Pakistan Journal of Statistics and Operation Research 2021-05-11T07:45:16+05:00 Editor PJSOR Open Journal Systems <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 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 variety of disciplines. Because of increasing submission rate, editoral board of PJSOR decided to publish it on quarterly basis from 2012. Brief chronicles is overseen by an Editorial Board comprised of academicians and scholars. We welcome you to submit your research for possible publication in PJSOR through our online submission system. Publication in PJSOR is absolutely free of charge.<br><a href=";tip=sid&amp;clean=0"><strong>ISSN : 1816 2711</strong></a>&nbsp; &nbsp;<strong>|&nbsp; &nbsp;<a href=";tip=sid&amp;clean=0">E- ISSN : 2220 5810</a></strong></p> The Extended Marshall-Olkin Burr III Distribution: Properties and Applications 2021-03-20T18:28:32+05:00 Muhammad Ahsan ul Haq Ahmed Z. Afify Hazem Al- Mofleh Rana Muhammad Usman Mohammed Alqawba Abdullah M. Sarg <p>We study a new continuous distribution called the Marshall-Olkin modified Burr III distribution. The density function of the proposed model can be expressed as a mixture of modified Burr III densities. A comprehensive account of its mathematical properties is derived. The model parameters are estimated by the method of maximum likelihood. The usefulness of the derived model is illustrated over other distributions using a real data set.</p> 2021-02-28T20:30:57+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Bivariate Transmuted Burr Distribution: Properties and Application 2021-03-20T07:10:59+05:00 Muhammad Qaiser Shahbaz Jumanah Ahmed Darwish Lutfiah Ismail Al Turk <p>The bivariate distributions are useful in simultaneous modeling of two random variables. These distributions provide a way of modeling complex joint phenomenon. In this article, a new bivariate distribution is proposed which is known as the bivariate transmuted Burr (<em>BTB</em>) distribution. This new bivariate distribution is extension of the univariate transmuted Burr (TB) distribution to two variables. The proposed &nbsp;<em>BTB</em> distribution is explored in detail and the marginal and conditional distributions for the distribution are obtained. Joint and conditional moments alongside hazard rate functions are obtained. The maximum likelihood estimation (MLE) for the parameters of the <em>BTB</em> distribution is also done. Finally, real data application of the <em>BTB</em> distribution is given. It is observed that the proposed <em>BTB</em> distribution is a suitable fit for the data used.</p> 2021-02-28T20:33:37+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Bibliomining and Comparison of Q4 and ESCI Indexed journals under Statistics and Probability Category 2021-03-19T18:27:37+05:00 Nadeem Shafique Butt <p>The field of ‘Statistics and Probability’ has expanded its scope over the last few decades and have become an integral part of many fields with continuously increasing demand. This manuscript aimed for at a bibliometric analysis and comparison of all published documents during 2015 – 2019, from journals in the study topic category of ‘Statistics and Probability’ for Q4 Impact Factor (IF) journals and Emerging Source Citation Index (ESCI) of Web of Science (WoS). Sources with incomplete data for study timeframe were excluded and 31 sources from Q4 IF and 32 from ESCI journals were selected yielding 12808 and 4294 documents respectively. After data extraction from WoS, the bibliometric analysis at; source, author and document levels, were performed using “Bibliometrix” R-package. Q4-IF sources produced around 3 times more documents than ESCI sources. Articles were the main document type for both categories. China and USA were leading countries for Q4-IF while India, USA and Korea were dominant among ESCI documents. Two authors, namely, ‘Cordeiro GM’ and ‘Alizadeh M’ were among the 10 most productive authors in both categories. Sources “Communications in Statistics-Theory and Methods” and “Korean Journal of Applied Statistics” were leading contributors for Q4-IF and ESCI category respectively. For both categories, mainly similar trends were observed for keywords and topic coverage. In both Q4-IF and ESCI journals ‘Maximum likelihood’ and ‘Ordered statistics’ were observed to be most predominant keywords. A consistent publication trend with few similarities was observed in terms of documents production over the years for these two categories.</p> 2021-02-28T20:35:39+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Reliability Estimation of Three Parameters Weibull Distribution based on Particle Swarm Optimization 2021-03-19T18:27:36+05:00 Zakariya Y Algamal Ghalia Basheer <p>The three-parameter Weibull distribution is a continuous distribution widely used in the study of reliability and life data. The estimation of the distribution parameters is an important problem that has received a lot of attention by researchers because of theirs effects in several measurements. In this research, we propose a particle swarm optimization (PSO) to estimate the three-parameter Weibull distribution and then to estimate the reliability and hazard functions. The real data results indicate that our proposed estimation method is significantly consistent in estimation compared to the maximum likelihood method. In terms of log likelihood and mean time to failure (MTTF).&nbsp;</p> 2021-02-28T20:58:44+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Relations for Moments of Generalized Order Statistics for Transmuted Exponential Distribution and Characterization 2021-03-19T07:10:55+05:00 Saman Shahbaz Mashail Al-Sobhi Rehan Ahmad Khan Sherwani <p>The relations for moments of generalized order statistics (<em>gos</em>) for transmuted exponential distribution are obtained. These include relations for single, inverse, product and ratio moments. These relations are useful in for recursive computation of moments of <em>gos</em> for transmuted exponential distribution. Some characterizations of the distribution, based on single and product moments of <em>gos</em>, are also obtained.</p> 2021-02-28T21:02:32+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Characterizations of Some Probability Distributions with Completely Monotonic Density Functions 2021-03-18T18:26:20+05:00 Mohammad Shakil Dr. Mohammad Ahsanullah Dr. B. M. G. Kibria Kibria <p>For a non-negative continuous random variable , Chaudhry and Zubair (2002, p. 19) introduced a probability distribution with a completely monotonic probability density function based on the generalized gamma function, and called it the Macdonald probability function. In this paper, we establish various basic distributional properties of Chaudhry and Zubair’s Macdonald probability distribution. Since the percentage points of a given distribution are important for any statistical applications, we have also computed the percentage points for different values of the parameter involved. Based on these properties, we establish some new characterization results of Chaudhry and Zubair’s Macdonald probability distribution by the left and right truncated moments, order statistics and record values. Characterizations of certain other continuous probability distributions with completely monotonic probability density functions such as Mckay, Pareto and exponential distributions are also discussed by the proposed characterization techniques.&nbsp; &nbsp;</p> 2021-03-02T17:52:38+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research The Efficiency Comparison and Application of Proposed Rhombus Ranked Set Sampling 2021-03-17T18:25:41+05:00 Mishal Choudri Nadia Saeed Kanwal Saleem <p>A new scheme ‘Rhombus Ranked Set Sampling’ (RRSS) is developed in this research together with its properties for estimating the population means. Mathematical validation along with the simulation evaluation is presented. The proposed method is an addition to the family of different sampling methods and generalization of ‘Folded Ranked Set Sampling’ (FRSS). For the simulation process, nine probability distributions are considered for the efficiency comparison of proposed scheme from which four are symmetric and rest are asymmetric among which Weibull and beta distributions which are used twice, unlike parametric values. (Al-Naseer, 2007 and Bani-Mustafa, 2011). Through simulation processes, it is observed that RRSS is competent and more reliable relative to simple random sampling (SRS), ranked set sampling (RSS) and folded ranked set sampling (FRSS). It is noted that for all the underlying distributions, an increase in the efficiency of Rhombus Ranked Set Sampling (RRSS) is achieved via increasing the size of the sample ‘<em>p</em>’<em>.</em> Besides the efficiency comparison, consistency of the proposed method is also valued by using Co-efficient of Variation (CV).&nbsp; Secondary data on zinc (Zn) concentration and lead (Pb) contamination in different parts and tissues of freshwater fish was collected to illustrate the evaluation of RRSS against SRS, RSS, FRSS and ERSS (extreme ranked set sampling). The results obtained through real life illustration defend the simulation study and hence indicates that the RRSS estimator is efficient substitute for existing methods (Al-Omari, 2011).</p> 2021-03-02T17:54:39+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Kumaraswamy regression modeling for Bounded Outcome Scores 2021-03-17T07:10:32+05:00 Soudabeh Hamedi-Shahraki Aliakbar Rasekhi Mir Saeed Yekaninejad Mohammad Reza Eshraghian Amir H Pakpour <p>In this paper, we use a regression model for modeling bounded outcome scores (BOS), where the outcome is Kumaraswamy distributed. Similar to the Beta distribution, this distribution can take a variety of shapes while being computationally easier to use. Thus, it is deemed as a suitable alternative distribution to the Beta in modeling bounded random processes. In the proposed model, the median of a bounded response is modeled by the linear predictors which is defined through regression parameters and explanatory variables. We obtained the maximum likelihood estimates (MLE) of the parameters, provided closed-form expressions for the score functions and Fisher information matrix, and presented some diagnostic measures. We conducted Monte Carlo simulations to investigate the finite-sample performance of the MLEs of the parameters. Finally, two practical applications of this model to the real data sets are presented and discussed.</p> 2021-03-02T17:57:20+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Efficient Rank-Based Analysis of Multilevel Models for the Family of Skew-t Errors 2021-03-17T18:25:41+05:00 Sehar Saleem Rehan Ahmad Khan Sherwani <p>Rank-based analysis of linear models is based on selecting an appropriate score function. The information about the shape of the underlying distribution is necessary for the optimal selection; leading towards asymptotically efficient analysis. In this study, we analyzed the multilevel model with cluster-correlated error terms following a family of skew-t distribution with the rank-based approach based on score function derived for the class of skew-normal distribution. The rank fit is compared with the Restricted Maximum Likelihood (REML) estimation in terms of validity and efficiency for different sample sizes. A Monte Carlo simulation study is carried out over skewed-t and contaminated-t distribution with a range of skewness parameter from moderately to highly skewed. The standard error of regression coefficients is significantly reduced in the rank-based approach and further reduces for a large sample size. Rank-based fit appeared asymptotically efficient than REML for each shape parameter of skewness in skew-t and contaminated-t distribution computed through a calculation of precision. The empirical validity of fixed effects is obtained up to the nominal level 0.95 in REML but not rank-based with skew-normal score function.</p> 2021-03-02T17:59:43+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research On A New Two Parameter Fréchet Distribution with Applications 2021-03-17T18:25:40+05:00 Rania Hassan Abd El Khaleq <p>A new ‡exible extension of the Fréchet model is proposed and studied. Some of its<br />fundamental statistical properties are derived. The importance of the new model is shown via two applications to real data sets. A simple type Copula based construction are also presented.</p><p>We assess the performance of the maximum likelihood estimations of the new distribution with respect to sample size n. The assessment was based on a simulation study.The new model is much better than other important competitive models.</p> 2021-03-02T18:02:05+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Estimating E-Bayesian of Parameters of Inverse Weibull Distribution Using an Unified Hybrid Censoring Scheme 2021-03-16T18:24:30+05:00 Shahram Yaghoobzadeh Shahrastani Iman Makhdoom <p>The combination of generalization Type-I hybrid censoring and generalization Type-II hybrid censoring schemes, scheme creates a new censoring called a Unified hybrid censoring scheme. Therefore, in this study, the E-Bayesian estimation of parameters of the inverse Weibull (IW) distribution is obtained under the unified hybrid censoring scheme, and the efficiency of the proposed method was compared with the Bayesian estimator using Monte Carlo simulation and a real data set.</p> 2021-03-02T18:04:21+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Length biased Weighted New Quasi Lindley Distribution: Statistical Properties and Applications 2021-03-16T18:24:31+05:00 Rashid A. Ganaie V. Rajagopalan <p>In this Paper, we have introduced a new version of new quasi lindley distribution known as the length-biased weighted new quasi lindley distribution (LBWNQLD). Length biased distribution is a special case of weighted distribution. The different structural properties of the newly proposed distribution are derived and the model parameters are estimated by using the method of maximum likelihood estimation and also the Fisher’s information matrix have been discussed. Finally, applications to real life two data sets are presented for illustration.</p> 2021-03-02T18:06:49+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Marshall–Olkin Alpha Power Lomax Distribution: Estimation Methods, Applications on Physics and Economics 2021-03-16T07:08:59+05:00 Hisham Mohamed Almongy Ehab Mohamed Almetwally Amaal Elsayed Mubarak <p>In this paper, we introduce and study a new extension of Lomax distribution with four-parameter named as the Marshall–Olkin alpha power Lomax (MOAPL) distribution. Some statistical properties of this distribution are discussed. Maximum likelihood estimation (MLE), maximum product spacing (MPS) and least Square (LS) method for the MOAPL distribution parameters are discussed. A numerical study using real data analysis and Monte-Carlo simulation are performed to compare between different methods of estimation. Superiority of the new model over some well-known distributions are illustrated by physics and economics real data sets. The MOAPL model can produce better fits than some well-known distributions as Marshall–Olkin Lomax, alpha power Lomax, Lomax distribution, Marshall–Olkin alpha power exponential, Kumaraswamy-generalized Lomax, exponentiated&nbsp; Lomax&nbsp; and power Lomax.</p> 2021-03-02T18:08:19+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research On Erlang-Truncated Exponential Distribution: Theory and Application 2021-03-16T18:24:30+05:00 Ibrahim Elbatal A. Aldukeel <span class="fontstyle0">In this article, we introduce a new distribution called the McDonald Erlangtruncated exponential distribution. Various structural properties including explicit expressions for the moments, moment generating function, mean deviation of the new distribution are derived. The estimation of the model parameters is performed by maximum likelihood method. The usefulness of the new distribution is illustrated by two real data sets. The new model is much better than other important competitive models in modeling relief times and survival times data sets.</span> 2021-03-02T18:10:50+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Estimation of the Exponential Pareto Distribution’s Parameters under Ranked and Double Ranked Set Sampling Designs 2021-03-13T07:07:04+05:00 Mohamed H. Sabry Ehab Mohamed Almetwally <p>In this paper, the derivation of the likelihood function for parameter estimation based on double ranked set sampling (DRSS) designs used by Sabry; (2019) for the estimation of the parameters of the power generalized Weibull distribution is considered. The developed likelihood function is then used for the estimation of the exponential Pareto distribution parameters. The maximum likelihood estimators (MLEs) are then investigated and compared to the corresponding ones based on simple random sampling (SRS) and ranked set sampling (RSS) designs. A Monte Carlo simulation is conducted and the absolute relative biases, mean square errors, and efficiencies are compared for the different designs. The relative efficiency of the DRSS estimates with respect to other designs was found to be higher in case of the exponential Pareto distribution (EP).</p> 2021-03-02T18:12:32+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Evaluation of Impact of Success Factors of Supply Chain Strategy and Flexibility on Supply Chain Performance 2021-03-07T18:21:48+05:00 Amit Chandak Sumit Chandak A. Dalpati <p>Due to rapid globalization, an introduction of Goods and Services Taxes (GST), ban on BS-III vehicles, and rapid technological advancement creates new dimensions for the automobile industry in India especially for those who come in the category of small and medium enterprise (SMEs).In the current changing scenario victorious execution of supply chain management (SCM) practices can provide competitive leverage to Automobile manufacturers over their rivals particularly those who are poorly implemented supply chain management practices. This paper takes two main SCM practices, i.e. supply chain flexibility (SCF) and supply chain strategy (SCS).Against this background, this paper acknowledged 5 success factors (SFs) for supply chain flexibility and 3 success factors (SFs) for supply chain strategy for proper implementation of SCM practices in the Automobile industry, and studied their impact on three factors of supply chain performance (SCP), and thereby on firms performance. The top management of any industry is focused on their core strength with commitment, long-term vision and provides resources for supply chain, and thereby developing effective and efficient SCS emerged as the most significant SFs. To measure the impact on supply chain performance, the author carefully measures different SFs of SCS related to customer-oriented strategy, innovation strategy and agile supply chain strategy on cost performance, logistics performance and customer satisfaction performance. Similarly to access the impact on supply chain performance, carefully measure different SFs of SCF on cost performance, logistics performance and customer satisfaction performance. Results are analyzed by testing research propositions using standard statistical tools.</p> <p>&nbsp;</p> 2021-03-02T18:19:04+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research A New Lifetime Model: Copulas, Properties and Real Lifetime Data Applications 2021-03-04T06:59:26+05:00 Wahid Shehata <p>A new four parameter lifetime model called the Weibullgeneralized Lomax is proposed and studied.&nbsp; The new density function can be "right skewed", "symmetric" and "left skewed" and its corresponding failure rate function can be "monotonically decreasing", " monotonically increasing" and "constant". The skewness of the new distribution can negative and positive. The maximum likelihood method is employed and used for estimating the model parameters. Using the "biases" and "mean squared errors", we performed simulation experiments for assessing the finite sample behavior of the maximum likelihood estimators. The new model deserved to be chosen as the best model among many well-known Lomax extension such as exponentiated Lomax, gamma Lomax, Kumaraswamy Lomax, odd log-logistic Lomax, Macdonald Lomax, beta Lomax, reduced odd log-logistic Lomax, reduced Burr-Hatke Lomax, reduced WG-Lx, special generalized mixture Lomax and the standard Lomax distributions in modeling the "failure times" and the "service times" data sets.</p> 2021-03-02T18:24:19+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Exact Moments of Generalized Pareto Distribution based on Generalized Order Statistics and Characterizations 2021-05-11T07:45:16+05:00 Bavita Singh Rafiqullah Khan Mohammad Azam Khan <p>In this paper, we present simple explicit expressions for single and product moments of generalized order statistics from generalized Pareto distribution. These relations are deduced for the moments of order statistics and record values and tabulated the mean and variance of this distribution. Further, conditional expectation, recurrence relations for single as well as for product moments of generalized order statistics and truncated moment are used to characterize this distribution.</p> 2021-03-02T18:26:41+05:00 Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research Remarks on and Characterizations of 2S-Lindley and 2D-Lindley Distributions Introduced by Chesneau et al. (2020) 2021-03-03T07:00:17+05:00 G.G. Hamedani Mahrokh Najaf <p>Chesneau et al.(2020) considered the distributions of sum and differences of two independent and identically distributed random variables with the common Lindley distribution. They derived, very nicely, the above mentioned distributions and provided certain important mathematical and statistical properties as well as simulations and applications of the new distributions. In this short note, we like to show that the assumption of ”independence” can be replaced with a much weaker assumption of ”sub-independence”. Then we present certain characterizations of the proposed distributions to complete, in someway, their work.</p> 2021-03-02T18:27:36+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Robust estimation of the extreme value index of Pareto-type distributions under random truncation with applications 2021-03-03T06:59:55+05:00 Yahia Djabrane Zahnit Abida Brahimi Brahim In this paper, we introduce a new robust estimator for the extreme value index of Pareto-type distributions under randomly right-truncated data and establish its consistency and asymptotic normality. Our considerations are based on the Lynden-Bell integral and a useful huberized M-functional and M-estimators of the tail index. A simulation study is carried out to evaluate the robustness and the nite sample behavior of the proposed estimator. Extreme quantiles estimation is also derived and applied to real data-set of lifetimes of automobile brake pads. 2021-03-02T18:31:14+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Bayes Estimation of the Modified Inverse Rayleigh Parameters Under Various Approximation Techniques 2021-03-19T07:10:33+05:00 Mehak Taufique Naila Bashir Naz Saud Muhammad Farooq <p>In this paper we proposed Bayes estimators for complete sample of the Modified Inverse<br>Rayleigh (MIR) parameters which was introduced by Khan (2014). Different approximation<br>methods with squared error loss function (SELF) have been used to develop the bayes<br>estimators for the unknown parameters. The proposed estimators are compared with the corresponding<br>maximum likelihood estimators by simulation study on the basis of mean square<br>error (MSE). To illustrate the usefulness and goodness of fit of Modified Inverse Rayleigh<br>distribution we considered two real data sets.</p> 2021-03-02T00:00:00+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research On nonparametric estimation of the ratio of two reliability function 2021-03-03T06:59:08+05:00 Zohreh Mohammadi <p>This study is concerned with estimating the reliability of residual lifetime random variable. A nonparametric estimator for the ratio of reliability of two independent residual lifetime random variables is derived and its asymptotic distribution is established. Then, the asymptotic confidence interval and the corresponding hypothesis testing procedure is obtained. Finally, the applicability of the proposed method is investigated through simulation and an example of real data.</p> 2021-03-02T19:11:37+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Calculating non-centrality parameter for power analysis under structural equation modelling: An alternative 2021-03-03T06:59:34+05:00 David Adedia Atinuke O. Adebanji Simon K. Appiah <p>Identifying the most parsimonious model in structural equation modelling is of utmost importance and the appropriate power estimation methods minimize the probabilities of Type I and Type II errors. The power of a test depends on the sample size, Type I error, degrees of freedom and effect size. In this study, a modified approach of using effect size in calculating the non-centrality parameter for power is proposed. This is compared to the approach in Maccallum et al. (1996) at different degrees of freedom and sample size specifications --- taken from 50 to 2000.&nbsp;As the sample size increased the difference between the power of a test for both methods reduced to zero. The results showed that the values for power of a test are the same for the modified and original approaches for large sample sizes and degrees of freedom. The findings also revealed that the sample discrepancy function ($\hat{F}$) is asymptotically unbiased.</p> 2021-03-02T18:36:09+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research A New Three-parameter Xgamma Fréchet Distribution with Different Methods of Estimation and Applications 2021-03-03T06:59:13+05:00 Mohamed Ibrahim Mohamed Laba Handique Subrata Chakraborty Nadeem Shafique Butt Haitham M. Yousof <p>In this article an attempt is made to introduce a new extension of the Fréchet model called the Xgamma Fréchet model. Some of its properties are derived. The estimation of the parameters via different estimation methods are discussed. The performances of the proposed estimation methods are investigated through simulations as well as real life data sets. The potentiality of the proposed model is established through modelling of two real life data sets. The results have shown clear preference for the proposed model compared to several know competing ones.</p> 2021-03-02T18:42:52+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research Poisson Transmuted-G Family of Distributions: Its Properties and Applications 2021-03-03T06:59:09+05:00 Laba Handique Subrata Chakraborty M.S. Eliwa Dr. G.G. Hamedani <p>In this article, an extension of the transmuted-G family is proposed, in the so-called Poison transmuted-G family of distributions. Some of its statistical properties including quantile function, moment generating function, order statistics, probability weighted moment, stress-strength reliability, residual lifetime, reversed residual lifetime, Rényi entropy and mean deviation are derived. A few important special models of the proposed family are listed. Stochastic characterizations of the proposed family based on truncated moments, hazard function and reverse hazard function, are also studied. The family parameters are estimated via the maximum likelihood approach. A simulation study is carried out to examine the bias and mean square error of the maximum likelihood estimators. The advantage of the proposed family in data fitting is illustrated by means of two applications to failure time data sets.</p> 2021-03-02T18:45:57+05:00 Copyright (c) 2021 Pakistan Journal of Statistics and Operation Research