Pakistan Journal of Statistics and Operation Research <p>Pakistan Journal of Statistics and Operation Research started in 2005 with the aim to promote and share scientific developments in the subject of statistics and its allied fields. Initially, PJSOR was a bi-annually double-blinded peer-reviewed publication containing articles about Statistics, Data Analysis, Teaching Methods, Operational Research, Actuarial Statistics, and application of statistical methods in a variety of disciplines. Because of the increasing submission rate, the editorial board of PJSOR decided to publish it on a quarterly basis from 2012. Brief chronicles are overseen by an <a title="PJSOR Editorial Board" href="">Editorial Board</a> comprised of academicians and scholars. We welcome you to <a title="Submissions" href="">submit</a> your research for possible publication in PJSOR through our online submission system. <strong>Publishing in PJSOR is absolutely free of charge (No Article Processing Charges)</strong>.<br><a href=""><strong>ISSN : 1816 2711</strong></a>&nbsp; &nbsp;<strong>|&nbsp; &nbsp;<a href="">E- ISSN : 2220 5810</a></strong></p> College of Statistical and Actuarial Sciences en-US Pakistan Journal of Statistics and Operation Research 1816-2711 <p><strong>Authors who publish with this journal agree to the following License</strong></p> <p><strong><a href=""><img class="alignleft" src="" width="118" height="41"></a><a href="">CC BY</a>:&nbsp;</strong>This license allows reusers to distribute, remix, adapt, and build upon the material in any medium or format, so long as attribution is given to the creator. The license allows for commercial use.</p> <p>&nbsp;</p> Sample size determination when the parameter of interest is the coefficient of variation under normality for the data <p>This study considers classical and Bayesian inference approaches for the coefficient of variation under normality for the data, especially on the determination of the sample size of a random sample needed in a second stage of an experiment. This topic has been explored by many authors in the last decades. The first goal of the study is to present simple formulations to get the inferences of interest for the coefficient of variation under normality and usual frequentist approach based on the asymptotic normality of the maximum likelihood estimators for the mean and standard deviation of the normal distribution and using the delta method to get the inferences of interest for the coefficient of variation. Simple hypothesis tests and determination of the sample size are discussed under the frequentist approach.The second goal of the study is to present a sample size determination under a Bayesian approach, where it is assumed a Jeffreys non-informative prior distribution of the parameters of the normal distribution assumed for the data and using standard Markov Chain Monte Carlo (MCMC) methods to get the posterior summaries of interest.</p> Jorge Alberto Achcar Emerson Barili Copyright (c) 2024 Pakistan Journal of Statistics and Operation Research 2024-06-05 2024-06-05 157 170 10.18187/pjsor.v20i2.4240 The Topp-Leone-Gompertz-G Power Series Class of Distributions with Applications <pre>A new generalized class of distributions called the Topp-Leone-Gompertz-G Power Series (TL-Gom-GPS) distribution is presented. Some mathematical and statistical properties of the new class of distributions are explored. For this new class of distributions, we derived the quantile function, moments and generating function, probability weighted moments, distribution of the order statistics and R\'enyi entropy. The maximum likelihood technique is used for estimating model parameters and Monte Carlo simulation is conducted to show the performance of the proposed model. Finally, the usefulness and flexibility of the new class of distributions is examined by means of applications to real data sets.<br><br></pre> Morongwa Gabanakgosi Broderick Oluyede Copyright (c) 2024 Pakistan Journal of Statistics and Operation Research 2024-06-05 2024-06-05 171 195 10.18187/pjsor.v20i2.4032 A new three-parameter discrete distribution to model over-dispersed count data <p>A novel discrete distribution with three parameters, referred to as the PoiNB distribution, is formulated through the convolution of a Poisson variable and an independently distributed negative binomial random variable. This distribution generalizes some well known count distributions and can be used for modelling over-dispersed as well as equi-dispersed count data. Numerous essential statistical properties of this proposed count model are thoroughly examined. Characterizations of this distribution in terms of conditional expectation and reverse hazard rate function are studied in detail. The estimation of the unknown parameters of this proposed distribution is carried out using the maximum likelihood estimation approach. Additionally, we introduce a count regression model based on the PoiNB distribution through the generalized linear model approach. Through two real-life modelling applications, it is demonstrated that the suggested distribution may offer practical utility for practitioners in modelling over-dispersed count data.</p> Anupama Nandi Partha Jyoti Hazarika Aniket Biswas G. G. Hamedani Copyright (c) 2024 Pakistan Journal of Statistics and Operation Research 2024-06-05 2024-06-05 197 215 10.18187/pjsor.v20i2.4554 A new class of probability distributions with an application in engineering science <p>In this manuscript, we introduced a new class of probability distributions called new exponentiated transformation(NET) that adds more flexibility to any baseline distribution without adding the complexity of an extra parameter. NET is then specialised on exponentiated exponential distribution and a new exponentiated exponential( NEE) distribution is obtained. The NEE distribution has wider flexibility in terms of density function and also has increasing, decreasing and bathtub hazard rate function. Several mathematical properties of NEE distribution are also highlighted. For applicability of proposed distribution, two engineering data sets are considered and it is sensed that NEE leads to a better fit than all models taken under consideration</p> Anwar hassan I. H Dar M. A Lone Copyright (c) 2024 Pakistan Journal of Statistics and Operation Research 2024-06-05 2024-06-05 217 231 10.18187/pjsor.v20i2.3845 A New Family of Heavy-Tailed Generalized Topp-Leone-G Distributions with Application <p>In this article, we introduce a robust generalization of the generalized Topp-Leone-G (GEN-TL-G) family of distributions via the heavy-tailed technique. The distribution is named heavy-tailed generalized Topp-Leone-G (HT-GEN-TL-G) family of distributions. Statistical properties of the HT-GEN-TL-G family of distributions including reliability functions, quantile function, density expansion, moments, moment generating function, incomplete moments, Rényi entropy, distribution of order statistics are derived. Different estimation methods including Maximum Likelihood, Anderson-Darling, Ordinary Least Squares, Weighted Least Squares, Cram\'er-von Mises and Maximum Product of Spacing are utilized to estimate the unknown parameters of the new distribution, and a simulation study is used to compare the results of the estimation methods. Risk measures for this distribution were also developed and finally the effectiveness of this new family of distributions was demonstrated using applications to two real data sets.</p> Thatayaone Moakofi Broderick Oluyede Bakang Tlhaloganyang Agolame Puoetsile Copyright (c) 2024 Pakistan Journal of Statistics and Operation Research 2024-06-05 2024-06-05 233 260 10.18187/pjsor.v20i2.4458 A discrete claims-model for the inflated and over-dispersed automobile claims frequencies data: Applications and actuarial risk analysis <p>This paper showcases the effectiveness of the discrete generalized Burr-Hatke distribution in analyzing insurance claims data, specifically focusing on scenarios with over-dispersed and zero-inflated claims. Key contributions include presenting foundational statistical theories with mathematical proofs to enrich the paper’s mathematical and statistical aspects. Through the application of this discrete distribution, the study conducted a thorough risk analysis across five diverse sets of insurance claims data, evaluating critical risk indicators at specified quantiles. These indicators provided detailed insights into potential losses across different risk levels, supporting effective risk management strategies. The research emphasizes the importance of selecting appropriate probability distributions when analyzing zero-inflated data, as commonly observed in insurance claims. The discrete distribution accommodated these unique data characteristics and facilitated a robust analysis of risk metrics, enhancing the accuracy of potential loss assessments and reducing associated uncertainties. Furthermore, the study highlights the practical relevance of the discrete distribution in addressing specific challenges inherent to insurance claims data. By leveraging this distribution, insurers and risk analysts can improve their risk modeling capabilities, leading to more informed decision-making and enhanced financial exposure management.</p> Haitham M. Yousof Mohammad Mehdi Saber Abdullah H. Al-Nefaie Nadeem Shafique Butt Mohamed Ibrahim Salwa L. Alkhayyat Copyright (c) 2024 Pakistan Journal of Statistics and Operation Research 2024-06-05 2024-06-05 261 284 10.18187/pjsor.v20i2.4535 Approximation Methods for the Bivariate Compound Truncated Poisson Gamma Distribution <p>In certain situations probability computations are required for some complex distributions; like a compound distribution. This can leads to some comptational complexities. In such situations, the problem can be simplified by using some approximation techniques like the “saddle-point” approximation. In this paper, we have first proposed a compound bivariate distribution; namly the bivariate compound truncated Poisson-Gamma distribution; by compounding the zero truncated Poisson distribution with independent Gamma variates. The bivariate saddle-point approximation for the distribution function of the proposed distribution is obtained. An illustrative example for the approximate computation is given. An extensive simulatin study has been conducted to see the performance of the proposed saddle-point approximation for the distribution function of the bivariate compound truncated Poisson-Gamma distribution. It is found that the proposed saddle-point approximation is reasonably good to approximate the distribution function of the bivariate compound truncated Poisson-Gamma distribution.</p> Amal Alhejaili Ateq Alghamedi Copyright (c) 2024 Pakistan Journal of Statistics and Operation Research 2024-06-05 2024-06-05 285 299 10.18187/pjsor.v20i2.4461 Approximations to the Moments of Order Statistics for Normal Distribution <p>Order statistics occupy an important place in statistical theory. They have an important place in many fields of applied statistics such as goodness of fit tests and parameter estimation. In addition, it is necessary to find the expected values of these order statistics in these application areas. However for some probability distributions, these expected values are very difficult to find such as the standard normal distribution. So the problem of finding the expected values of the order statistics in statistical theory is of importance. In this study, two novel approximation methods are proposed for the expected values of the order statistics of the standard normal distribution. Also, the true values with previously given approximations, simulation results and our proposed approximations are compared by using mean square error (MSE), mean absolute error (MAE) and maximum error (ME) criteria. Furthermore, to evaluate the performances of all approximation methods, we compute the differences between exact values and approximation values. Then, the plot of these differences against the exact values is given.&nbsp; Based on both the plots and the comparison results, novel approximations fit the true values better than the other approximations presented in this paper.</p> Asuman Yılmaz Mahmut Kara Copyright (c) 2024 Pakistan Journal of Statistics and Operation Research 2024-06-05 2024-06-05 301 309 10.18187/pjsor.v20i2.4411 Using the Single-Exponential-Smoothing Time Series Model under the Additive Holt-Winters Algorithm with Decomposition and Residual Analysis to Forecast the Reinsurance-Revenues Dataset <p>Time series analysis plays a pivotal role in the strategic planning and risk management of reinsurance companies. It is an indispensable tool for gaining insights into the future utilization of reinsurance revenues. To effectively safeguard against substantial financial losses stemming from anticipated claims, reinsurance businesses must have a thorough understanding of the expected values of these claims. The ability to estimate the potential value of future claims is paramount, as it empowers reinsurance companies to proactively prepare and allocate resources, ensuring that they are well-equipped to cover likely future claims. Our research incorporates an innovative approach to estimate reinsurance revenues, leveraging the power of time series analysis. By applying the proposed paradigm to an original time series dataset, we aim to showcase its practical value and effectiveness in predicting future revenue trends. To assess the accuracy of these predictions, we employ the Box-Ljung statistical test, a statistical test commonly used in time series analysis. The corresponding p-value generated from this test provides a quantitative measure of the ability to analyze, capture and explain the underlying patterns in the data, thereby aiding reinsurance companies in providing an informed decisions and managing their financial risks effectively. In summary, the integration of time series analysis, single exponential smoothing (SEXS), and advanced forecasting techniques forms a critical foundation for enhancing the predictive capabilities of reinsurance businesses and ensuring their financial stability in the face of uncertain future claims.</p> Murtadha Mansour Abdullah Copyright (c) 2024 Pakistan Journal of Statistics and Operation Research 2024-06-05 2024-06-05 311 340 10.18187/pjsor.v20i2.4409 Characterizations of the Recently Introduced Discrete Distributions <p>Certain characterizations of 26 recently introduced discrete distributions are presented in three directions: (i) based on an appropriate function of the random variable; (ii) in terms of the reverse hazard function and (iii) in terms of the hazard function.</p> G. G. Hamedani Amin Roshani Copyright (c) 2024 Pakistan Journal of Statistics and Operation Research 2024-06-06 2024-06-06 341 367 10.18187/pjsor.v20i2.4624