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>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>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>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> The Negative Binomial – Weighted Garima Distribution: Model, Properties and Applications <p>In this paper, a new mixed negative binomial (NB) distribution named as negative binomial-weighted Garima (NB-WG) distribution has been introduced for modeling count data. Two special cases of the formulation distribution including negative binomial- Garima (NB-G) and negative binomial-size biased Garima (NB-SBG) are obtained by setting the specified parameter. Some statistical properties such as the factorial moments, the first four moments, variance and skewness have also been derived. Parameter estimation is implemented using maximum likelihood estimation (MLE) and real data sets are discussed to demonstrate the usefulness and applicability of the proposed distribution.</p> Winai Bodhisuwan Pornpop Saengthong Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-02-27 2020-02-27 1 10 10.18187/pjsor.v16i1.3013 Bayesian Estimates Based On Record Values Under Weighted LINEX Loss Function <p>In this paper, we developed linear exponential (LINEX) loss function by emerging weights to produce weighted linear exponential (WLINEX) loss function. Then we utilized WLINEX to derive scale parameter and reliability function of the Weibull distribution based on record values when the shape parameter is known. After, we estimated scale parameter and reliability function of Weibull distribution by using maximum likelihood (ML) estimation and by several Bayes estimations. &nbsp;The Bayes estimates were obtained with respect to symmetric loss function (squared error loss (SEL)), asymmetric loss function (LINEX) and asymmetric loss function (WLINEX). The ML and the different Bayes estimates are compared via a Monte Carlo simulation study. The result of simulation mentioned that the proposed WLINEX loss function is promising and can be used in real environment especially at the case of underestimate where it revealed better performance than LINEX loss function for estimating scale parameter.</p> Fuad Al-Duais Mohammed Alhagyan Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-02-27 2020-02-27 11 19 10.18187/pjsor.v16i1.2854 Marshall-Olkin Generalized Pareto Distribution: Bayesian and Non Bayesian Estimation <p class="Abstract"><span lang="EN-GB">A new generalization of generalized Pareto Distribution is obtained using the generator Marshall-Olkin distribution (1997). The new distribution MOGP is more flexible and can be used to model non-monotonic failure rate functions. MOGP includes six different sub models: Generalized Pareto, Exponential, Uniform, Pareto type I, Marshall-Olkin Pareto and Marshall-Olkin exponential distribution. We consider different estimation procedures for estimating the model parameters, namely: Maximum likelihood estimator, Maximum product spacing, Least square method, weighted least square method and Bayesian Method. The Bayesian Method is considered under quadratic loss function and Linex loss function. Simulation analysis using MCMC technique is performed to compare between the proposed point estimation methods. The usefulness of MOGP is illustrated by means of real data set, which shows that this generalization is better fit than Pareto, GP and MOP distributions.</span></p> Hanan Haj AHmad Ehab Almetwally Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-02-27 2020-02-27 21 33 10.18187/pjsor.v16i1.2935 Extended Reciprocal Rayleigh Distribution: Copula, Properties and Real Data Modeling <strong>Abstract:</strong> A new extension of the reciprocal Rayleigh distribution is introduced. Simple type copula-based construction is presented for deriving and many bivariate and multivariate type distributions of the reciprocal Rayleigh model. The new reciprocal Rayleigh model generalizes another three reciprocal Rayleigh distributions. The performance of the estimation method is assessed using a graphical simulation study. The new model is better than some other important competitive models in modeling different real data sets. Hoda Ragab Rezk Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-03-10 2020-03-10 35 52 10.18187/pjsor.v16i1.3112 Odd Generalized N-H Generated Family of Distributions with Application to Exponential Model <p>A new family of distributions called the odd generalized N-H is introduced and studied. Four new special models are presented. Some mathematical properties of the odd generalized N-H family are studied. Explicit expressions for the moments, probability weighted, quantile function, mean deviation, order statistics and Rényi entropy are investigated. Characterizations based on the truncated moments, hazard function and conditional expectations are presented for the generated family. Parameter estimates of the family are obtained based on maximum likelihood procedure. Two real data sets are employed to show the usefulness of the new family.</p> Zubair Ahmad M. Elgarhy G.G. Hamedani Nadeem Shafique Butt Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-02-27 2020-02-27 53 71 10.18187/pjsor.v16i1.2295 A comparison of different methods for building Bayesian kriging models Kriging is a statistical approach for analyzing computer experiments. Kriging models can be used as fast running surrogate models for computationally expensive computer codes. Kriging models can be built using different methods, the maximum likelihood estimation method and the leave-one-out cross validation method. The objective of this paper is to evaluate and compare these different methods for building kriging models. These evaluation and comparison are achieved via some measures that test the assumptions that are used in building kriging models. We apply kriging models that are built based on the two different methods on a real high dimensional example of a computer code. We demonstrate our evaluation and comparison through some measures on this real computer code. Younus Hazim Al-Taweel Najlaa Sadeek Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-03-06 2020-03-06 73 82 10.18187/pjsor.v16i1.2921 Transmuted Probability Distributions: A Review <p>Transmutation is the functional composition of the cumulative distribution function (cdf) of one distribution with the inverse cumulative distribution function (quantile function) of another. Shaw and Buckley(2007), first apply this concept and introduced quadratic transmuted family of distributions. In this article, we have presented a review about the transmuted families of distributions. We have also listed the transmuted distributions, available in the literature along with some concluding remarks.</p> Md. Mahabubur Rahman Bander Al-Zahrani Saman Hanif Shahbaz Muhammad Qaiser Shahbaz Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-03-06 2020-03-06 83 94 10.18187/pjsor.v16i1.3217 A New Compound Version of the Generalized Lomax Distribution for Modeling Failure and Service Times <p style="margin-top: 0; margin-bottom: 0;">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.<br style="font-style: normal; font-variant: normal; font-weight: normal; letter-spacing: normal; line-height: normal; orphans: 2; text-align: -webkit-auto; text-indent: 0px; text-transform: none; white-space: normal; widows: 2; word-spacing: 0px; -webkit-text-size-adjust: auto; -webkit-text-stroke-width: 0px;"><br></p> S. Ansari Rezk H. Haitham Yousof Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-03-06 2020-03-06 95 107 10.18187/pjsor.v16i1.2851 The Log-Balakrishnan-Alpha-Skew-Normal Distribution and its Applications <p>In this paper, a new form of log-alpha-skew distribution is proposed by the same methodology of Venegas et al. (2016) and investigated some of its related distributions. The moments and distributional properties of the proposed distribution are also discussed. Also, the appropriateness of this distribution are checked by performing the data fitting experiment and comparing the values of Akaike Information Criterion (AIC) and Bayesian Information Criterion (BIC) with the values of some other known distributions. Likelihood ratio test is used for discriminating between normal and the proposed distributions.</p> Sricharan Shah Subrata Chakraborty Partha Jyoti Hazarika M Masoom Ali Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-03-06 2020-03-06 109 117 10.18187/pjsor.v16i1.3080 Power Comparisons of Five Most Commonly Used Autocorrelation Tests <p>In regression analysis, autocorrelation of the error terms violates the ordinary least squares assumption that the error terms are uncorrelated. The consequence is that the estimates of coefficients and their standard errors will be wrong if the autocorrelation is ignored. There are many tests for autocorrelation, we want to know which test is more powerful. We use Monte Carlo methods to compare the power of five most commonly used tests for autocorrlation, namely Durbin-Watson, Breusch-Godfrey, Box–Pierce, Ljung Box, and Runs tests in two different linear regression models. The results indicate the Durbin-Watson test performs better in the regression model without lagged dependent variable, although the advantage over the other tests reduce with increasing autocorrelation and sample sizes. For the model with lagged dependent variable, the Breusch-Godfrey test is generally superior to the other tests.<br />R code for Power Comparison of the Five Autocorrelation Tests is provided.</p> Stanislaus S. Uyanto Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-03-06 2020-03-06 119 130 10.18187/pjsor.v16i1.2691 The Weibull-G Poisson Family for Analyzing Lifetime Data <p><span class="fontstyle0">We study a new family of distributions defined by the minimum of the Poisson<br>random number of independent identically distributed random variables having a general Weibull-G distribution (see Bourguignon et al. (2014)). Some mathematical properties of the new family including ordinary and incomplete moments, quantile and generating functions, mean deviations, order statistics, reliability and entropies are derived. Maximum likelihood estimation of the model parameters is investigated. Three special models of the new family are discussed. We perform three applications to real data sets to show the potentiality of the<br>proposed family.</span></p> Haitham Yousof Muhammad Mansoor Morad Alizadeh Ahmed Afify Indranil Ghosh Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-03-06 2020-03-06 131 148 10.18187/pjsor.v16i1.2840 Estimation Methods of Alpha Power Exponential Distribution with Applications to Engineering and Medical Data <p>This paper addresses the estimation of the unknown parameters of the alpha<br />power exponential distribution (Mahdavi and Kundu, 2017) using nine frequentist estimation methods. We discuss the nite sample properties of the parameter<br />estimates of the alpha power exponential distribution via Monte Carlo simulations. The potentiality of the distribution is analyzed by means of two real data<br />sets from the elds of engineering and medicine. Finally, we use the maximum<br />likelihood method to derive the estimates of the distribution parameters under<br />competing risks data and analyze one real data set.</p> Mazen Nassar Ahmed Z. Afify Mohammed Shakhatreh Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-03-06 2020-03-06 149 166 10.18187/pjsor.v16i1.3129 A zero truncated discrete distribution: Theory and applications to count data <p>The analysis and modeling of zero truncated count data is of primary interest in many elds such as engineering, public health, sociology, psychology, epidemiology. Therefore, in this article we have proposed a new and simple structure model, named a zero truncated discrete Lindley distribution. The<br>distribution contains some submodels and represents a two-component mixture of a zero truncated geometric distribution and a zero truncated negative binomial distribution with certain parameters. Several properties of the distribution are obtained such as mean residual life function, probability generating function, factorial moments, negative moments, moments of residual life function, Bonferroni and Lorenz curves, estimation of parameters, Shannon and Renyi entropies, order statistics with the asymptotic distribution of their extremes and range, a characterization, stochastic ordering and stress-strength parameter. Moreover, the collective risk model is discussed by considering the<br>proposed distribution as primary distribution and exponential and Erlang distributions as secondary ones. Test and evaluation statistics as well as three real data applications are considered to assess the peformance of the distribution among the most frequently zero truncated discrete probability models.</p> Tassaddaq Hussain Kiani Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-03-06 2020-03-06 167 190 10.18187/pjsor.v16i1.2133 Impact of Different Repetitive Sampling Schemes on the Performance of X-bar Control Chart <p>Repetitive sampling has been found very popular in improving the control chart techniques for last couple of years. In repetitive sampling based control charts, there are two additional control limits inside the usual upper control limit (UCL) and lower control limit (LCL). If any subgroup crosses these limits but remain inside outer limits, it is deferred and replaced with another selection. Process is said to be out of control if any subgroup falls outside UCL/LCL. In this article, the technique has been modified by introducing a relation between outer and inner control limits in terms of a ratio and need of this modification has also been justified by highlighting a gap in the existing technique. By using Monte Carlo simulation, several results have been generated relevant to different sample sizes and introduced ratios. The results have been described with the help of average run length (ARL) tables that how the efficiency of control chart is effected by using different ratios. The modification in the technique also provides variety of alternatives within the scope of repetitive based control charts. All the discussed options have summarized to one table to see that how the control limits under this technique behave and impact on detecting shifts in the process average. The schemes have been interpreted in the light of above ratio and their comparison has been described under different sample sizes that facilitate the user to select most appropriate scheme for a desired process control. An example has been included by choosing one of the proposed schemes to show the application and performance of the proposed control chart in a manufacturing process.&nbsp;</p> Muhammad Anwar Mughal Muhammad Azam Muhammad Aslam Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-03-06 2020-03-06 191 201 10.18187/pjsor.v16i1.1876 Transient Analysis of M[X1],M[X2]/G1,G2/1 retrial queueing system with priority services, working breakdown, start up/close down time, Bernoulli vacation, reneging and balking <p>This paper considers&nbsp; M[X1],M[X2]/G1,G2/1&nbsp;general retrial queueing system with priority services. Two types of customers from different classes arrive at the system in different independent compound Poisson processes. The server follows the pre-emptive priority rule subject to working breakdown, startup/closedown time and Bernoulli vacation with general (arbitrary) vacation periods. After completing the service, if there are no priority customers present in the system the server may go for a vacation or close down the system. On completion of the close down, the server needs some time to set up the system. The priority customers who find the server busy are queued in the system. A low-priority customer who find the server busy are routed to a retrial (orbit) queue that attempts to get the service. The system may breakdown at any point of time when it is in operation. However, when the system fails, instead of stopping service completely, the service is continued only to the high priority customers at a slower rate. We consider balking to occur to the low priority customer while the server is busy or idle, and reneging to occur at the high priority customers during server’s vacation, start up/close down time. Using the supplementary variable technique, we derive the joint distribution of the server state and the number of customers in the system. Finally, some performance measures and numerical examples are presented.</p> Govindhan Ayyappan Udayageetha J Copyright (c) 2020 Pakistan Journal of Statistics and Operation Research 2020-03-06 2020-03-06 203 216 10.18187/pjsor.v16i1.2181