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A Special Issue of Mathematics (ISSN 2227-7390) belonging to the section "D1: Probability and Statistics".

Deadline for manuscript submissions: 31 October 2026 | Viewed by 2935

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Prof. Dr. Fernando Rojas

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Centro de Micro-Bio Innovación, Escuela de Nutrición y Dietética, Facultad de Farmacia, Universidad de Valparaíso, Gran Bretaña 1093, Valparaíso 2340000, Chile
Interests: applied probability; mathematical statistics; stochastic processes; statistical modeling; high-dimensional data analysis; penalized and regularized methods; statistical learning; uncertainty quantification; resampling and bootstrap methods; Bayesian and frequentist inference; causal and predictive modeling; applied mathematics

Special Issue Information

Dear Colleagues,

Applied probability and statistics constitute a fundamental pillar of modern mathematical research, with growing relevance across scientific and engineering disciplines. The increasing availability of complex, high-dimensional, and structured data has stimulated the development of new probabilistic models and statistical methodologies that extend classical frameworks and address contemporary analytical challenges.

This Special Issue aims to provide a forum for recent advances in applied probability and statistics; it will emphasize both theoretical developments and methodological innovations with practical relevance. We welcome contributions that propose new probabilistic models, statistical inference techniques, or computational methods, as well as studies that apply established approaches in novel or challenging contexts.

Topics of interest include, but are not limited to, the following: stochastic processes, probabilistic modeling, statistical learning, high-dimensional inference, regularization and variable selection, resampling and bootstrap methods, uncertainty quantification, Bayesian and frequentist inference, and modern approaches to prediction and modeling. Contributions addressing complex data structures such as dependence, heterogeneity, or non-standard sampling schemes are particularly encouraged.

Applications may arise in diverse fields, including, but not limited to, data science, engineering, economics, environmental sciences, public health, and decision-support systems. By bringing together methodological and applied contributions, this Special Issue seeks to highlight the versatility of probabilistic and statistical tools for addressing complex real-world problems while maintaining strong mathematical foundations.

We invite researchers to submit original research articles and comprehensive reviews that will advance the theory and application of probability and statistics in contemporary scientific research.

Prof. Dr. Fernando Rojas
Guest Editor

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Keywords

applied probability
mathematical statistics
stochastic processes
statistical modeling
high-dimensional data
statistical learning
uncertainty quantification
resampling methods
Bayesian and frequentist inference
applied mathematics

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Published Papers (7 papers)

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Research

16 pages, 288 KB

Open AccessArticle

Behavior of Galton–Watson Processes on the Brink of Extinction

by Junping Li and Yicen Mao

Mathematics 2026, 14(18), 3253; https://doi.org/10.3390/math14183253 - 8 Sep 2026

Viewed by 243

Abstract

This paper investigates the stochastic behavior of Galton–Watson processes at the brink of extinction. Let

{Z_{n} : n \geq 0}

denote a Galton–Watson process. We define the brink time of extinction as [...] Read more.

This paper investigates the stochastic behavior of Galton–Watson processes at the brink of extinction. Let

{Z_{n} : n \geq 0}

denote a Galton–Watson process. We define the brink time of extinction as

τ = inf \{n - 1 : Z_{n} = 0\}

, and set

S_{n} = \sum_{k = 0}^{n} Z_{k}

. We study the joint probability distributions and associated properties of the random pairs

(τ, Z_{τ})

) and

(τ, S_{τ})

. We further derive the joint distribution linking the brink time of extinction to the total number of particles that produce no offspring throughout their lifetimes. By constructing iterative bivariate probability generating functions, we obtain exact joint probability generating functions, probability mass functions and expectations for these pairs in the subcritical and critical regimes. These exact results complement the classical marginal laws of total progeny and leaf counts, as well as the asymptotic joint limit theorems available in the literature. Full article

(This article belongs to the Special Issue Applied Probability and Statistics: Theory, Methods, and Applications)

42 pages, 1058 KB

Open AccessArticle

The Bivariate Inverted Topp–Leone Distribution: Distributional Properties and Statistical Inference

by Daya K. Nagar, Edwin Zarrazola and Santiago Echeverri-Valencia

Mathematics 2026, 14(17), 3199; https://doi.org/10.3390/math14173199 - 4 Sep 2026

Viewed by 150

Abstract

In this article, we propose a new bivariate generalization of the inverted Topp–Leone distribution. First, we express its joint cumulative distribution and survival functions in series forms using special functions, which yields the bivariate hazard rate, reversed hazard rate, and the exact distributions [...] Read more.

\min {X, Y}

. We then establish several essential properties, such as marginal and conditional distributions, joint moments, entropy, and the Fisher information matrix. After proving that the distribution exhibits positive likelihood ratio dependence, we derive the exact distributions for the transformations

X + Y

X / (X + Y)

, and

X Y

when X and Y follow an inverted bivariate Topp–Leone distribution. Finally, we round out the statistical framework by presenting parameter estimation techniques, a simulation study, and Bayesian inference. Full article

(This article belongs to the Special Issue Applied Probability and Statistics: Theory, Methods, and Applications)

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40 pages, 907 KB

Open AccessArticle

A Parsimonious Quadratic-Exponential Submodel of the Kummer–Beta-G Family: Properties and Regression Modeling

by Shaykhah Aldossari, Hugo S. Salinas, Hassan S. Bakouch and Çağatay Çetinkaya

Mathematics 2026, 14(17), 3102; https://doi.org/10.3390/math14173102 - 28 Aug 2026

Viewed by 253

Abstract

Bounded continuous data arise in many areas of applied probability and statistics, particularly when the underlying variable is restricted to a finite interval and the density is expected to vanish at the endpoints. This paper studies a parsimonious fixed-shape submodel of the Kummer–beta-G family, referred to as the asymmetric quadratic-exponential bounded generator model, abbreviated as AQEB-G. The model is obtained by fixing the two beta shape parameters at

a = b = 2

, which yields the unit quadratic-exponential kernel

w_{β} (u) = u (1 - u) exp (- β u)

0 < u < 1

, where

β \in R

is a dimensionless tilting parameter. Composing its normalized distribution function with an absolutely continuous baseline cdf G produces the corresponding fixed-shape Kummer–beta-G specialization on the support inherited from G. The bounded AQEB distribution on

(0, α)

is obtained by using the uniform baseline

G (x; α) = x / α

and the endpoint-scale parametrization

β = λ α

. We derive the cumulative distribution, density, survival, hazard and reversed hazard functions. We also show that, after standardization, the bounded AQEB model is exactly the natural exponential tilt of a beta

(2, 2)

distribution, and, hence, its version on

(0, α)

is a scaled exponentially tilted beta

(2, 2)

model. Several mathematical properties are obtained, including ordinary and incomplete moments, generating functions, entropy measures, Lorenz and Bonferroni curves, shape properties, stochastic representations and ordering results. Likelihood-based inference is also discussed with attention to the support-dependent endpoint parameter. The finite-sample behavior of the estimators is examined through a Monte Carlo simulation study, and two empirical applications illustrate the practical use of the AQEB model. In the examples considered, the AQEB model performs competitively relative to the evaluated alternatives according to the reported likelihood-based criteria, goodness-of-fit statistics, and residual diagnostics. Full article

(This article belongs to the Special Issue Applied Probability and Statistics: Theory, Methods, and Applications)

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33 pages, 601 KB

Open AccessArticle

Phase-Tagged Fluctuation Analysis of Cumulative Shock Reliability Systems with Phase-Type Inter-Shock Times

by Lotfi Tadj

Mathematics 2026, 14(11), 1920; https://doi.org/10.3390/math14111920 - 1 Jun 2026

Cited by 2 | Viewed by 358

Abstract

We develop a closed-form analysis of the joint distribution for cumulative shock reliability systems with phase-type inter-shock times. The analytical literature on shock-driven reliability has hitherto been split into two largely separate traditions: scalar fluctuation theory, which delivers closed-form joint distributions of pre-failure and failure-time observables but cannot accommodate matrix phase structure; and matrix-analytic methods, which handle phase-type dynamics naturally but focus on stationary indicators rather than first-passage distributions. We bridge these traditions by introducing a matrix-valued reliability functional

Φ_{ν} (ξ, u, v, ϑ, θ)

that encodes the joint distribution of the failure index, pre-failure damage and time, failure-time damage and time, and the operational phase at the moment of failure. We derive

Φ_{ν}

in closed form via Sherman–Morrison reduction of the matrix Laplace–Stieltjes transform together with the Dshalalow

D

-operator, and establish a span-reduction theorem showing that

Φ_{ν}

lies in a three-dimensional matrix subspace generated by the identity and two matrix LSTs. The functional simultaneously generalizes the scalar fluctuation functional of Dshalalow and White and the phase-tagged first excess functional of Tadj, recovering both as projections. We extract twelve closed-form reliability indices, including the reliability function, mean time to failure, mean overshoot, joint pre-failure and failure transforms, and, new to the cumulative shock literature, the phase distribution at failure and the phase-resolved failure-time distribution. Two structural identities of Wald type emerge as corollaries. The framework reduces to elementary arithmetic for rational model primitives and is verified against

2 \times 10^{5}

Monte Carlo trajectories in a worked example. Full article

(This article belongs to the Special Issue Applied Probability and Statistics: Theory, Methods, and Applications)

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18 pages, 873 KB

Open AccessFeature PaperArticle

The Touchard Process for Count Data with Dependent Increments

by Moisés Lima, Gladston Da Silva, Regina Da Fonseca and Raul Matsushita

Mathematics 2026, 14(11), 1798; https://doi.org/10.3390/math14111798 - 22 May 2026

Viewed by 318

Abstract

This paper introduces the Touchard process, a flexible two-parameter stochastic framework for modeling count data that depart from the classical Poisson assumptions. In contrast to standard Poisson processes, the proposed model allows for both nonstationary and dependent increments, enabling the representation of overdispersion, underdispersion, and temporal dependence within a unified structure. The main contribution lies in extending weighted Poisson models to a stochastic-process setting through recursively defined transition probabilities associated with Touchard marginal distributions. We derive key theoretical properties, including admissibility conditions and a recursive formulation for the transition probabilities, and propose an efficient simulation algorithm. Maximum likelihood estimation is developed for parameter inference, and a likelihood ratio framework is used for model comparison. An empirical application to motor vehicle crash data illustrates the ability of the model to capture dynamic patterns that are not adequately described by classical Poisson-based approaches. Full article

(This article belongs to the Special Issue Applied Probability and Statistics: Theory, Methods, and Applications)

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32 pages, 594 KB

Open AccessArticle

Design-Aware Predictive and Causal Modeling of Cardiovascular Risk in Chronic Kidney Disease Using Penalized and Double Machine Learning Approaches

by Fernando Rojas, Axa Tapia and Hilda Espinoza

Mathematics 2026, 14(9), 1554; https://doi.org/10.3390/math14091554 - 4 May 2026

Viewed by 519

Abstract

We develop a design-aware framework that combines penalized prediction and causal inference for finite populations observed through complex survey designs. The framework integrates survey-weighted pseudo-likelihoods,

ℓ_{1}

-penalized estimation, Neyman-orthogonal moment functions, and a bootstrap procedure that resamples primary sampling units within strata. [...] Read more.

ℓ_{1}

-penalized estimation, Neyman-orthogonal moment functions, and a bootstrap procedure that resamples primary sampling units within strata. Methodologically, the contribution is an explicit pipeline that supports design-based inference while separating predictive associations from structurally adjusted effects in high-dimensional, clustered data. We illustrate the framework using data from the Chilean National Health Survey (ENS) 2016–2017 to study the relationship between chronic kidney disease (CKD) and high cardiovascular (CV) risk. In the ENS adult population, the survey-weighted prevalence of CKD was 3.1% (95% CI: 2.4–3.8), and the prevalence of high CV risk was 23.9% (95% CI: 21.5–26.3). High CV risk was markedly more frequent among individuals with CKD than among those without CKD (90.9% versus 21.5%). Predictive and associational analyses combined survey-weighted penalized logistic regression (LASSO) with refitted unpenalized models. In conventional survey-weighted logistic regressions, CKD showed a strong association with high CV risk (odds ratio = 5.66; 95% CI: 2.71–11.82;

p < 0.001

), and effect sizes remained stable after LASSO-based variable selection. To assess causal relevance under confounding and potential endogeneity, we implemented two endogeneity-aware estimators: two-stage residual inclusion (2SRI) and double/debiased machine learning (DML). The DML estimator, defined as the primary causal estimand, reports an orthogonalized estimate of the average treatment effect of CKD on the probability of high CV risk. After adjustment for age and major cardiometabolic comorbidities, the DML estimate was attenuated and statistically non-significant (average treatment effect =

- 0.094

; 95% CI:

[- 0.409, 0.220]

). The 2SRI approach yielded unstable estimates with wide confidence intervals, consistent with the limited effective sample size of CKD cases (

n_{C K D} \approx 190

in a sample with n ≈ 6233) and weak identification conditions under low-prevalence settings. Simulation experiments under ENS-like complex sampling suggest that naive predictive associations may overestimate the structural contribution of CKD under confounding, whereas orthogonalized estimators yield more conservative estimates when identification holds. The causal interpretation relies on a conditional mean independence assumption given observed covariates and survey design, while control-function specifications are treated as diagnostic sensitivity analyses due to the absence of credible exclusion-based instruments. Overall, the results demonstrate a fundamental divergence between predictive relevance and causal importance in finite-population settings, underscoring the need for design-aware and endogeneity-robust methods in statistical modeling. Full article

(This article belongs to the Special Issue Applied Probability and Statistics: Theory, Methods, and Applications)

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29 pages, 3654 KB

Open AccessArticle

The Baker Type-I Model: Theory, Comprehensive Inference, and Empirical Evidence from Complex Reliability and Biomedical Data

by Ohud A. Alqasem and Ahmed Elshahhat

Mathematics 2026, 14(9), 1419; https://doi.org/10.3390/math14091419 - 23 Apr 2026

Cited by 1 | Viewed by 430

Abstract

Recently, two novel extensions of the Weibull distribution have been introduced through Manly’s exponential transformation, offering a flexible mechanism for modeling skewness, tail behavior, and complex hazard rate structures. In this study, we develop a comprehensive theoretical and inferential framework for one of these models, referred to as the Baker–T1 distribution, to establish it as a mature and practically viable lifetime model for reliability and survival analysis. While the Baker–T1 model exhibits remarkable flexibility in capturing skewness, tail behavior, and complex hazard rate shapes, its statistical properties and practical performance have not yet been systematically investigated. To bridge this gap, we derive a wide range of fundamental distributional characteristics, including reliability measures, hazard and reversed-hazard functions, quantiles, moments, skewness, kurtosis, dispersion indices, and order statistics, establishing the model’s analytical tractability and structural richness. An extensive inferential framework is introduced by implementing eight classical estimation techniques, and their finite-sample behavior is rigorously examined through a large-scale Monte Carlo simulation study under diverse parameter configurations. The practical relevance of the Baker–T1 model is further demonstrated using two genuine datasets from biomedical and engineering domains, where it consistently outperforms thirteen competing lifetime distributions according to likelihood-based and information-theoretic criteria. Full article

(This article belongs to the Special Issue Applied Probability and Statistics: Theory, Methods, and Applications)

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