a reparameterized exponentiated odd log-logistic Weibull distribution, and obtain some of its structural properties. It includes as sub-models some known regression models that can be utilized in many areas. The maximum likelihood method is adopted to estimate the parameters, and several simulations are performed to study the finite sample properties of the maximum likelihood estimators. The applicability of the proposed regression model is well justified by means of a gastric carcinoma dataset. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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15 pages, 344 KiB

Open AccessArticle

RKHS Representations for Augmented Quaternion Random Signals: Application to Detection Problems

by Antonia Oya

Mathematics 2022, 10(23), 4432; https://doi.org/10.3390/math10234432 - 24 Nov 2022

Cited by 2 | Viewed by 940

Abstract

The reproducing kernel Hilbert space (RKHS) methodology has shown to be a suitable tool for the resolution of a wide range of problems in statistical signal processing both in the real and complex domains. It relies on the idea of transforming the original [...] Read more.

The reproducing kernel Hilbert space (RKHS) methodology has shown to be a suitable tool for the resolution of a wide range of problems in statistical signal processing both in the real and complex domains. It relies on the idea of transforming the original functional data into an infinite series representation by projection onto an specific RKHS, which usually simplifies the statistical treatment without any loss of efficiency. Moreover, the advantages of quaternion algebra over real-valued three and four-dimensional vector algebra in the modelling of multidimensional data have been proven useful in much relatively recent research. This paper accordingly proposes a generic RKHS framework for the statistical analysis of augmented quaternion random vectors, which provide a complete description of their second order characteristics. It will allow us to exploit the full advantages of the RKHS theory in widely linear processing applications, such as signal detection. In particular, we address the detection of a quaternion signal disturbed by additive Gaussian noise and the discrimination between two quaternion Gaussian signals in continuous time. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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20 pages, 1467 KiB

Open AccessArticle

A Credibility Theory-Based Robust Optimization Model to Hedge Price Uncertainty of DSO with Multiple Transactions

by Li-Peng Shao, Jia-Jia Chen, Lu-Wen Pan and Zi-Juan Yang

Mathematics 2022, 10(23), 4420; https://doi.org/10.3390/math10234420 - 23 Nov 2022

Cited by 3 | Viewed by 996

Abstract

This paper addresses the deregulated electricity market arising in a distribution system with an electricity transaction. Under such an environment, the distribution system operator (DSO) with a distributed generator faces the challenge of electricity price uncertainty in a spot market. In this context, [...] Read more.

This paper addresses the deregulated electricity market arising in a distribution system with an electricity transaction. Under such an environment, the distribution system operator (DSO) with a distributed generator faces the challenge of electricity price uncertainty in a spot market. In this context, a credibility theory-based robust optimization model with multiple transactions is established to hedge the uncertain spot price of the DSO. Firstly, on the basis of credibility theory, the spot price is taken as a fuzzy variable and a risk aversion-based fuzzy opportunity constraint is proposed. Then, to exploit the resiliency of multiple transactions on hedging against uncertain spot price, the spot market, option contract and bilateral contract integrating power flow constraints are studied, because it is imperative for DSO to consider the operational constraints of the local network in the electricity market. Finally, the clear equivalence class is adopted to transform the risk aversion constraint into a deterministic robust optimization one. Under the premise of considering the expected cost of the DSO, the optimal electricity transaction strategy that maximizes resistance to uncertain spot price is pursued. The rationality and effectiveness of the model are verified with a modified 15-node network. The results show that the introduction of option contracts and bilateral contracts reduces the electricity transaction cost of DSO by USD 28.5. In addition, under the same risk aversion factor, the cost of the proposed model is reduced by USD 195.18 compared with robust optimization, which avoids the over-conservatism of traditional robust optimization. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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35 pages, 2832 KiB

Open AccessArticle

Economic Performance and Stock Market Integration in BRICS and G7 Countries: An Application with Quantile Panel Data and Random Coefficients Modeling

by José Clemente Jacinto Ferreira, Ana Paula Matias Gama, Luiz Paulo Fávero, Ricardo Goulart Serra, Patrícia Belfiore, Igor Pinheiro de Araújo Costa and Marcos dos Santos

Mathematics 2022, 10(21), 4013; https://doi.org/10.3390/math10214013 - 28 Oct 2022

Cited by 3 | Viewed by 1507

Abstract

The interest in studies aimed at understanding the integration of the stock market with the economic performance of countries has been growing in recent years, perhaps driven by the recent economic crises faced by the world. Although several studies on the topic have [...] Read more.

The interest in studies aimed at understanding the integration of the stock market with the economic performance of countries has been growing in recent years, perhaps driven by the recent economic crises faced by the world. Although several studies on the topic have been carried out, the results are still far from a meaningful conclusion. In this sense, this paper considered the dual objective of investigating whether there is significant variance in the economic performance of developed and emerging markets’ countries and whether the global risk factors are statistically significant in explaining the variations in their future economic performance over time. From a sample of (i) gross domestic products from BRICS and G7 countries (total of twelve countries), and (ii) returns of the risk factors of developed and emerging stock markets for the period 1993 to 2019, we applied longitudinal regression modeling for five distinct percentiles, and random coefficients modeling (RCM) with repeated measures. We found that risk factors explain the future economic performance, there is significant variation in economic performance over time among countries, and the temporal variation in the random effects of intercepts can be explained by RCM. The results of this study confirm that stock markets follow an integration process and that moderately integrated markets may have the same risk factors. Furthermore, considering that risk factors are related to future GDP growth, they act as proxies for unidentified state variables. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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20 pages, 331 KiB

Open AccessArticle

On the Residual Lifetime and Inactivity Time in Mixtures

by Francisco Germán Badía and María Dolores Berrade

Mathematics 2022, 10(15), 2795; https://doi.org/10.3390/math10152795 - 06 Aug 2022

Cited by 1 | Viewed by 1366

Abstract

In this paper we study the aging characteristics in mixtures of distributions, providing characterizations for their derivatives that explain the smooth behavior of the mixture. The classical preservation results for the reversed hazard rate, mean residual life and mean inactivity time are derived [...] Read more.

In this paper we study the aging characteristics in mixtures of distributions, providing characterizations for their derivatives that explain the smooth behavior of the mixture. The classical preservation results for the reversed hazard rate, mean residual life and mean inactivity time are derived under a different approach than in previous studies. We focus on the variance of both the residual life and inactivity time in mixtures, obtaining some preservation properties. We also state conditions for weak and strong bending properties for the variance of the residual life and the inactivity time in mixtures. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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19 pages, 1141 KiB

Open AccessArticle

A Geologic-Actuarial Approach for Insuring the Extraction Tasks of Non-Renewable Resources by One and Two Agents

by Rigoberto Real-Miranda and José Daniel López-Barrientos

Mathematics 2022, 10(13), 2242; https://doi.org/10.3390/math10132242 - 26 Jun 2022

Cited by 5 | Viewed by 1183

Abstract

This work uses classic stochastic dynamic programming techniques to determine the equivalence premium that each of two extraction agents of a non-renewable natural resource must pay to an insurer to cover the risk that the extraction pore explodes. We use statistical and geological [...] Read more.

This work uses classic stochastic dynamic programming techniques to determine the equivalence premium that each of two extraction agents of a non-renewable natural resource must pay to an insurer to cover the risk that the extraction pore explodes. We use statistical and geological methods to calibrate the time-until-failure distribution of extraction status for each agent and couple a simple approximation scheme with the actuarial standard of Bühlmann’s recommendations to charge the extracting agents a variance premium, while the insurer earns a return on its investment at risk. We test our analytical results through Monte Carlo simulations to verify that the probability of ruin does not exceed a certain predetermined level. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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28 pages, 2067 KiB

Open AccessArticle

Optimal Control with Partially Observed Regime Switching: Discounted and Average Payoffs

by Beatris Adriana Escobedo-Trujillo, Javier Garrido-Meléndez, Gerardo Alcalá and J. D. Revuelta-Acosta

Mathematics 2022, 10(12), 2073; https://doi.org/10.3390/math10122073 - 15 Jun 2022

Cited by 2 | Viewed by 1233

Abstract

We consider an optimal control problem with the discounted and average payoff. The reward rate (or cost rate) can be unbounded from above and below, and a Markovian switching stochastic differential equation gives the state variable dynamic. Markovian switching is represented by a [...] Read more.

We consider an optimal control problem with the discounted and average payoff. The reward rate (or cost rate) can be unbounded from above and below, and a Markovian switching stochastic differential equation gives the state variable dynamic. Markovian switching is represented by a hidden continuous-time Markov chain that can only be observed in Gaussian white noise. Our general aim is to give conditions for the existence of optimal Markov stationary controls. This fact generalizes the conditions that ensure the existence of optimal control policies for optimal control problems completely observed. We use standard dynamic programming techniques and the method of hidden Markov model filtering to achieve our goals. As applications of our results, we study the discounted linear quadratic regulator (LQR) problem, the ergodic LQR problem for the modeled quarter-car suspension, the average LQR problem for the modeled quarter-car suspension with damp, and an explicit application for an optimal pollution control. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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15 pages, 1612 KiB

Open AccessEditor’s ChoiceArticle

A Preventive Replacement Policy for a System Subject to Bivariate Generalized Polya Failure Process

by Hyunju Lee, Ji Hwan Cha and Maxim Finkelstein

Mathematics 2022, 10(11), 1833; https://doi.org/10.3390/math10111833 - 26 May 2022

Cited by 2 | Viewed by 1368

Abstract

Numerous studies on preventive maintenance of minimally repaired systems with statistically independent components have been reported in reliability literature. However, in practice, the repair can be worse-than-minimal and the components of a system can be statistically dependent. The existing literature does not cover [...] Read more.

Numerous studies on preventive maintenance of minimally repaired systems with statistically independent components have been reported in reliability literature. However, in practice, the repair can be worse-than-minimal and the components of a system can be statistically dependent. The existing literature does not cover this important in-practice setting. Therefore, our paper is the first to deal with these issues by modeling dependence in the bivariate set up when a system consists of two dependent parts. We employ the bivariate generalized Polya process to model the corresponding failure and repair process. Relevant stochastic properties of this process have been obtained in order to propose and further discuss the new optimal bivariate preventive maintenance policy with two decision parameters: age and operational history. Moreover, introducing these two parameters in the considered context is also a new feature of the study. Under the proposed policy, the long-run average cost rate is derived and the optimal replacement policies are investigated. Detailed numerical examples illustrate our findings and show the potential efficiency of the obtained results in practice. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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17 pages, 384 KiB

Open AccessArticle

On Robustness for Spatio-Temporal Data

by Alfonso García-Pérez

Mathematics 2022, 10(10), 1785; https://doi.org/10.3390/math10101785 - 23 May 2022

Cited by 2 | Viewed by 2037

Abstract

The spatio-temporal variogram is an important factor in spatio-temporal prediction through kriging, especially in fields such as environmental sustainability or climate change, where spatio-temporal data analysis is based on this concept. However, the traditional spatio-temporal variogram estimator, which is commonly employed for these [...] Read more.

The spatio-temporal variogram is an important factor in spatio-temporal prediction through kriging, especially in fields such as environmental sustainability or climate change, where spatio-temporal data analysis is based on this concept. However, the traditional spatio-temporal variogram estimator, which is commonly employed for these purposes, is extremely sensitive to outliers. We approach this problem in two ways in the paper. First, new robust spatio-temporal variogram estimators are introduced, which are defined as M-estimators of an original data transformation. Second, we compare the classical estimate against a robust one, identifying spatio-temporal outliers in this way. To accomplish this, we use a multivariate scale-contaminated normal model to produce reliable approximations for the sample distribution of these new estimators. In addition, we define and study a new class of M-estimators in this paper, including real-world applications, in order to determine whether there are any significant differences in the spatio-temporal variogram between two temporal lags and, if so, whether we can reduce the number of lags considered in the spatio-temporal analysis. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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38 pages, 598 KiB

Open AccessArticle

Robust Parametric Identification for ARMAX Models with Non-Gaussian and Coloured Noise: A Survey

by Jesica Escobar and Alexander Poznyak

Mathematics 2022, 10(8), 1291; https://doi.org/10.3390/math10081291 - 13 Apr 2022

Cited by 4 | Viewed by 1483

Abstract

In this paper the Cramer-Rao information bound for ARMAX (Auto-Regression-Moving-Average-Models-with-Exogenuos-inputs) under non-Gaussian noise is derived. It is shown that the direct application of the Least Squares Method (LSM) leads to incorrect (shifted) parameter estimates. This inconsistency can be corrected by the implementation of [...] Read more.

In this paper the Cramer-Rao information bound for ARMAX (Auto-Regression-Moving-Average-Models-with-Exogenuos-inputs) under non-Gaussian noise is derived. It is shown that the direct application of the Least Squares Method (LSM) leads to incorrect (shifted) parameter estimates. This inconsistency can be corrected by the implementation of the parallel usage of the MLMW (Maximum Likelihood Method with Whitening) procedure, applied to all measurable variables of the model, and a nonlinear residual transformation using the information on the distribution density of a non-Gaussian noise, participating in Moving Average structure. The design of the corresponding parameter-estimator, realizing the suggested MLMW-procedure is discussed in details. It is shown that this method is asymptotically optimal, that is, reaches this information bound. If the noise distribution belongs to some given class, then the Huber approach (min-max version of MLM) may be effectively applied. A numerical example illustrates the suggested approach. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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23 pages, 950 KiB

Open AccessArticle

Statistical Analysis of the Lifetime Distribution with Bathtub-Shaped Hazard Function under Lagged-Effect Step-Stress Model

by Zihui Zhang and Wenhao Gui

Mathematics 2022, 10(5), 674; https://doi.org/10.3390/math10050674 - 22 Feb 2022

Cited by 2 | Viewed by 1748

Abstract

In survival analysis, applying stress is often used to accelerate an experiment. Stress can be discontinuous, and the step-stress model is applied widely due to its flexibility. However, in reality, when new stress is applied, it often does not take effect immediately, but [...] Read more.

In survival analysis, applying stress is often used to accelerate an experiment. Stress can be discontinuous, and the step-stress model is applied widely due to its flexibility. However, in reality, when new stress is applied, it often does not take effect immediately, but there will be a lagged effect. Under the lagged-effect step-stress model, the statistical inference of the Chen distribution is discussed. The Chen distribution is an important life distribution as its risk function is bathtub-shaped with certain parameters. In this paper, the maximum likelihood estimators are presented and the Newton–Raphson algorithm is used. According to the form of risk function under this model, the explicit expressions of least squares estimators are obtained. The calculation methods of asymptotic confidence intervals and coverage probabilities are proposed by using the observed Fisher matrix. Finally, to evaluate the performance of the above estimation methods, a Monte Carlo simulation study is provided. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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6 pages, 221 KiB

Open AccessArticle

On Consistency of the Bayes Estimator of the Density

by Agustín G. Nogales

Mathematics 2022, 10(4), 636; https://doi.org/10.3390/math10040636 - 18 Feb 2022

Cited by 1 | Viewed by 935

Abstract

Under mild conditions, strong consistency of the Bayes estimator of the density is proved. Moreover, the Bayes risk (for some common loss functions) of the Bayes estimator of the density (i.e., the posterior predictive density) goes to zero as the sample size goes [...] Read more.

Under mild conditions, strong consistency of the Bayes estimator of the density is proved. Moreover, the Bayes risk (for some common loss functions) of the Bayes estimator of the density (i.e., the posterior predictive density) goes to zero as the sample size goes to ∞. In passing, a similar result is obtained for the estimation of the sampling distribution. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Modeling with Applications)

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