Pakistan Journal of Statistics and Operation Research

A Comprehensive Study of the Bernoulli-Transmuted Geometric Distribution: Theory and Applications

Wed, 03 Jun 2026 10:41:36 +0000

This paper presents the BerTG distribution, an innovative three-parameter discrete probability model developed by convolving a Bernoulli random variable with an independently distributed transmuted geometric random variable. The suggested distribution constitutes a significant and adaptable generalization that en- compasses various established count distributions as specific instances, thereby offering a cohesive framework for modeling a range of count data formats. The BerTG distribution is notable for its exceptional ability to handle many types of dispersion, including as overdispersion, underdispersion, and equidispersion, com- monly found in real-world count data, thereby overcoming a significant weakness of numerous conventional discrete models. A thorough examination of the distributional and structural characteristics of the BerTG model is conducted, including the probability mass function, cumulative distribution function, moments, moment generating function, factorial moments, probability generating function, and index of dispersion, among other aspects. Special emphasis is placed on reliability-theoretic attributes, encompassing the hazard rate function, survival function, reverse hazard rate function, and conditional expectation, which are metic- ulously generated and examined. Additionally, essential actuarial metrics, including the stop-loss premium, value-at-risk, and tail value-at-risk, are analysed to illustrate the model’s appropriateness for risk-theoretic applications. Model parameters are estimated by maximum likelihood estimation, and the asymptotic prop- erties of the resultant estimators are determined. A comprehensive simulation analysis is performed to assess the finite-sample performance of the estimators concerning bias, mean squared error, and consistency across diverse parameter configurations and sample sizes, thereby validating the reliability and accuracy of the estimation technique. The practical applicability of the BerTG distribution is evidenced through real-world data applications, wherein the model is applied to several empirical count datasets displaying diverse dispersion traits. Comparative analyses with various competing discrete distributions demonstrate that the BerTG model consistently attains superior goodness-of-fit performance, as indicated by standard model selection criteria such as the Akaike information criterion, Bayesian information criterion, and chi- square goodness-of-fit statistics. The results combined demonstrate that the BerTG distribution is a very competitive, manageable, and adaptable instrument for the statistical modelling of count data across several applicable fields.

Atypical Functional Brain Regions Identification in Children with Autism Spectrum Disorder Using One-Class SVM and fMRI-Derived Graph-Theoretic Parameters

Muhammad Farooq, Muhammad Kashif Saleem, Tahir Abbas — Wed, 03 Jun 2026 09:40:15 +0000

Autism spectrum disorder (ASD) is a complex neurodevelopmental condition that typically emerges in early childhood and persists throughout life, making early and objective detection crucial. This study integrates graph theoretic approach with anomaly detection approaches to identify atypical functional brain regions in children aged up to 5 years with ASD. Resting-state fMRI data of 53 ASD and 53 healthy subjects were used to construct functional connectivity matrices were across 32 functional regions of interest, from which graph theoretic features were extracted. The one-class SVM achieved an AUC of 0.733 in identifying atypical regions. Atypicality was observed in the salience network , specifically, in the supramarginal gyrus, anterior cingulate cortex and left anterior insula. In the visual network, medial occipital and laterl occipital regions were identified as atypical. The language network showed atypical regions in the right inferior frontal gyrus and the left posterior superior temporal gyrus. The dorsal attention network exhibited atypicality in the right frontal eye field region. Graph-theoretic analysis to regional atypicality highlighted disruptions in integration, segregation, and hub-related characteristics.

Evaluating Time Series Imputation Algorithms: Kalman Smoothing vs Seasonal Trend Decomposition Using LOESS on Climate Data

Cevira Dhiya Ayuni, Gustriza Erda, Nidia Mindiyarti, Felia Rizky Aulia — Wed, 03 Jun 2026 09:41:58 +0000

Missing data refers to a condition where part of the dataset is unavailable due to unrecorded information during the data collection process. Missing data problems must be addressed because they can affect the accuracy of the analysis. This study aims to evaluate the performance of univariate imputation methods, namely Kalman Smoothing and STL Decomposition, on time series data obtained from a weather station in Lampung during the period 2001–2024. This study also incorporates the application of missing data mechanisms, specifically MCAR (Missing Completely at Random), MAR (Missing at Random), and MNAR (Missing Not at Random), with missing rates of 5%, 10%, 20%, 30%, 40%, and 50%. The variables used in this analysis include average temperature, relative humidity, total rainfall, solar radiation, and wind speed. The results show that STL Decomposition is more effective and accurate than Kalman Smoothing, indicated by lower Root Mean Square Error values. This method performs better at missing rates of 5%, 10%, and 20%, and demonstrates superior performance in handling MCAR, MAR, and MNAR missing data conditions. Although Kalman Smoothing produces stable results, STL Decomposition provides more precise estimates for missing data.

Some New Double and Multiple Acceptance Sampling Plans Using the X-gamma Distribution: Theory and Evaluation with Some Applications

Wed, 03 Jun 2026 09:53:48 +0000

This study develops double and multiple (three-stage) acceptance sampling plans based on the X-gamma distribution, a flexible model widely used in reliability engineering. The plans are built around truncated life tests, meaning testing stops either after a predetermined time or once a set number of failures occur. For the two-stage (double) sampling approach, we determine the smallest practical sample sizes and calculate the average sample number (ASN). We then expand the framework to a three-stage plan, outlining how it works and specifying the sample sizes needed at each step. To evaluate performance, we derive operating characteristic (OC) curves for all plan types, showing how well they distinguish between good and poor-quality lots across different quality levels. We also identify the minimum ratio of actual to specified mean life required to keep producer’s risk within acceptable limits. Additional efficiency metrics like average total inspection (ATI) and average outgoing quality (AOQ) are calculated to give a complete picture of how each plan performs in practice. Real-life numerical examples are included to walk through how these plans guide lot-acceptance decisions using the X-gamma model. Ultimately, this work gives quality control professionals reliable, distribution-specific tools for designing smarter, more efficient acceptance sampling procedures.

A Two-Sided CUSUM Control Chart with Repetitive Sampling with Industrial Application

Ambreen Shafqat, Dr. Muhammad Aslam, Chi-Hyuck Jun — Wed, 03 Jun 2026 09:59:27 +0000

Recently, the practice of employing control charts to monitor manufacturing processes has gained significant interest in the field of statistical process control (SPC). A properly designed control chart is capable of promptly detecting any shifts in the process. In this article, a novel two-sided CUSUM control chart with repetitive sampling is introduced for monitoring the process mean. To evaluate the performance of the proposed CUSUM control chart, various statistical measures such as the average, standard deviation, and percentiles (including the median) of run lengths are utilized. These measures are assessed under different distribution scenarios. Furthermore, a comparison is made between the performance of the proposed chart and several existing control charts. By presenting the new two-sided CUSUM control chart with repetitive sampling, this article contributes to the advancement of process monitoring techniques in SPC. The evaluation of its performance using various statistical measures and comparison with existing control charts provide valuable insights into its effectiveness.

Spatial Prediction Simulation with Nonlinear Multicovariate

Geneveve Parreño-Lachica — Wed, 03 Jun 2026 10:00:55 +0000

Cokriging is a multivariate spatial method used to predict the observed value for a primary variable in an unknown location with the help of a spatially correlated secondary variable. The existence of two or more nonlinear secondary variables in predicting spatial data usually arises, especially in cokriging. Therefore, a method that can improve the model's predictive power by adding the interaction of variables is proposed. The proposed method can be effectively used, especially when the primary and secondary variables have a nonlinear relationship. By transforming the nonlinear variables, a higher correlation can be attained. This study used principal component analysis with interaction (PCAI) method among secondary variables to reduce two or more secondary variables into one dimension as a secondary variable in the cokriging technique. The proposed method was tested and verified through simulation and real data using the 2015 South Korea Air Pollution, a dataset known for its complex spatial patterns and high variability, to prove its validity and usefulness. The predicted residual error sum of squares (PRESS) statistic was used for cross-validations. Computations were done using the R Project for Statistical Computing software. PCAI as a secondary variable gives the lowest PRESS value compared to only one secondary variable or principal component analysis (PCA). Considering the criterion, the lowest value of PRESS indicates the best model. Thus, PCAI cokriging outperformed PCA cokriging. Using PCAI as a secondary variable may be a better method than PCA for cokriging with nonlinear multicovariates.

A New Bivariate Weibull Distribution: Properties and Application

Amani Abdullah Alsalafi, Saman Hanif Shahbaz, Lutfiah Ismail Al-Turk — Wed, 03 Jun 2026 10:20:32 +0000

Lifetime distributions play a key role in statistical modeling, with extensive applications across biostatistics, reliability engineering, and survival analysis. This paper introduces a novel and flexible bivariate lifetime model, termed the Bi-variate Cubic Transmuted Weibull Distribution (BCTWD), which extends the transmuted Weibull framework proposed by Alsalafi et al. (2025) by incorporating a cubic transmutation mechanism to enhance modeling flexibility and capture complex dependence structures. Existing bivariate Weibull models cannot simultaneously accommodate flexible marginal tail behavior and complex dependence structures, limiting their applicability in scenarios with heterogeneous failure patterns. The theoretical foundations of the proposed BCTWD are rigorously developed, including its joint and marginal probability density and cumulative distribution functions, along with essential statistical and reliability properties. Parameter estimation is performed using both the Maximum Likelihood (ML) and Inference Functions for Margins (IFM) methods, whose performances are systematically evaluated through simulation experiments. The simulation outcomes indicate that the estimators are, For n=200, the maximum absolute bias for shape parameters are 0.048, and the maximum MSE is 0.29, indicating satisfactory finite-sample performance, particularly for the shape parameters under heavy-tailed scenarios. An empirical application to bilateral eye failure time data from a diabetic retinopathy study demonstrates the practical utility of the proposed model. Based on the maximum likelihood estimates and model selection criteria, including AIC, AICc, and BIC, the BCTWD achieves superior goodness-of-fit compared with the Bivariate Transmuted Weibull (BTW) and Bivariate Weibull (BW) distributions. While the BCTWD exhibits slightly greater parameter variability due to its added flexibility, it provides the most accurate representation of the data, confirming its effectiveness in modeling dependent lifetimes. Overall, the BCTWD enriches the family of multivariate lifetime distributions by offering enhanced adaptability and interpretability, making it a valuable tool for applications in reliability analysis, biostatistics, and survival modeling.

A New G family of Probability Distributions: Characterizations and Applications under USA Social Security Administration Disability Beneficiaries and UK Insurance Claims Data Sets

Wed, 03 Jun 2026 10:37:10 +0000

This paper introduces the logarithmic Topp–Leone-G (LTL-G) family, a novel generalized class engineered to significantly enhance the modeling capacity of baseline distributions for complex empirical data exhibiting pronounced skewness, heavy tails, and non-monotonic hazard structures. We rigorously establish the theoretical foundations of the proposed specification, deriving explicit linear expansions for the probability density function, closed-form expressions for ordinary and incomplete moments, and comprehensive distributional characterizations based on truncated moments, reverse hazard functions, and conditional expectations. To critically evaluate inferential reliability, we designed an extensive Monte Carlo simulation protocol comparing six competing estimation methodologies including the maximum likelihood (MLE), ordinary least squares (OLS), Cramér–von Mises (CVM), Anderson–Darling (ADE), right-tail ADE (RTADE), and left-tail ADE (LTADE) across systematically varied sample sizes and challenging parameter configurations. The finite-sample performance is rigorously quantified through bias, root mean squared error, and Kolmogorov–Smirnov diagnostics, with dedicated attention to the convergence behavior of key risk indicators (KRIs). We compute Value-at-Risk (VaR), Tail Value-at-Risk (TVaR), Tail Variance (TV), Tail Mean Variance (TMV), and expected shortfall (ELq) to demonstrate the framework’s superior capacity for extreme-tail quantification under data scarcity. Empirical validation is conducted using two high-stake real-life datasets: Social Security Administration (SSA) disability beneficiary records and a UK motor non-comprehensive claims development triangle. The analytical results consistently reveal that the new family specification accurately accommodates extreme dispersion and temporal claim dependencies, while delivering a statistically rigorous foundation for modern actuarial reserving and evidence-based capital allocation.

On Estimating Reliability of a Multicomponent Stress–Strength Model for Kumaraswamy Inverse Weibull Distribution

Muhammad Qaiser Shahbaz, Saeed A. Dobbah, Ahmed Ibrahim Shawky, Khusnoor Khan — Wed, 03 Jun 2026 10:40:17 +0000

In this paper, we have discussed the reliability estimation for a multicomponent stress-strength (MCSS) model when stress and strength follow the Kumaraswamy inverse Weibull distribution, given by Shahbaz et al. (2012). We have obtained the maximum likelihood estimate of the reliability alongside the asymptotic distribution of the parameters involved. Also, the asymptotic confidence intervals have been obtained. An extensive simulation study has been conducted to assess the performance of the estimates. A real data application has also been given. It is found that the reliability increases with an increase in one of the shape parameters of stress distribution.