Axioms

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16 pages, 332 KB

Open AccessArticle

A Common Generalization of the (a,b)- and (s,t)-Transformations of Probability Measures

by Ghadah Alomani and Raouf Fakhfakh

Axioms 2026, 15(5), 374; https://doi.org/10.3390/axioms15050374 - 16 May 2026

Viewed by 65

Abstract

This paper presents two analytic mappings defined on probability measures that extend and unify the

(a, b)

- and

(s, t)

-deformations arising in free probability for s,

b > 0

and a, [...] Read more.

This paper presents two analytic mappings defined on probability measures that extend and unify the

(a, b)

- and

(s, t)

-deformations arising in free probability for s,

b > 0

and a,

t \in R

. These unified operators, denoted

U_{(a, b, s, t)}^{'}

and

U_{(a, b, s, t)}^{''}

, are characterized by a functional equation involving the Cauchy–Stieltjes transform, providing a transform-based formulation of measure deformation. They reduce to the

(a, b)

-transformation when

s = t = 1

and to the

(s, t)

-transformation when

a = b = 1

. Working in the framework of Cauchy–Stieltjes kernel families, we study the induced effect of these transformations on the associated variance functions and obtain explicit transformation formulas. These results yield a stability theorem showing that the free Meixner class is stable under both operators. In addition, we derive two properties of the semicircle law via the restricted deformations

U_{(a, b, 1 / b, t)}^{'}

and

U_{(a, b, 1 / b, t)}^{''}

, thereby emphasizing the structural role of symmetry in measure transformations and in the preservation of canonical measures. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)

18 pages, 425 KB

Open AccessArticle

ARIMA Model Selection and Prediction Intervals

by W. A. Dhanushka M. Welagedara, Mulubrhan G. Haile and David J. Olive

Axioms 2026, 15(3), 228; https://doi.org/10.3390/axioms15030228 - 19 Mar 2026

Viewed by 614

Abstract

Inference after model selection is a very important problem. Model selection algorithms for ARIMA time series, with criteria such as AIC and BIC, tend to select an inconsistent model with positive probability, making data-splitting inference for testing and confidence intervals unreliable. One technique [...] Read more.

Inference after model selection is a very important problem. Model selection algorithms for ARIMA time series, with criteria such as AIC and BIC, tend to select an inconsistent model with positive probability, making data-splitting inference for testing and confidence intervals unreliable. One technique was fairly reliable for sample sizes greater than 600, and a modification also worked. Model selection is often useful for prediction, since the selected submodel tends to have fitted values and residuals that are highly correlated with those of the full model. A few prediction intervals perform fairly well even after model selection. A useful technique for handling outliers is to replace the outliers with missing values. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)

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13 pages, 1056 KB

Open AccessArticle

A New Index for Quantifying the Statistical Difference Between Two Probability Distributions

by Hening Huang

Axioms 2026, 15(2), 150; https://doi.org/10.3390/axioms15020150 - 18 Feb 2026

Viewed by 707

Abstract

In many scientific fields (e.g., statistics, data science, machine learning, and image processing), effectively quantifying the statistical difference between two probability distributions is an important task. Although a wide variety of measures have been proposed in the literature, some of them (such as [...] Read more.

In many scientific fields (e.g., statistics, data science, machine learning, and image processing), effectively quantifying the statistical difference between two probability distributions is an important task. Although a wide variety of measures have been proposed in the literature, some of them (such as the chi-square divergence and the Kullback–Leibler divergence) do not satisfy one or both of two key axioms: normalization and symmetry. This paper proposes a new index for quantifying the statistical difference between two probability distributions, called the distribution discrepancy index (DDI). The proposed DDI is based on the recently developed concepts of informity and cross-informity in informity theory. Its value ranges from 0 to 1, with values close to 1 indicating a large discrepancy and values close to 0 indicating minimal discrepancy. The DDI satisfies the two key axioms and is applicable to both discrete and continuous distributions. This paper also proposes the distribution similarity index (DSI) as a complement to the DDI. Three examples are presented to compare the DDI with three existing discrepancy measures (the Hellinger distance, total variation distance, and Jensen–Shannon divergence) and the DSI with two existing similarity measures (the Bhattacharyya coefficient and overlapping index). Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)

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43 pages, 5548 KB

Open AccessArticle

A Novel Probabilistic Model for Streamflow Analysis and Its Role in Risk Management and Environmental Sustainability

by Tassaddaq Hussain, Enrique Villamor, Mohammad Shakil, Mohammad Ahsanullah and Bhuiyan Mohammad Golam Kibria

Axioms 2026, 15(2), 113; https://doi.org/10.3390/axioms15020113 - 4 Feb 2026

Viewed by 821

Abstract

Probabilistic streamflow models play a pivotal role in quantifying hydrological uncertainty and form the backbone of modern risk management strategies for flood and drought forecasting, water allocation planning, and the design of resilient infrastructure. Unlike deterministic approaches that yield single-point estimates, these models [...] Read more.

Probabilistic streamflow models play a pivotal role in quantifying hydrological uncertainty and form the backbone of modern risk management strategies for flood and drought forecasting, water allocation planning, and the design of resilient infrastructure. Unlike deterministic approaches that yield single-point estimates, these models provide a spectrum of possible outcomes, enabling a more realistic assessment of extreme events and supporting informed, sustainable water resource decisions. By explicitly accounting for natural variability and uncertainty, probabilistic models promote transparent, robust, and equitable risk evaluations, helping decision-makers balance economic costs, societal benefits, and environmental protection for long-term sustainability. In this study, we introduce the bounded half-logistic distribution (BHLD), a novel heavy-tailed probability model constructed using the T–Y method for distribution generation, where T denotes a transformer distribution and Y represents a baseline generator. Although the BHLD is conceptually related to the Pareto and log-logistic families, it offers several distinctive advantages for streamflow modeling, including a flexible hazard rate that can be unimodal or monotonically decreasing, a finite lower bound, and closed-form expressions for key risk measures such as Value at Risk (VaR) and Tail Value at Risk (TVaR). The proposed distribution is defined on a lower-bounded domain, allowing it to realistically capture physical constraints inherent in flood processes, while a log-logistic-based tail structure provides the flexibility needed to model extreme hydrological events. Moreover, the BHLD is analytically characterized through a governing differential equation and further examined via its characteristic function and the maximum entropy principle, ensuring stable and efficient parameter estimation. It integrates a half-logistic generator with a log-logistic baseline, yielding a power-law tail decay governed by the parameter

β

, which is particularly effective for representing extreme flows. Fundamental properties, including the hazard rate function, moments, and entropy measures, are derived in closed form, and model parameters are estimated using the maximum likelihood method. Applied to four real streamflow data sets, the BHLD demonstrates superior performance over nine competing distributions in goodness-of-fit analyses, with notable improvements in tail representation. The model facilitates accurate computation of hydrological risk metrics such as VaR, TVaR, and tail variance, uncovering pronounced temporal variations in flood risk and establishing the BHLD as a powerful and reliable tool for streamflow modeling under changing environmental conditions. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)

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16 pages, 291 KB

Open AccessArticle

General Convergence Rates by the Delayed Sums Method

by Cheng Hu, Shangshang Yang and Tonghui Wang

Axioms 2026, 15(2), 92; https://doi.org/10.3390/axioms15020092 - 26 Jan 2026

Viewed by 409

Abstract

In this study, we propose a delayed sums method to investigate the convergence rates of partial sums. This approach enables general and systematic treatment of the convergence behavior of partial sums, encompassing and extending classical results such as the law of large numbers, [...] Read more.

In this study, we propose a delayed sums method to investigate the convergence rates of partial sums. This approach enables general and systematic treatment of the convergence behavior of partial sums, encompassing and extending classical results such as the law of large numbers, the law of logarithm, and the law of the iterated logarithm, as well as convergence with respect to the general norming factors. By establishing almost sure convergence of appropriately defined delayed sums, the proposed method yields explicit convergence rates across a wide range of probabilistic settings. As a result, many convergence problems that were previously treated in isolation can be analyzed within a single coherent theoretical structure. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)

15 pages, 851 KB

Open AccessArticle

Partially Observed Two-Phase Point Processes

by Olivier Jacquet, Walguen Oscar and Jean Vaillant

Axioms 2026, 15(1), 59; https://doi.org/10.3390/axioms15010059 - 15 Jan 2026

Viewed by 434

Abstract

In this paper, a two-phase spatio-temporal point process (STPP) defined on a countable metric space and characterized by a conditional intensity function is introduced. In the first phase, the process is memoryless, generating completely random point patterns. In the second phase, the location [...] Read more.

In this paper, a two-phase spatio-temporal point process (STPP) defined on a countable metric space and characterized by a conditional intensity function is introduced. In the first phase, the process is memoryless, generating completely random point patterns. In the second phase, the location and occurrence time of each event depend on the spatial configuration of previous events, thereby inducing spatio-temporal correlation. Theoretical results that characterize the distributional properties of the process are established, enabling both efficient numerical simulation and Bayesian inference. A statistical inference framework is developed, for the setting in which the STPP is observed at discrete calendar dates while the spatial locations of events are recorded, their exact occurrence times are unobserved, i.e., interval-censored. This partial observation scheme commonly arises in ecological and epidemiological applications, such as the monitoring of plant disease or insect pest spread across a spatial grid over time. The methodology is illustrated through an analysis of the spatio-temporal spread of sugarcane yellow leaf virus (SCYLV) in an initially disease-free sugarcane plot in Guadeloupe, FrenchWest Indies. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)

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11 pages, 624 KB

Open AccessArticle

Estimating Covariances and Goodness of Fit Plots for Accelerated Failure Time Models

by David J. Olive and Sanjuka Johana Lemonge

Axioms 2026, 15(1), 15; https://doi.org/10.3390/axioms15010015 - 25 Dec 2025

Viewed by 432

Abstract

Let the response variable Y be the time until an event such as death. Assume that there are p predictors

x_{1}, \dots, x_{p}

and that the response variable is right censored. Several survival regression models, including accelerated failure time [...] Read more.

Let the response variable Y be the time until an event such as death. Assume that there are p predictors

x_{1}, \dots, x_{p}

and that the response variable is right censored. Several survival regression models, including accelerated failure time models, have the form

Z = log (Y) = α_{Z} + x_{i}^{T} β_{Z} + e

. This paper gives a simple method for estimating the covariances

C o v (x_{i}, Z)

for some of these models. Plots are given for checking the goodness of fit of accelerated failure time models. Plots for checking the proportional hazards regression model are often harder to use. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)

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24 pages, 344 KB

Open AccessArticle

Bayesian Updating for Stochastic Processes in Infinite-Dimensional Normed Vector Spaces

by Serena Doria

Axioms 2025, 14(12), 927; https://doi.org/10.3390/axioms14120927 - 17 Dec 2025

Viewed by 611

Abstract

In this paper, we introduce a generalized framework for conditional probability in stochastic processes taking values in infinite-dimensional normed spaces. Classical definitions, based on measurability with respect to a conditioning

σ

-algebra, become inadequate when the available information is restricted to a

σ

[...] Read more.

In this paper, we introduce a generalized framework for conditional probability in stochastic processes taking values in infinite-dimensional normed spaces. Classical definitions, based on measurability with respect to a conditioning

σ

-algebra, become inadequate when the available information is restricted to a

σ

-algebra generated by a finite or countable family of random variables. In such settings, many events of interest are not measurable with respect to the conditioning

σ

-field, preventing the standard definition of conditional probability. To overcome this limitation, we propose an extension of the coherent conditioning model through the use of Hausdorff measures. The key idea is to exploit the non-equivalence of norms in infinite-dimensional spaces, which gives rise to distinct metric structures and corresponding Hausdorff dimensions for the same events. Conditional probabilities are then defined relative to families of Hausdorff outer measures parameterized by their dimensional exponents. This geometric reformulation allows the notion of conditionality to depend explicitly on the underlying metric and topological properties of the space. The resulting model provides a flexible and coherent framework for analyzing conditioning in infinite-dimensional stochastic systems, with potential implications for Bayesian inference in functional spaces. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)

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28 pages, 464 KB

Open AccessArticle

Analysis of a Retrial Queueing System Suitable for Modeling Operation of Ride-Hailing Platforms with the Dynamic Service Pricing

by Alexander Dudin, Sergei Dudin and Olga Dudina

Axioms 2025, 14(9), 714; https://doi.org/10.3390/axioms14090714 - 22 Sep 2025

Cited by 1 | Viewed by 992

Abstract

Effective operation of any service system requires optimal organization of the sharing of resources between the users (customers). To this end, it is necessary to elaborate on the mechanisms that allow for the mitigation of congestion, i.e., the accumulation of many users requiring [...] Read more.

Effective operation of any service system requires optimal organization of the sharing of resources between the users (customers). To this end, it is necessary to elaborate on the mechanisms that allow for the mitigation of congestion, i.e., the accumulation of many users requiring service. Due to the randomness of the user’s arrival process, congestions can occur even when an arrival rate is constant, e.g., the arrivals are described by the stationary Poisson process, which is assumed in the majority of existing papers. However, congestions can be more severe if the possibility of fluctuation of the instantaneous arrival rate exists. Such a possibility is an inherent feature of many systems and can be taken into account via the description of arrivals by the Markov arrival process (MAP). This makes the problem of congestion avoidance drastically more challenging. In many real-world systems, there exists the possibility of customer admission control via dynamic pricing. We propose a novel predictive mechanism of dynamic pricing. Decision moments coincide with the transition moments of the underlying process of the MAP. A customer may join or balk the system or postpone joining the system depending on the current cost. We illustrate the application of this mechanism in a multi-server retrial queueing model with dynamic service pricing. The behavior of the system is described by a multidimensional Markov chain with state-inhomogeneous transitions. Its stationary distribution is computed and may be used for solving the various problems of system revenue maximization via the choice of the proper pricing strategy. Full article

(This article belongs to the Special Issue Probability Theory and Stochastic Processes: Theory and Applications)

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