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 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> 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 terms:</strong></p> <ul> <li class="show">Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a <a href="" target="_new">Creative Commons Attribution License</a> that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.&nbsp;</li> <li class="show">Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.</li> <li class="show">Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See <a href="" target="_new">The Effect of Open Access</a>).</li> </ul> <p>&nbsp;</p> Bayesian and Classical Estimation for the One Parameter Double Lindley Model <p>The main motivation of this paper is to show how the different frequentist estimators of the new distribution perform for different sample sizes and different parameter values and to raise a guideline in choosing the best estimation method for the new model. The unknown parameters of the new distribution are estimated using the maximum likelihood method, ordinary least squares method, weighted least squares method, Cramer-Von-Mises method and Bayesian method. The obtained estimators are compared using Markov Chain Monte Carlo simulations and we observed that Bayesian estimators are more efficient compared to other the estimators.</p> Mohamed Ibrahim Wahhab Mohammed Haitham M. Yousof Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-27 2020-08-27 409 420 10.18187/pjsor.v16i3.2966 A General Class of Dual to Ratio Estimator <p>In this paper we have considered the problem of estimating the population mean using auxiliary information in sample surveys. A class of dual to ratio estimators has been defined. Exact expressions for bias and mean squared error of the suggested class of dual to ratio estimator have been obtained. In particular, properties of some members of the proposed class of dual to ratio estimators have been discussed. It has been shown that the proposed class of estimators is more efficient than the sample mean, ratio estimator, dual to ratio estimator and some members of the suggested class of estimators in some realistic conditions. Some numerical illustrations are given in support of the present study.</p> Housila Prasad Singh Pragati Nigam Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 421 431 10.18187/pjsor.v16i3.2936 The Modified Kumaraswamy Weibull Distribution: Properties and Applications in Reliability and Engineering Sciences <p>We introduce the Kumaraswamy alpha power-G (KAP-G) family which extends the alpha power family (Mahdavi and Kundu, 2017) and some other families. We consider the Weibull as baseline for the KAP family and generate Kumaraswamy alpha power Weibull distribution which has right-skewed, left-skewed, symmetrical, and reversed-J shaped densities, and decreasing, increasing, bathtub, upside-down bathtub, increasing-decreasing–increasing, J shaped and reversed-J shaped hazard rates. The proposed distribution can model non-monotone&nbsp; and monotone failure rates which are quite common in engineering and reliability studies. Some basic mathematical properties of the new model are derived. The maximum likelihood estimation method is used to evaluate the parameters and the observed information matrix is determined. The performance and flexibility of the proposed family is illustrated via two real data applications.</p> M. E. Mead Ahmed Afify Nadeem Shafique Butt Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 433 446 10.18187/pjsor.v16i3.3306 The Flexible Weibull Extension-Burr XII Distribution: Model, Properties and Applications <p>This paper is devoted to study a new four- parameter additive model. The newly suggested model is referred to as the flexible Weibull extension-Burr XII distribution. It is derived by considering a serial system with one component following a flexible Weibull extension distribution and another following a Burr XII distribution. The usefulness of the model stems from the flexibility of its failure rate which accommodates bathtub and modified bathtub among other risk patterns. These two patterns have been widely accepted in several fields, especially reliability and engineering fields. In addition, the importance of the new distribution is that it includes new sub-models which are not known in the literature. Some statistical properties of the proposed distribution such as quantile function, the mode, the rth moment, the moment generating function and the order statistics are discussed. Moreover, the method of maximum likelihood is used to estimate the parameters of the model. Also, to evaluate the performance of the estimators, a simulation study is carried out. Finally, the performance of the proposed distribution is compared through a real data set to some well-known distributions including the new modified Weibull, the additive Burr and the additive Weibull distributions. It is shown that the proposed model provides the best fit for the used real data set.&nbsp;&nbsp;</p> Rania M. Kamal Moshira A. Ismail Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 447 460 10.18187/pjsor.v16i3.2957 Extended Poisson Lomax Distribution <p>The main goal of this article is to introduce a new extension of the continuous Lomax distribution with a strong physical motivation. Some of its statistical properties such as moments, incomplete moments, moment generating function, quantile function, random number generation, quantile spread ordering and moment of the reversed residual life are derived. Two applications are provided to illustrate the importance and flexibility of the new model.</p> Mohamed Sowilem Hamed Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 461 470 10.18187/pjsor.v16i3.2837 A New Unit Distribution Based On The Unbounded Johnson Distribution Rule: The Unit Johnson SU Distribution <p>This paper proposes a new probability distribution, which belongs a member of the exponential family, defined on (0,1) unit interval. The new unit model has been defined by relation of a random variable defined on unbounded interval with respect to standard logistic function. Some basic statistical properties of newly defined distribution are derived and studied. The different estimation methods and some inferences for the model parameters have been derived. We assess the performance of the estimators of these estimation methods based on the three different simulation scenarios. The analysis of three real data examples which one is related to the coronavirus data, show better fit of proposed distribution than many known distributions on the unit interval under some comparing criteria.</p> Selim Gündüz Mustafa Ç. Korkmaz Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 471 490 10.18187/pjsor.v16i3.3421 Identification of High Leverage Points in Linear Functional Relationship Model <p>In a standard linear regression model the explanatory variables, , are considered to be fixed and hence assumed to be free from errors. But in reality, they are variables and consequently can be subjected to errors. In the regression literature there is a clear distinction between outlier in the - space or errors and the outlier in the <strong><em>X</em></strong>-space. The later one is popularly known as high leverage points. If the explanatory variables are subjected to gross error or any unusual pattern we call these observations as outliers in the - space or high leverage points. High leverage points often exert too much influence and consequently become responsible for misleading conclusion about the fitting of a regression model, causing multicollinearity problems, masking and/or swamping of outliers etc. Although a good number of works has been done on the identification of high leverage points in linear regression model, this is still a new and unsolved problem in linear functional relationship model. In this paper, we suggest a procedure for the identification of high leverage points based on deletion of a group of observations. The usefulness of the proposed method for the detection of multiple high leverage points is studied by some well-known data set and Monte Carlo simulations.</p> Abu Sayed Md. Al Mamun A.H.M. R. Imon A. G. Hussin Y. Z. Zubairi Sohel Rana Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 491 500 10.18187/pjsor.v16i3.2620 Transmuted Topp-Leone Weibull Lifetime Distribution: Statistical Properties and Different Method of Estimation <p>In this work we focus on proposing a new lifetime Weibull type model called the transmuted Topp-Leone Weibull and studying its properties. We derive some new bivariate and multivariate transmuted Topp-Leone Weibull versions using “Farlie Gumbel Morgenstern (FGM) Copula”, “modified FGM Copula”, “Clayton Copula” and “Renyi's entropy Copula”. The estimation of its unknown parameters is carried out by considering different method of estimation. The statistical performances of all methods are studied by two real data sets and a numerical Monte Carlo simulation. The Cramer-Von Mises method is the best method for modeling the carbon fibers data. The maximum likelihood method is the best method for modeling the Leukemia data, however all other methods performed well.</p> Mohamed Ibrahim Haitham Yousof Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 501 515 10.18187/pjsor.v16i3.2811 Exponentiated Exponential-Exponentiated Weibull Linear Mixed Distribution: Properties and Applications <p>In this article we have discussed linear mixing of two exponentiated distribution. The proposed model is named as exponentiated exponential-exponentiated Weibull (EE-EW) distribution. The proposed distribution generalize several existing distributions. We study several characteristics of the proposed distribution including moment, moment generating function, reliability and hazard rate functions. An empirical study is presented for mean, variance, coefficient of skewness, and coefficient of kurtosis. The method of maximum likelihood is used for the estimation of parameters. For the illustration purpose, we have use two real-life data set for application. The results justify the capability of the new model.</p> Salman Abbas Muhammad Mohsin Saman Hanif Shahbaz Muhammad Qaiser Shahbaz Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 517 527 10.18187/pjsor.v16i3.3415 The Generalized Odd Generalized Exponential Fréchet Model: Univariate, Bivariate and Multivariate Extensions with Properties and Applications to the Univariate Version <p>A new univariate extension of the Fréchet distribution is proposed and studied. Some of its fundamental statistical properties such as stochastic properties, ordinary and incomplete moments, moments generating functions, residual life and reversed residual life functions, order statistics, quantile spread ordering, Rényi, Shannon and q-entropies are derived. A simple type Copula based construction using Morgenstern family and via Clayton Copula is employed to derive many bivariate and multivariate extensions of the new model. We assessed the performance of the maximum likelihood estimators using a simulation study. The importance of the new model is shown by means of two applications to real data sets.</p> Hisham Abdel Hamid Elsayed Haitham M. Yousof Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 529 544 10.18187/pjsor.v16i3.2953 Robust Estimation methods of Generalized Exponential Distribution with Outliers <p>This paper discussed robust estimation for point estimation of the shape and scale parameters for generalized exponential (GE) distribution using a complete dataset in the presence of various percentages of outliers. In the case of outliers, it is known that classical methods such as maximum likelihood estimation (MLE), least square (LS) and maximum product spacing (MPS) in case of outliers cannot reach the best estimator. To confirm this fact, these classical methods were applied to the data of this study and compared with non-classical estimation methods. The non-classical (Robust) methods such as least absolute deviations (LAD), and M-estimation (using M. Huber (MH) weight and M. Bisquare (MB) weight) had been introduced to obtain the best estimation method for the parameters of the GE distribution. The comparison was done numerically by using the Monte Carlo simulation study. The two real datasets application confirmed that the M-estimation method is very much suitable for estimating the GE parameters. We concluded that the M-estimation method using Huber object function is a suitable estimation method in estimating the parameters of the GE distribution for a complete dataset in the presence of various percentages of outliers.</p> Hisham Mohamed Almongy Ehab M. Almetwally Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 545 559 10.18187/pjsor.v16i3.3016 A Discrete Event Simulation Analysis of the Bullwhip Effect in a Multi-Product and Multi-Echelon Supply Chain of Fast Moving Consumer Goods <p>Timely delivery is the major issue in Fast Moving Consumer Good (FMCG) since it depends on the lead time which is stochastic and long due to several reasons; e.g., delay in processing orders and transportation. Stochastic lead time can cause inventory inaccuracy where echelons have to keep high product stocks. Such performance inefficiency reflects the existence of the bullwhip effect (BWE), which is a common challenge in supply chain networks. Thus, this paper studies the impact of stochastic lead time on the BWE in a multi-product and multi-echelon supply chain of FMCG industries under two information-sharing strategies; i.e., decentralized and centralized. The impact was measured using a discrete event simulation approach, where a simulation model of a four-tier supply chain whose echelons adopt the same lead time distribution and continuous review inventory policy was developed and simulated. Different lead time cases under the information-sharing strategies were experimented and the BWE was measured using the standard deviation of demand ratios between echelons. The results show that the BWE cannot be eliminated but can be reduced under centralized information sharing. All the research analyses help the practitioners in FMCG industries get insight into the impact of sharing demand information on the performance of a supply chain when lead time is stochastic.</p> Ramsha Ali Ruzelan Khalid Shahzad Qaiser Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 561 576 10.18187/pjsor.v16i3.3088 Statistical Inference for Burr Type X Distribution using Geometric Process in Accelerated Life Testing Design for Time censored data <p>In accelerated life testing researcher generally use a life stress relationship between life characteristic and stress to estimate the parameters of failure time distributions at use condition which is just a re-parameterization of original parameters but from statistical point of view it is easy and reasonable to deal with original parameters of the distribution directly instead of developing inference for the parameters of the life stress relationship. So, an attempt is made here to estimate the parameters of Burr Type X life distribution directly in accelerated life testing by assuming that the lifetimes at increasing stress levels forms a geometric process. A mathematical model for the analysis of constant stress accelerated life testing for type-I censored data is developed and the estimates of parameters are obtained by using the maximum likelihood method. Also a Fisher information matrix is constructed in order to get the asymptotic variance and interval estimates of the parameters. Lastly, a simulation study is performed to illustrate the statistical properties of the parameters and the confidence intervals.</p> Ahmadur Rahman Tabassum Naz Sindhu Showkat Ahmad Lone Mustafa Kamal Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 577 586 10.18187/pjsor.v16i3.2252 Goodness of Fit Tests for Marshal-Olkin Extended Rayleigh Distribution <p>A class of goodness of fit tests for Marshal-Olkin Extended Rayleigh distribution with estimated parameters is proposed. The tests are based on the empirical distribution function. For determination of asymptotic percentage points, Kolomogorov-Sminrov, Cramer-von-Mises, Anderson-Darling,Watson, and Liao-Shimokawa test statistic are used. This article uses Monte Carlo simulations to obtain asymptotic percentage points for Marshal-Olkin extended Rayleigh distribution. Moreover, power of the goodness of fit test statistics is investigated for this lifetime model against several alternatives.</p> Naz Saud Sohail Chand Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2019-09-16 2019-09-16 587 598 10.18187/pjsor.v16i3.2624 Characterizations of Discrete Weibull Distributions <p>Seven versions of the discrete Weibull distribution are characterized via conditional expectation of function of the random variable as well as based on the hazard or reverse hazard function.</p> G.G. Hamedani Nadeem Shafique Butt Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 599 607 10.18187/pjsor.v16i3.3464 Characterizations of Three (2020) Introduced Discrete Distributions <p>The problem of characterizing a probability distribution is an important problem which has attracted the attention of many researchers in the recent years. To understand the behavior of the data obtained through a given process, we need to be able to describe this behavior via its approximate probability law. This, however, requires to establish conditions which govern the required probability law. In other words we need to have certain conditions under which we may be able to recover the probability law of the data. So, characterization of a distribution plays an important role in applied sciences, where an investigator is vitally interested to find out if their model follows the selected distribution. In this short note, certain characterizations of three recently introduced discrete distributions are presented to complete, in some way, the works ofHussain(2020), Eliwa et al.(2020) and Hassan et al.(2020).</p> Shirin Nezampour G. G. Hamedani Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 609 616 10.18187/pjsor.v16i3.3288 Inverse Odd Weibull Generated Family of Distribution <p>This work proposes an inverse odd Weibull (IOW) family of distributions for a lifetime distributions. Some mathematical properties of this family of distribution were derived. Survival, hazard, quantiles, reversed hazard, cumulative, odd functions, kurtosis, skewness, order statistics and entropies of this new family of distribution were examined. The parameters of the family of distributions were obtained by maximum likelihood. The behavior of the estimators were studied through simulation. The flexibility and importance of the distribution by means of real data set applications were emphasized.</p> Joseph Thomas Eghwerido John David Ikwuoche Obinna Damian Adubisi Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-08-28 2020-08-28 617 633 10.18187/pjsor.v16i3.2760