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> Generalized Linear Models for Loss Calculation in General Insurance <p>In most cases, loss in non-life insurance is calculated based on claim severity and frequency and an assumption of independence. However, in some cases, claim severity depends upon the claim frequency. This paper presents the derivation of aggregate loss calculation by modeling claim severity and frequency as the assumption of independence is eliminated. The authors modeled average claim severity using claim frequency as the covariate to induce the dependence among them. For that purpose, we use the generalized linear model. After doing parameters estimation, we will obtain the calculated loss.</p> Dian Lestari Raymond Tanujaya Rahmat Al Kafi Sindy Devila Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-07 2022-09-07 505 510 10.18187/pjsor.v18i3.3676 A Generalized Form of Power Transformation on Exponential Family of Distribution with Properties and Application <p>In this paper, we proposed a new generalized family of distribution namely new alpha power Exponential (NAPE) distribution based on the new alpha power transformation (NAPT) method by Elbatal <em>et al.</em> (2019). Various statistical properties of the proposed distribution are obtained including moment, incomplete moment, conditional moment, probability weighted moments (PWMs), quantile function, residual and reversed residual lifetime function, stress-strength parameter, entropy and order statistics. The percentage point of NAPE distribution for some specific values of the parameters is also obtained. The method of maximum likelihood estimation (MLE) has been used for estimating the parameters of NAPE distribution. A simulation study has been performed to evaluate and execute the behavior of the estimated parameters for mean square errors (MSEs) and bias.&nbsp; Finally, the efficiency and flexibility of the new proposed model are illustrated by analyzing three real-life data sets.</p> Seema Chettri Bhanita Das Imliyangba Imliyangba P. J. Hazarika Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-07 2022-09-07 511 535 10.18187/pjsor.v18i3.3883 Expanding the Nadarajah Haghighi Model: Copula, Censored and Uncensored Validation, Characterizations and Applications <p>A new three-parameter Nadarajah Haghighi model is introduced and studied. The new density has various shapes such as the right skewed, left skewed and symmetric and its corresponding hazard rate shapes can be increasing, decreasing, bathtub, upside down and constant. Characterization results are obtained based on two truncated moments and in terms of the hazard function. Validation via a modified chi-squared goodness-of-fit test is presented under the new model. A simple type Copula based construction is employed in deriving many bivariate and multivariate type distributions. The potentiality uncensored and censored real data sets. We constructed a modified Nikulin-Rao-Robson chi-square goodness-of-fit type test for the new model. This modi…ed chi-square test takes into account both unknown parameters and censorship. Validation in case of right censoring and all the elements constituting the test criteria. The censored aluminum reduction cells data is analyzed for validation.</p> Mohamed Ibrahim G.G. Hamedani Nadeem Shafique Butt Haitham Yousof Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-09 2022-09-09 537 553 10.18187/pjsor.v18i3.3420 Some improvements for existing simple Approximations of the Normal Distribution Function <p>Due to the widespread applicability and use of the normal distribution, a need has arisen to approximate its cumulative distribution function (cdf). In this article, five new simple approximations to the standard normal cdf are developed. In order to assess the accuracy of the proposed approximations, both maximum absolute error and mean absolute error were used.&nbsp; The maximum absolute errors of the proposed approximations lie between 0.00095 and 0.00946, which is highly accurate if compared to the existing simple approximations and quite sufficient for many real-life applications. Even though simple approximations may not as accurate as complicated ones, they are, though, fairly good when judged vis-a-vis their simplicity.</p> Ahmad Hanandeh Omar M. Eidous Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-09 2022-09-09 555 559 10.18187/pjsor.v18i3.4007 The Effect of Different Similarity Distance Measures in Detecting Outliers Using Single-Linkage Clustering Algorithm for Univariate Circular Biological Data <p>The procedure of outliers detection in univariate circular data can be developed using clustering algorithm. In clustering, it is necessary to calculate the similarity measure in order to cluster the observations into their own group. The similarity measure in circular data can be determined by calculating circular distance between each point of angular observation. In this paper, clustering-based procedure for outlier detection in univariate circular biological data with different similarity distance measures will be developed and the performance will be investigated. Three different circular similarity distance measures are used for the outliers detection procedure using single-linkage clustering algorithm. However, there are two similarity measures namely Satari distance and Di distance that are found to have similarity in formula for univariate circular data. The aim of this study is to develop and demonstrate the effectiveness of proposed clustering-based procedure with different similarity distance measure in detecting outliers. Therefore, in this study the circular similarity distance of SL-Satari/Di and another similarity measure namely SL-Chang will be compared at certain cutting rule. It is found that clustering-based procedure using single-linkage algorithm with different similarity distances are applicable and promising approach for outlier detection in univariate circular data, particularly for biological data. The result also found that at a certain condition of data, the SL-Satari/Di distance seems to overperform the performance of SL-Chang distance.</p> Nur Syahirah Zulkipli Siti Zanariah Satari Wan Nur Syahidah Wan Yusoff Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-09 2022-09-09 561 573 10.18187/pjsor.v18i3.3982 The Performance of Bayesian Analysis in Structural Equation Modelling to Construct The Health Behaviour During Pandemic COVID-19 <p>Originating from Wuhan, China, COVID-19 is spreading rapidly throughout the world. The epidemiological model is required to provide evidence for public health policymakers to reduce the spread of COVID-19. Health behaviour is assumed could reduce the spread of this virus.&nbsp; This study purposes to construct an acceptable model of health behaviour. To achieve this goal, a Bayesian structural equation modelling (SEM) is implemented. This current study is also purposed to evaluate the performance of Bayesian SEM, including the sensitivity, adequacy, and the acceptability of parameters estimated with the result that the acceptable model is obtained. The sensitivity of the Bayesian SEM estimator is evaluated by choosing several types of prior and the model results are compared. The adequacy of the Bayesian SEM estimate is checked by doing the convergence test of the corresponding model parameters. The acceptability of the Bayesian approach and its associated algorithm in recovering the true parameters are monitored by the Bootstrap simulation study. The Bayesian SEM applies the Gibbs sample approach in estimating model parameters. This method is applied to the primary data gathered from an online survey from March to May 2020 during COVID-19 to individuals living in West Sumatera, Indonesia. It is found that health motivation is significantly related to health behaviour. Whereas socio-demographic and perceived susceptibility has no significant effect on health behaviour.&nbsp;</p> Ferra Yanuar Aidinil Zetra Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-09 2022-09-09 575 587 10.18187/pjsor.v18i3.4096 Optimization of Backpropagation Using Harmony Search for Gold Price Forecasting <p>Gold is a precious metal often used for investment, due to its cash-in ease and yearly value increase. This indicates that price forecasting is used to determine the prospect of future gold prices. Strong gold price forecasting is highly desired by investors to make decisions. That is why technical indicators are very important used for forecasting. By using technical indicators the information obtained can be more informative than using pure gold prices. One of the commonly used methods is Backpropagation (BP). BP has been shown to have good performance in dealing with nonlinear problems. However, due to the random determination of the parameters of neurons in the hidden layer BP requires a number of neurons in the hidden layer to get optimal results. Therefore, this study aims to analyze the optimization of Backpropagation (BP) through the Harmony Search (HS) algorithm by evaluating the use of relevant technical indicators for forecasting gold prices. In the HS-BP model, this method is used to determine input variables and neurons in the hidden layer. HS with the principle of musicians with the aim of finding the best harmony. This technique is used based on the results of the fitness function. In this research, the fitness function used is Mean Square Error (MSE). HS aims to optimize BP in such a way that the forecasting system provides the lowest MSE and improves the forecasting performance of gold prices. Based on this research, the input variables used are Moving Average, Relative Strength Index, and Bollinger Bands. Next, the selected variables and neurons are applied to the BP algorithm. Where the implementation uses gold closing price data for January 2020-2021. The results showed that the proposed method has better results in forecasting accuracy and convergence error. HS-BP provides a better level of gold price forecasting than the regular BP model.</p> Yuni Kurniawati Muhammad Muhajir Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-09 2022-09-09 589 599 10.18187/pjsor.v18i3.3915 A New Compound Lomax Model: Properties, Copulas, Modeling and Risk Analysis Utilizing the Negatively Skewed Insurance Claims Data <p>Analyzing the future values of anticipated claims is essential in order for insurance companies to avoid major losses caused by prospective future claims. This study proposes a novel three-parameter compound Lomax extension. The new density can be "monotonically declining", "symmetric", "bimodal-asymmetric", "asymmetric with right tail", "asymmetric with wide peak" or "asymmetric with left tail". The new hazard rate can take the following shapes: "J-shape", "bathtub (U-shape)", "upside down-increasing", "decreasing-constant", and "upside down-increasing". We use some common copulas, including the Farlie-Gumbel-Morgenstern copula, the Clayton copula, the modified Farlie-Gumbel-Morgenstern copula, Renyi's copula and Ali-Mikhail-Haq copula to present some new bivariate quasi-Poisson generalized Weibull Lomax distributions for the bivariate mathematical modelling. Relevant mathematical properties are determined, including mean waiting time, mean deviation, raw and incomplete moments, residual life moments, and moments of the reversed residual life. Two actual data sets are examined to demonstrate the unique Lomax extension's usefulness. The new model provides the lowest statistic testing based on two real data sets. The risk exposure under insurance claims data is characterized using five important risk indicators: value-at-risk, tail variance, tail-value-at-risk, tail mean-variance, and mean excess loss function. For the new model, these risk indicators are calculated. In accordance with five separate risk indicators, the insurance claims data are employed in risk analysis. We choose to focus on examining these data under five primary risk indicators since they have a straightforward tail to the left and only one peak. All risk indicators under the insurance claims data are addressed for numerical and graphical risk assessment and analysis.</p> Mohamed S. Hamed Gauss M. Cordeiro Haitham M. Yousof Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-09 2022-09-09 601 631 10.18187/pjsor.v18i3.3652 Continuous wavelet estimation for multivariate fractional Brownian motion <p>&nbsp;In this paper, we propose a method using continuous wavelets to study the multivariate fractional Brownian motion through the deviations of the transformed random process to find an efficient estimate of Hurst exponent using eigenvalue regression of the covariance matrix. The results of simulations experiments shown that the performance of the proposed estimator was efficient in bias but the variance get increase as signal change from short to long memory the MASE increase relatively. The estimation process was made by calculating the eigenvalues for the variance-covariance matrix of Meyer’s continuous wavelet details coefficients.</p> Munaf Y. Hmood Amjed Hibatallah Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-10 2022-09-10 633 641 10.18187/pjsor.v18i3.3657 The Construction of Some New Quasi Rees Neighbor Designs Using Cyclic Shifts <p>Many popular neighbor designs are used in serology, agriculture, and forestry which manifest neighbor effects very much. If every treatment appears as a neighbor with other (<em>v</em>-2) treatments once but emerges twice with only one treatment, such designs are called Quasi Rees neighbor designs (QRNDs) in k size of circular blocks. These designs were used for counterbalancing the neighboring effects for the cases for which minimal neighbor designs cannot be constructed. In this article, various generators are constructed to obtain circular binary NDs, using cyclic shifts.</p> Saira Sharif Rashid Ahmed Qaiser Mehmood Muhammad Rizwan Shahid Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-10 2022-09-10 643 648 10.18187/pjsor.v18i3.3501 The Generalized Odd Log-Logistic Fréchet Distribution for Modeling Extreme Values <p>We introduce a new extension of the Fréchet distribution for modeling the extreme values. The new model generalizes eleven distributions at least, five of them are quite new. Some important mathematical properties of the new model are derived. We assess the performance of the maximum likelihood estimators (MLEs) via a simulation study. The new model is better than some other important competitive models in modeling the breaking stress data, the glass fibers data and the relief time data.</p> Rania Hassan Abd El Khaleq Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-10 2022-09-10 649 674 10.18187/pjsor.v18i3.2902 Bayesian Life Analysis of Generalized Chen's Population Under Progressive Censoring <p>Chen's model with bathtub shape provides an appropriate conceptual for the hazard rate of various industrial products and clinical cases. This article deals with the problem of estimating the model parameters, reliability and hazard functions of a three-parameter Chen distribution based on progressively Type-II censored sample have been obtained. Based on the s-normal approximation to the asymptotic distribution of the maximum likelihood estimates and log-transformed maximum likelihood estimates, the approximate confidence intervals for the unknown parameters, and any function of them, are constructed. Using independent gamma conjugate priors, the Bayes estimators of the unknown parameters and reliability characteristics are derived under different versions of a symmetric squared error loss functions. However, the Bayes estimators are obtained in a complex form, so we have been used Metropolis-Hastings sampler procedure to carry out the Bayes estimates and also to construct the corresponding credible intervals. To assess the performance of the proposed estimators, numerical results using Monte Carlo simulation study were reported. To determine the optimum censoring scheme among different competing censoring plans, some optimality criteria have been considered. A practical example using real-life data set, representing the survival times of head and neck cancer patients, is discussed to demonstrate how the applicability of the proposed methods in real phenomenon.</p> Ahmed Elshahhat Manoj Kumar Rastogi Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-10 2022-09-10 675 702 10.18187/pjsor.v18i3.3766 Alpha Power Exponentiated New Weibull-Pareto Distribution: Its Properties and Applications <p>In this paper, a new five-parameter model called alpha power exponentiated new Weibull-Pareto distribution is introduced based on a new developing technique. We derived some properties relating to the proposed distribution, including moments, moment generating function, quantile function, mean residual life and mean waiting time, and order statistics of the new model. The model parameters are estimated using the maximum likelihood method. Some simulation studies are performed to investigate the effectiveness of the estimates. Finally, we used three real-life data sets to show the flexibility of the introduced distribution.</p> Wedad Aljuhani Hadeel S. Klakattawi Lamya A. Baharith Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-10 2022-09-10 703 720 10.18187/pjsor.v18i3.3937 The The Gamma Odd Burr X-G Family of Distributions with Applications <p>A new family of distributions called Gamma Odd Burr X-G (GOBX-G) distribution is introduced in this paper. Its structural properties such as the density expansion, quantile function, moments and generating functions, incomplete moments, probability weighted moments, R´enyi entropy and order statistics were derived. Maximum likelihood technique is used to estimate the parameter of this model and simulation results are provided. The flexibility and applicability of this model is demonstrated using real life datasets.</p> Bakang Percy Tlhaloganyang Whatmore Sengweni Broderick Oluyede Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-10 2022-09-10 721 746 10.18187/pjsor.v18i3.4045 Remarks on the Papers by Coelho-Barros et al. (2017), Usman et al. (2021) and Obeid and Kadry (2022) <p>We would like to point out that the formula for the cumulative distribution given in Coelho-Barros et al. (2017) and a similar version of it given in Usman et al. (2021) are not cumulative distribution functions as these functions do not satisfy the one or more necessary and sufficient conditions for a function to be a cumulative distribution function. We would also like to mention that formulas for the cumulative distribution functions of product and ratio of two independent Pareto and Exponential random variables given by Obeid and Kadry (2022) are not cumulative distribution functions either. We do not believe that these formulas can be fixed to be cumulative distribution functions. In this short article, we provide mathematical justification in support of these claims.</p> G.G. Hamedani I. Ghosh A. Saghir Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-10 2022-09-10 747 748 10.18187/pjsor.v18i3.4180 A 2-Phase Method for Solving Transportation Problems with Prohibited Routes <p>The Transportation Problem (TP) is a mathematical optimization technique which regulates the flow of items along routes by adopting an optimum guiding principle to the total shipping cost. However, instances including road hazards, traffic regulations, road construction and unexpected floods sometimes arise in transportation to ban shipments via certain routes. In formulating the TPs, potential prohibited routes are assigned a large penalty cost, M, to prevent their presence in the model solution. The arbitrary usage of the big M as a remedy for this interdiction does not go well with a good solution. In this paper, a two-phase method is proposed to solve a TP with prohibited routes. The first phase is formulated as an All-Pairs Least Cost Problem (APLCP) which assigns respectively a non-discretionary penalty cost M<sup><span style="font-family: symbol;">*</span></sup><sub>ij</sub> <span style="font-family: symbol;">&lt;=</span> M to each of n prohibited routes present using the Floyd<span style="font-family: symbol;">¢</span>s method. At phase two, the new penalty values are substituted into the original problem respectively and the resulting model is solved using the transportation algorithm. The results show that, setting this modified penalty cost ( M<sup><span style="font-family: symbol;">*</span></sup>) logically presents a good solution. Therefore, the discretionary usage of the M <span style="font-family: symbol;">&lt;=</span> ∞&nbsp;is not a guarantee for good model solutions. The modified cost M<sup><span style="font-family: symbol;">*</span></sup><span style="font-family: symbol;">&lt;=</span> M so attained in the sample model, is relatively less than the Big M ( <span style="font-family: symbol;">&lt;=</span> ∞) and gives a good solution which makes the method reliable.</p> Joseph Ackora Prah Valentine Acheson Benedict Barnes Ishmael Takyi Emmanuel Owusu-Ansah Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-10 2022-09-10 749 758 10.18187/pjsor.v18i3.3911 Three Parameters Quasi Gamma Distribution and with Properties and Applications <p>This paper introduced a new life time data analysis distribution name three parameters quasi gamma distribution discussed about its some properties including moment generating function, rth moment about origin and mean, mean deviations, reliability measurements, Bonferroni and Lorenz curve, Order statistics, Renyi entropy, also discussed about maximum likelihood method and real-life data applications.</p> Qaisar Rashid Dr. Hafiz Muhammad Yaseen Muhammad Uzair Muhammad Tariq Jamshaid Copyright (c) 2022 Pakistan Journal of Statistics and Operation Research 2022-09-10 2022-09-10 759 773 10.18187/pjsor.v18i3.3759