Advances in Symmetrical and Asymmetrical Distributions for Modern Statistics and Data Science

The study of symmetrical and asymmetrical distributions is one of the core topics of modern statistics and data science. Symmetry has traditionally provided a foundation for many classical models (most notably the normal distribution) and enables elegant mathematical formulations and powerful inferential tools. Yet in real-world applications, perfectly symmetric data are the exception rather than the rule. Asymmetry, heavy tails, skewness, multi-modality and structural deviations often emerge naturally in empirical datasets across domains such as finance, environmental sciences, engineering and medicine. This Special Issue (SI), Symmetrical and Asymmetrical Distributions in Statistics and Data Science II, aims to address this interplay by collecting recent advances that deepen the understanding of distributional structure and its implications for modeling, inference and data-driven decision-making.

The articles in this SI highlight a variety of innovative perspectives on how symmetric and asymmetric behaviors arise, how they can be characterized, and how statistical methodologies can be adapted to accommodate them. Some contributions develop new theoretical distributions or refine existing ones to better capture asymmetry, kurtosis and other features commonly encountered in practice. Others focus on estimation techniques, goodness-of-fit methods or computational strategies designed to handle data that diverge from classical assumptions. Collectively, they demonstrate that distributional flexibility is not merely a technical detail but a crucial factor that shapes the performance and reliability of statistical procedures.

This second edition of the SI reflects the growing recognition that the boundary between symmetry and asymmetry is not a binary distinction but a continuum that offers rich opportunities for methodological development. The papers included here not only expand the theoretical toolkit available to statisticians and data scientists but also highlight practical applications where distributional innovation leads to improved real-world outcomes.

The response from the scientific community to our call for papers was remarkable: 49 papers were submitted for consideration and 11 papers were finally accepted after a rigorous peer-review process. These publications are listed below, ordered by date of publication:

• Attwa, R.A.E.-W.; Radwan, T. Applying Generalized Type-II Hybrid Censored Samples on Generalized and q-Generalized Extreme Value Distributions under Linear Normalization. Symmetry 2023, 15, 1869. https://doi.org/10.3390/sym15101869.
• Chen, W.; Zhao, X.; Zhou, M.; Chen, H.; Ji, Q.; Cheng, W. Statistical Inference and Application of Asymmetrical Generalized Pareto Distribution Based on Peaks-Over-Threshold Model. Symmetry 2024, 16, 365. https://doi.org/10.3390/sym16030365.
• Srisuradetchai, P.; Niyomdecha, A. Bayesian Inference for the Gamma Zero-Truncated Poisson Distribution with an Application to Real Data. Symmetry 2024, 16, 417. https://doi.org/10.3390/sym16040417.
• Bal, A. Improving the Robustness of the Theil-Sen Estimator Using a Simple Heuristic-Based Modification. Symmetry 2024, 16, 698. https://doi.org/10.3390/sym16060698.
• Alsolmi, M.M.; Almulhim, F.A.; Amine, M.M.; Aljohani, H.M.; Alrumayh, A.; Belouadah, F. Statistical Analysis and Several Estimation Methods of New Alpha Power-Transformed Pareto Model with Applications in Insurance. Symmetry 2024, 16, 1367. https://doi.org/10.3390/sym16101367.
• Veljkovic, K. Comparing Confidence Intervals for the Mean of Symmetric and Skewed Distributions. Symmetry 2024, 16, 1424. https://doi.org/10.3390/sym16111424.
• El-Morshedy, M.; El-Dawoody, M.; El-Faheem, A.A. Symmetric and Asymmetric Expansion of the Weibull Distribution: Features and Applications to Complete, Upper Record, and Type-II Right-Censored Data. Symmetry 2025, 17, 131. https://doi.org/10.3390/sym17010131.
• Abd El-Raheem, A.E.-R.M.; Hosny, M. On the Distribution of the Random Sum and Linear Combination of Independent Exponentiated Exponential Random Variables. Symmetry 2025, 17, 200. https://doi.org/10.3390/sym17020200.
• Elshahhat, A.; Abo-Kasem, O.E.; Mohammed, H.S. The Inverted Hjorth Distribution and Its Applications in Environmental and Pharmaceutical Sciences. Symmetry 2025, 17, 1327. https://doi.org/10.3390/sym17081327.
• Filipiak, K.; Markiewicz, A.; Krajewski, P.; Ćwiek-Kupczyńska, H. Estimation and Sufficiency Under the Mixed Effects Extended Growth Curve Model with Compound Symmetry Covariance Structure. Symmetry 2025, 17, 1901. https://doi.org/10.3390/sym17111901.
• Hussain, Z.; Yeganeh, A.; Vilakati, S.; Koning, F.F.; Shongwe, S.C. Improving the Detection Ability of Binary CUSUM Risk-Adjusted Control Charts with Run Rules. Symmetry 2025, 17, 2114. https://doi.org/10.3390/sym17122114.

The remainder of this editorial contains a summary of the contributions to this SI.

Attwa and Radwan (1.) deal with parameter estimation methods for both the generalized extreme value distribution under linear normalization and its q-analogue, which offers additional flexibility through the parameter q. Using maximum likelihood estimation (MLE) within the framework of generalized type-II hybrid censored samples, the authors derive estimators and construct confidence intervals for all model parameters. A real data application and simulations demonstrate the accuracy, robustness and comparative performance of the proposed estimators.

Chen et al. (2.) introduce an estimation method for the generalized Pareto distribution within the peaks-over-threshold framework that aims to improve the accuracy of Value-at-Risk (VaR) estimation. The method extends generalized probability-weighted moments by integrating them with nonlinear weighted least squares, using exceedances and iterative weighting to better capture tail behavior. Through Monte Carlo simulations and analysis of a real heavy-tailed dataset, the proposed estimator demonstrates good performance, especially for high-confidence VaR levels.

Srisuradetchai and Niyomdecha (3.) develop Bayesian estimation procedures for the gamma zero-truncated Poisson (GZTP) and complementary GZTP distributions. They derive the Jeffreys prior for the one-parameter case and employ both informative and non-informative priors within a random-walk Metropolis framework to obtain posterior samples. Simulation studies demonstrate that Bayesian estimators outperform MLE, especially for small samples, and that Bayesian credible intervals offer higher coverage and shorter average lengths than traditional Wald intervals.

Bal (4.) proposes the Robustified Theil–Sen (RTS) estimator, a modification of the classical Theil–Sen regression method designed to enhance robustness against asymmetric and influential outliers. By incorporating a heuristic selection procedure, the RTS approach reduces the number of initial estimates contaminated by outliers. Numerical experiments demonstrate that RTS consistently outperforms both the original Theil–Sen and the repeated median estimators in robustness. The findings further indicate that the RTS estimator achieves a breakpoint comparable to that of high-breakdown robust estimators.

Alsolmi et al. (5.) discuss a three-parameter distribution derived via an alpha power transformation, which offers higher flexibility for modeling skewed, complex, symmetric and asymmetric data. The authors study key distributional properties and derive multiple parameter estimation methods, including MLE-based and distance-based estimators. A Monte Carlo simulation assesses the performance of these estimators and two insurance applications demonstrate the model’s strong empirical fit.

Veljkovic (6.) is concerned with confidence intervals for the population mean that account for both skewness and kurtosis using Edgeworth expansion, addressing challenges in highly skewed distributions. The author compares these intervals with existing methods across a variety of symmetric and skewed distributions through Monte Carlo simulations. Performance is evaluated in terms of coverage probability, mean interval length and variability of interval length. The results show that the proposed Edgeworth-based bootstrap intervals consistently provide better results than the alternatives.

El-Morshedy et al. (7.) consider the Odd Flexible Weibull–Weibull (OFW-W) distribution, a three-parameter lifetime model capable of modeling both symmetric and asymmetric data with diverse hazard rate shapes. Key properties, including moments, quantiles, reliability and entropies, are explored, and parameters are estimated using eight methods evaluated via simulation. Applications to four real datasets and comparisons with five existing distributions demonstrate the OFW-W model’s high flexibility and effectiveness across various data types.

Abd El-Raheem and Hosny (8.) develop saddlepoint approximations for the distribution of random sums, linear combinations and the reliability index $R=P(X_2<X_1)$ for exponentiated exponential variables with unequal scale parameters. These approximations fill gaps in the existing literature where exact distributions were previously unknown. Numerical studies demonstrate that the proposed method is both accurate and computationally efficient compared to traditional simulation approaches.

Elshahhat et al. (9.) introduce the inverted Hjorth (IH) distribution, a three-parameter asymmetric model capable of fitting positively skewed datasets with inverted bathtub-shaped hazard rates. Key properties are derived, and parameters are estimated using both MLE and Bayesian methods. Interval estimation is performed via asymptotic confidence intervals and Bayesian credible intervals, and type-II censoring is considered to assess incomplete data. Simulation studies and practical applications demonstrate the utility of the IH model in reliability and survival analysis.

Filipiak et al. (10.) study an extended growth curve model that incorporates both fixed and random effects and derive MLEs for the fixed effects and dispersion matrix under multivariate normality. Estimation is considered both with and without assumptions on the covariance structure, including the case of compound symmetry, using rules for differentiating symmetric matrices. In balanced complete block designs, closed-form expressions for the estimators are obtained, and the sufficiency of the statistics for fixed effects is shown to extend to models with random nuisance parameters.

Hussain et al. (11.) propose a run-rules method integrated with a risk-adjusted cumulative sum control chart designed for monitoring binary clinical outcomes, such as post-surgery mortality. Their approach exhibits symmetry in evaluating deviations from expected performance, allowing balanced detection of both improvements and deteriorations in clinical results. Simulation studies based on the average run length confirm its robustness and consistent behavior under beta-distributed conditions, with a practical example illustrating its real-world applicability.

Finally, we hope that this collection will inspire further research into flexible distributional models, encourage deeper examination of asymmetry in empirical data and foster collaboration between theoretical and applied communities. We would like to thank all contributing authors for sharing their insights and advancing the discussion in this SI. The diversity of contributions demonstrates the vitality of this area and its central role in the evolving landscape of statistics and data science.