Pakistan Journal of Statistics and Operation Research

The Kth-Order Equilibrium Rayleigh Distribution: Characterization and Estimation

Nuzhat Ahad — 2026-03-03

This paper introduces a new extension of Rayleigh distribution named as the Kth-order equilibrium Rayleigh distribution (KERD), by employing the concept of Kth order equilibrium method. Various statistical properties of the new distribution, including its aging behavior and stochastic ordering relations, are analyzed. Explicit expressions are derived for moments, conditional moments, incomplete moments, the mean residualfunction, the mean waiting function, entropy measures, and order statistics. Distribution characterization has been examined. Maximum likelihood estimation method is used to estimate the parameters. A simulation study using the Anderson–Darling test statistic is carried out to analyze the asymptotic behavior of maximum likelihood estimators. The behaviors of bias and mean square error are observed with the increase in sample size. The applications of new distribution are demonstrated using two different real life datasets. Ultimately, a comparison is conducted among
KERD and its sub-models regarding their fit using Information Criterion tools.

On the Uniqueness and Structural Identification of Some Univariate Continuous Probability Distributions

Partha Jyoti Hazarika — 2026-03-03

This paper examines the characterizations of five recent univariate continuous probability distributions (2022-2025) that were proposed relatively recently. These characterizations are based on: (i) a simple relationship
between two truncated moments; (ii) reverse hazard function. It should be mentioned that for the characterization
(i) the cumulative distribution function need not have a closed form and depends on the solution
of a first order differential equation, which provides a bridge between probability and differential equation

A closed-form estimator of R = P(X < Y) based on ranked set sampling for a family of statistical distributions with application in agriculture

Hossein Pasha-Zanoosi — 2026-03-03

In this article, we will derive a closed-form estimator for the probability R = P (X < Y) based on a ranked set sampling (RSS) scheme when the he random variables X and Y are assumed to follow the Lehmann Type-II (L-II) family of distributions. Estimating R through the maximum likelihood (ML) method within the RSS framework does not yield an analytical solution because of the non-linear components present in the likelihood equations. In this context, we employ a modified maximum likelihood (MML) estimation approach to derive a closed-form estimator for R. Estimates of R under both ML and MML techniques along with their corresponding asymptotic confidence intervals are determined and compared in a simulation study under one of the distributions of the L-II family called the inverse Topp-Leone distribution. At the end, the simulation results are strengthened using a real example in the field of agriculture.

Parameter Estimation for the Bivariate Compound Zero-Truncated Poisson-Gamma Model under Different Data Scenarios

Amal Alhejaili — 2026-03-04

The bivariate compound zero-truncated Poisson-gamma distribution models the sum of a random number of bivariate Gamma variables, where the count follows a zero-truncated Poisson distribution, which makes it well-suited for applications in actuarial science, climatology, and reliability engineering, where zero outcomes are inherently absent. Owing to the intractable nature of the probability density function, which involves an infinite sum, the direct maximum likelihood estimation is computationally challenging. In this study, we used a standard (exact) maximum likelihood estimation when event counts were observed (complete data and Scenario~A) and employed the saddle-point approximation only when counts were latent (Scenario~B). We developed a stable maximum likelihood estimation based on saddle-point approximation. We derived the cumulative distribution function from the cumulant generating function and obtained the probability density function using numerical differentiation. Detailed derivations, implementation guidelines in the \textsf{R} programming language, and a parameter initialization strategy using the method of moments are provided. A simulation study using various sample sizes demonstrated the accuracy, consistency, and superiority of this method over the moment-based estimators. Computational challenges and limitations are discussed, along with potential extensions to model the dependence structures using copulas. In addition, we develop a likelihood ratio test and a formal symmetry test (for example, $H_0:\alpha_1=\alpha_2,\ \beta_1=\beta_2$) to compare nested specifications, enabling principled inference on symmetry and overall model adequacy.

An Alternative Exponential Model for Skewed Real Data: Characterizations, Bayesian, Non- Bayesian Estimation and Distributional Validation Testing

Mohamed Ibrahim Mohamed — 2026-03-04

This paper presents a novel exponential model with two parameters, placing particular attention on its practical applications to skewed data as the central area of investigation. The mathematical characteristics of this atypical distribution are established, in a lucid and succinct manner, by the discoveries made in this investigation. Furthermore, it is worth noting that there exist three distinct approaches to describing the distribution. The process of estimating the parameters of the novel model involves employing a range of established methodologies, including the Bayesian technique. When confronted with censored data, the maximum likelihood technique is commonly considered as a viable approach. Pitman's closeness criteria areemployed as the comparative tool when assessing the probability estimate in relation to Bayesian estimation approaches. During the computation of Bayesian estimations, three distinct loss functions, namely generalized quadratic, Linex, and entropy, are employed. A multitude of simulated experiments are conducted to assess the efficacy of various estimation methodologies. The BB algorithm is employed to facilitate the comparison and contrast between the Bayesian technique and the censored maximum likelihood strategy. The Nikulin-Rao-Robson (NKRR) statistic was derived by conducting two empirical studies using real-world data sets characterized by skewed distributions, along with simulation research conducted in an unfiltered environment. Furthermore, this paper delineates two other uses within the same context. The study's findings illustrate the efficacy of the approaches presented for the purposes of distribution and estimation.

A Novel Chen Extension for Risk Analysis with MOOP and PORT-VAR Assessments under Hydrological Flow Data and Financial Case Study

Mohamed Ibrahim — 2026-03-04

This paper introduces a new extension of the Chen distribution, designed to better model extreme low-flow events in hydrology and rare events in the medical field. The proposed model incorporates asymmetrical and heavy-tailed behavior, making it particularly useful for analyzing extreme values in complex real datasets. We derive the mathematical properties of the BGC distribution and apply two advanced analytical techniques: the Mean-of-Order-P (MOOP) method to determine the optimal value of P (referred to as Opt-P), and the Peaks Over Threshold Value-at-Risk (PORT-VaR) approach to identify and assess critical extreme events. These methods are applied to real datasets including relief times, minimum river flow data from the Cuiabá River, and U.S. indemnity losses from general liability claims. The MOOP analysis shows that increasing the order P leads to reduced Mean Squared Error (MSE) and Bias, indicating improved estimation accuracy. For example, in the relief times dataset, MSE decreases from 0.64 at P=1 to 0.3844 at P=5. Similarly, for the minimum flow data, MSE drops from 4402.88 to 3684.27 with increasing P, highlighting the benefits of higher-order statistics in capturing central tendencies. Using PORT-VaR, we analyze extreme peaks under varying confidence levels (50%, 70%, 90%, and 99%) and compute key risk indicators such as Value-at-Risk (VaR) , Tail Value-at-Risk (TVaR) , Mean Excess Loss (MEXL) , Tail Variance (TV) , and Tail Mean Variance (TMV) . In the relief times dataset, VaR increases from 1.70 at 50% confidence to 3.055 at 99% confidence, demonstrating growing risk exposure at higher confidence levels. For the minimum flow data, VaR rises from 115.925 at 50% to 157.169 at 99%, underscoring the importance of adaptive risk thresholds in managing water scarcity and dam safety. A financial case study using U.S. indemnity loss data further validates the robustness of the BGC model in capturing tail behavior and estimating extreme risks. At the 99% confidence level, VaR reaches 170400 (in thousands of USD), and MEXL is 203411, illustrating the nonlinear growth of risk in heavy-tailed insurance claims. Finally, a comparative study under a historical financial claims data through an application.

Identifying an Emerging HIV Epidemic in Punjab, Pakistan: Forecasting Trends using Prophet Model and Classical ARIMA Model (2020–2025)

Iqra Hamid Khan — 2026-03-04

Pakistan has witnessed concerning shifts in HIV epidemic especially in Punjab, where HIV and AIDS incidence continues to rise. This study compares the predictive accuracy of the Prophet model, machine learning model with classical ARIMA configurations for monthly HIV and AIDS case forecasting in Punjab. Methods: Monthly surveillance data (January 2020–October 2025) from Punjab AIDS Control Program (PACP) was used to train and validate Prophet and multiple ARIMA models. The modelling performance was assessed using RMSE, MAE, MAPE, BIC and also Ljung Box Q tests. Forward forecasts were generated for HIV reactive and AIDS (CD4 < 200) cases through 2026. Results: Machine learning model (Prophet) outperformed all ARIMA models in forecasting HIV reactive cases by achieving the lowest RMSE (132.6) and MAPE (16.4%), for AIDS cases projection, all models exhibited high error rates (Prophet MAPE > 300%) with ARIMA (0,1,0)(0,1,1)₁₂ better performance (MAPE ~174%). Forecasted outputs estimates approximately 8,490 new HIV cases in 2026 with uncertainty bounds reaching nearly 15,000 cases, indicating a continued upward trajectory and for AIDS the count in 2026 may rise to 25,596 new cases, thou, forecasting AIDS remains a challenge. The results demonstrate superior ability of Prophet model to capture non-linear trends and seasonality in HIV surveillance data.
Conclusion: Prophet model superior performance reflects its ability to model nonlinear and seasonally irregular HIV surveillance data. Integration of machine learning techniques such as Prophet model into provincial HIV programs can enhance planning and accelerate progress toward achieving UNAIDS 95-95-95 targets.