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• Open Access   Article

  An Inventory Model for Perishable Goods with Controllable Deterioration, Partially Backlogged Shortage and Advanced Payment

  Anima Bag, Rojalini Patro

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.10 , Issue.5 , pp.1-6, Oct-2023

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  Abstract
  The purpose of the paper is to develop an optimal strategy for the inventory system for perishable goods. Discount facility is adopted here for advanced payment. Demand is partially backlogged. Preservation technology is applied in the proposed model to reduce the rate of deterioration. An inventory model has been developed and the optimum order quantity as well as the time at which shortage occurs are determined which maximizes the total profit. To observe the practical utility of the proposed model, three numerical illustrations are provided which are solved by Mathematica 5.1 software. Concavity of the average profit is examined graphically. Finally a post optimality analysis is performed to show the effects of inventory parameters on optimal policy. The proposed model can be used for decaying items like fruits, vegetables, food grains and medicines that are subject to preservation technology to maximize the total profit.

  Key-Words / Index Term
  Inventory, Decaying Items, Preservation Technology, Partial Backlogging, Advance Payment

  References
  [1] A.K., Maiti, A.K. Bhunia, M. Maiti, “An application of real coded genetic algorithm (RCGA) for mixed integer non-linear programming in two-storage multi-item inventory model with discount policy”, Applied Mathematics and Computation. Vol.183, Issue2, pp 903–915, 2006.
  [2] Anima Bag, P.K. Tripathy, “Decision support model with price and time induced demand under multilevel conditional deliveries” published in the journal, Investigacion operacional, Vol 40, Issue 4, pp 441-451, 2019.
  [3] M.A. Khan, A.A. Shaikh,., G.C. Panda, I., Konstantaras, L.E. Ca´rdenas-Barro´n, “The effect of advance payment with discount facility on supply decisions of deteriorating products whose demand is both price and stock dependent”, International Transactions in Operational Research, Vol 27, Issue 3, pp1343–1367, 2020.
  [4] A.Thangam, “Optimal price discounting and lot-sizing policies for perishable items in a supply chain under the advance payment scheme and two-echelon trade credits”, International Journal of Production Economics, Vol 139, Issue 2, pp 459–472, 2012.
  [5] M.A. Khan, S. Ahmed, M.S. Babu, N. Sultana, “Optimal lot-size decision for deteriorating items with price-sensitive demand, Hybrid price and stock dependent inventory model for perishable goods with advance 3463 linearly time-dependent holding cost under all-units discount environment”, International Journal of Systems Science: Operations & Logistics Vol 9, Issue 2, pp 1–14, 2020.
  [6] L.N. De, “A study of inventory model for deteriorating items with price and stock dependent demand under the joined effect of preservation technology and price discount facility”, Journal of Mathematics and Computer Science, Vol 10, Issue 5, pp 1481–1498, 2020.
  [7] Q. Zhang, D. Zhang, Y.-C. Tsao, J. Luo, “Optimal ordering policy in a two-stage supply chain with advance payment for stable supply capacity”, International Journal of Production Economics, Vol 177, pp 34–43, 2016.
  [8] R. Patro, M. Acharya, and M. M. Nayak, “An EOQ model for fuzzy defective rate with allowable proportionate discount”, Opsearch , Vol 26, Issue 1, pp 191-215, 2019.
  [9] S. Sana, K.S. Chaudhuri, “A stock-review EOQ model with stock-dependent demand, quadratic deterioration rate”, Advanced Modeling and Optimization, Vol 6, Issue 2, pp 25–32, 2004.
  [10] Z.T. Balkhi, L. Tadj, “A generalized economic order quantity model with deteriorating items and time varying demand, deterioration and costs”, International Transactions in Operational Research, Vol 15, Issue 4, pp 509–517, 2008.
  [11] R. Maihami, I. Nakhai Kamalabadi, “Joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging and time and price dependent demand”, International Journal of Production Economics, Vol 136, Issue1, pp 116–122, 2012.
  [12] A.A. Shaikh, G.C. Panda, S. Sahu, A.K. Das, “Economic order quantity model for deteriorating item with preservation technology in time dependent demand with partial backlogging and trade credit”, International Journal of Logistics Systems and Management, Vol 32, Issue 1, 2019.
  [13] H. Hsu, H.M. Wee, H.M. Teng, “Preservation technology investment for deteriorating inventory”, International Journal of Production Economics, Vol 124 ,Issue 2, pp 388–394, 2010.
  [14] Y.-P. Lee, C.-Y. Dye, “An inventory model for deteriorating items under stock-dependent demand and controllable deterioration rate”, Computer and Industrial Engineering, Vol 63, Issue 2, pp 474–482, 2012.
  [15] Q. Zhang, Y.-C. Tsao, T.-H. Chen, “Economic order quantity under advance payment”, Applied Mathematical Modeling, Vol 38 , Issue 24, pp 5910–5921, 2014.
  [16] U. Mishra, L.E. Ca´rdenas-Barro´n, S. Tiwari, A.A. Shaikh, G. Trevin˜o-Garza, “An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment”, Annals of Operations Research, Vol 254 (1-2), pp165–190, 2017.
  [17] C.-T. Yang, C.-Y. Dye, J.-F. Ding, “Optimal dynamic trade credit and preservation technology allocation for a deteriorating inventory model”, Computer and Industrial Engineering, Vol 87, pp 356–369, 2015.
  [18] Q. Zhang, D. Zhang, Y.-C. Tsao, J. Luo, “Optimal ordering policy in a two-stage supply chain with advance payment for stable supply capacity”, International Journal of Production Economics, Vol. 177, pp. 34–43, 2016.
  [19] U. Mishra, L.E. Ca´rdenas-Barro´n, S. Tiwari, A.A. Shaikh, G. Trevin˜o-Garza, “An inventory model under price and stock dependent demand for controllable deterioration rate with shortages and preservation technology investment”, Annals of Operations Research, Vol 254 (1-2), pp. 165–190, 2017.
  [20] S.C. Das, A.M. Zidan, A.K. Manna, A.A. Shaikh, A.K. Bhunia, “An application of preservation technology in inventory control system with price dependent demand and partial backlogging”, Alexandria Engineering Journal, Vol. 59(3), pp.1359-1369, 2020.
  [21] H.L. Yang, J.T. Teng, M.S. Chern, “ An inventory model under inflation for deteriorating items with stock-dependent consumption rate and partial backlogging shortages”, International Journal of Production Economics, Vol 123, Issue1, pp 8–19, 2010.
  [22] P.K. Tripathy, A.Bag, “Optimal disposal strategy with controllable deterioration and shortage” published in the journal, International Journal of Agricultural and Statistical Sciences, Vol. 15,Issue1, pp 271-279, 2019.
  [23] A. Bag, P.K. Tripathy, “Decision support model with price and time induced demand under multilevel conditional deliveries” published in the journal, Investigacion operacional, Vol. 40, Number 4, pp. 441-451, 2019.
  [24] P.K. Tripathy, A. Bag, “Decision support model with default risk under conditional delay” published in the journal, International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol. 5, Issue 2, pp. 40-45, 2018.
  [25] R. Patro, M. Acharya, M. M. Nayak, and S. Patnaik, “A fuzzy inventory model with time dependent weibull deterioration, quadratic demand and partial backlogging”, International Journal of Management and Decision Making, Vol.16, Issue3, pp. 243-279, 2017.

  Citation
  Anima Bag, Rojalini Patro, "An Inventory Model for Perishable Goods with Controllable Deterioration, Partially Backlogged Shortage and Advanced Payment," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.10, Issue.5, pp.1-6, 2023

• Open Access   Article

  Break Points in Life Table Mean Residual Life Times and Estimated Hazard Rates

  Abhijeet Jadhav, Rohit R Mutkekar, S. B. Munoli

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.10 , Issue.5 , pp.7-13, Oct-2023

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  Abstract
  The paper is notably focused on statistics and technical aspects of Mean Residual Life Time (MRLT) and Hazard Rate (HR). A polynomial with appropriate degree is fit to the life table MRLT data of countries belonging to different income groups and geological regions. Break points of MRLTs at various age of human life cycle are studied through segmented regression. HRs are estimated using respective MRLTs, break points in estimated HRs are analyzed.

  Key-Words / Index Term
  Mean Residual Life Time, Hazard Rate, Life Table, Segmented Regression, Polynomial Regression, Break Points.

  References
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  [13]. Lai, Chin-Diew, Min Xie, “Mean Residual Life - Concepts and Applications in Reliability Analysis,” Stochastic Ageing and Dependence for Reliability, Springer, New York, NY, pp.109-138. 2006.
  [14]. Vito M. R. Muggeo, “Segmented: an R package to fit regression models with broken-line relationships,” R news, Vol.8, Issue.1, pp.20-25, 2008.
  [15]. Melody S Goodman, Yi Li, and Ram C Tiwari, “Detecting multiple change points in piecewise constant hazard functions,” Journal of applied statistics, Vol.38, Issue.11, pp.2523-2532, 2011. https://doi.org/10.1080%2F02664763.2011.559209.
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  Citation
  Abhijeet Jadhav, Rohit R Mutkekar, S. B. Munoli, "Break Points in Life Table Mean Residual Life Times and Estimated Hazard Rates," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.10, Issue.5, pp.7-13, 2023

• Open Access   Article

  Calibration Ratio Estimator for Estimating Population Mean in Stratified Random Sampling Design Using Properties of Two Supplementary Variables

  Etebong Clement, Idorenyin Etukudoh

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.10 , Issue.5 , pp.14-21, Oct-2023

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  Abstract
  The performances of most improved ratio estimators may depend on some optimality conditions. This study introduced calibration weightings that produce more efficient ratio estimator that is independent on any optimality conditions. Calibration ratio estimator for estimating population mean when two auxiliary variables are available is proposed under the stratified random sampling design. The bias and mean square error expressions for the proposed estimator are formulated using the principle of large sample approximation. The method was adopted to guarantee optimal performances of the proposed estimator in terms of minimum mean square error and efficient bias reduction. Analytical and numerical illustrations were carried out to compare the performances of the proposed estimator with existing ratio and regression-type estimators of population mean under study. Analytically, the proposed estimator under certain prescribed conditions has been showed to perform better than all existing estimators under review. The numerical results also showed that the proposed estimator is substantially superior in terms of efficiency to all existing estimators under study including the classical multivariate regression estimator of population mean by Cochran [31]. This is against an established fact in survey sampling literature that generally the regression estimator is more efficient than the ratio and product estimators of population mean. Analysis and evaluation are presented.

  Key-Words / Index Term
  Bias reduction, calibration weightings, efficiency, mean square error, ratio estimation

  References
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  [2] Sharma, B. and Tailor, R. ‘A new ratio-cum-dual to ratio estimator of finite population mean in simple random sampling’. Global Journal of Science, Vol.10, Issue.1, pp.27-31, 2010.
  [3] Diana, G., Giordan, M and Perri P.F. ‘Improved class of estimators for the population mean’, Statistical methods and Applications, Vol.20, pp.123-140, 2011.
  [4] Solanki, R. S., Singh, H. P. and Rathour, A. ‘An alternative estimator for estimating the finite population mean using auxiliary information in sample surveys’, ISRN Probability and Statistics Article ID 657682, doi:10.5402/2012/657682, 2012
  [5] Swain, A.K.P.C. ‘On classes of modified ratio type and regression-cum-ratio type estimators in sample surveys using two auxiliary variables’. Statistics in Transition-New Series, Vol.13, Issue.3, pp.473-494, 2012.
  [6] Haq, A. and Shabbir, J. ‘Improved family of ratio estimators in simple and stratified random Sampling’, Communications in Statistics Theory and Methods Vol.42, Issue.5, pp.782-799, 2013.
  [7] Singh, R. V. K. and Audu, A. ‘Efficiency of ratio estimators in stratified random sampling using information on auxiliary attribute’, International Journal of Engineering and Innovative Technology, Vol.2, Issue.1, pp.166-172, 2013.
  [8] Shittu, O. I. and Adepoju, K. A. ‘On the efficiency of modified ratio estimator based on linear combination of kurtosis, median and quartile deviation’, International Journal of Modern Mathematics Sciences. Vol.11, Issue.2, pp.103-107, 2014.
  [9] Clement, E. P. ‘An Improved Ratio Estimator for Population Mean in Stratified Random Sampling’ European Journal of Statistics and Probability’ Vol.4, Issue.4, pp.12-17, 2016.
  [10] Clement, E. P. ‘Efficient Exponential Estimators of Population Mean in Survey Sampling’. International Journal of Mathematics and Computation Vol.28, Issue.3, pp.94-106, 2017.
  [11] Kamlesh Kumar and Mukesh Kumar ”Two-phase sampling exponential type estimator for ratio and product of two population means in the presence of non-response” International Journal of Scientific Research in Mathematical and Statistical Sciences,Vol.4, Issue.6, pp.26-34, 2017.
  [12] Clement, E. P. ‘Improved Family of Ratio Estimators of Finite Population Variance in Stratified Random Sampling’ Biostatistics and Biometrics Open Access Journal Vol.5, Issue.2, No: 555659, 2018a.DOI:10.19080/BBOAJ.2018.04.555659. 2018a.
  [13] Nikita and Malik S. ”Generalized Logarithmic Ratio and Product type estimators in Simple Random Sampling (SRS)” International Journal of Scientific Research in Mathematical and Statistical Sciences,Vol.9, Issue.6, pp.56-60, 2022.
  [14] Deville, J.C. and Sarndal, C. E. ‘Calibration estimators in survey sampling’ Journal of the American Statistical Association, Vol.87, pp.376-382, 1992.
  [15] Kim, J.M., Sungur, E.A. and Heo, T.Y. ‘Calibration approach estimators in stratified sampling’. Statistics and Probability Letters, Vol.77, Issue.1, pp.99-103. 2007.
  [16] Kim, J.K. and Park, M. ‘Calibration estimation in survey sampling’. International Statistical Review, Vol.78, Issue.1, pp.21-29, 2010.
  [17] Rao, D, Khan, M. G. M. and Khan, S. ‘Mathematical programming on multivariate calibration estimation in stratified sampling’, World Academy of Science, Engineering and Technology, Vol.72, pp.12-27, 2012.
  [18] Koyuncu, N., and Kadilar, C. ‘Calibration weighting in stratified random sampling’. Communications in Statistics: Simulation and Computation Vol.45, Issue.7, pp.2267-2275, 2016.
  [19] Clement, E. P., Udofia, G.A. and Enang, E. I. ‘Estimation for domains in stratified random sampling in the presence of non-response’. American Journal of Mathematics and Statistics Vol.4 Issue.2 pp.65-71, 2014.
  [20] Clement, E. P. and Enang, E. I. ‘Calibration approach alternative ratio estimator for population mean in stratified sampling’. International Journal of Statistics and Economics, Vol.16, Issue.1, pp 83-93, 2015.
  [21] Clement, E. P. and Enang, E. I. ‘On the Efficiency of Ratio Estimator over the Regression Estimator’, Communication in Statistics: Theory and Methods, Vol.46 Issue.11, pp.5357-5367, 2017.
  [22] Clement, E. P. ‘A Note on Calibration Weightings in Stratified Double Sampling with Equal Probability’ Communication in Statistics: Theory and Methods, Vol.47. Issue12 pp.2835-2847. 2018b
  [23] Clement, E. P. ‘On the Improvement of Multivariate Ratio Method of Estimation in Sample Surveys by Calibration Weightings’ Asian Journal of Probability and Statistics, Vol.10 Issue.1, pp.1-12, 2020, DOI: 10.9734/AJP AS/2020/v10i130236.
  [24] Clement, E. P, ‘Ratio-Product Estimator in Stratified Double Sampling based on the coefficient of skewness of the Auxiliary Variable’ International Journal of Statistics and Applied Mathematics, Vol.6 Issue 1, pp.24-28, 2022
  [25] Clement, E. P. ‘On the Precision of Estimators in Sampling Surveys by Multi-Parametric Calibration weightings’, Thailand Statistician Vol. 20 Issue 3, pp. 655-668, 2022
  [26] Clement, E. P. and Inyang, E. J.‘A Class of Modified Calibration Ratio Estimators of Population Mean with Known Coefficient of Kurtosis in Stratified Double Sampling’ Journal of modern Mathematics and Statistics Vol.14 Issue 4, pp.55-60, 2020.
  [27] Clement, E. P. and Inyang, E. J. ‘Improving the Efficiency of Ratio Estimators by Calibration Weightings’ International Journal of Statistics and Mathematics; Vol.8 Issue.1, pp.164–172, 2021.
  [28] Enang, E. I. and Clement, E. P., ‘An efficient class of calibration Ratio Estimators of Domain Mean in Survey Sampling’ Communication in Mathematics and Statistics Vol. 8: pp 279-293, 2020. DOI: 10.1007/s40304-018-00174-z
  [29] Clement, E.P., Ekpoese, E.D. and Inyang, E.J. ‘Improving the Efficiency of Estimators of Population Mean in Systematic Sampling Using Inverse Exponentiation’ International Journal of Statistics and Applied Mathematics; Vol.6, Issue.3, pp.87-97.2021
  [30] Olkin, I. ‘Multivariate ratio estimation for finite populations’. Biometrika, Vol.45, Issue1–2, pp.154–165, 1958, doi:10.1093/biomet/45.1-2.154.
  [31] Cochran, W. G. ‘Sampling techniques’. New York: Wiley and Sons. 1977
  [32] Abu-Dayyeh, W. A., Ahmad, M. S., Ahmad, R. A., et al. (2003). Some estimators of a finite population mean using auxiliary information. Applied Mathematics and Computation, Vol.139, pp.287–298, 2003.
  [33] Singh, H. P. and Kumar, S. ‘Estimation of mean in presence of non-response using two phase sampling scheme’, Statistical Papers, Vol.51, pp.559–582, 2010.
  [34] Riaz, N, Noor - ul - Amin, M. and Hanif, M. ‘Regression-Cum-exponential Ratio-type Estimators for population mean’, Middle – East journal of Scientific Research, Vol.19, Issue.12, pp.1716–1721, 2014.
  [35] Lu, J., Yan, Z., and Aegerter, C. M. ‘A class of ratio estimators of a finite population mean using two auxiliary variables’. PLOS ONE, Vol.9, Issue.2:e89538. 2014, doi:10.1371/journal. pone.0089538

  Citation
  Etebong Clement, Idorenyin Etukudoh, "Calibration Ratio Estimator for Estimating Population Mean in Stratified Random Sampling Design Using Properties of Two Supplementary Variables," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.10, Issue.5, pp.14-21, 2023

• Open Access   Article

  Cordiality in the Context of Duplication in Quadrilateral Snake Graph

  U. M. Prajapati, Rohoullah Asad, R. M. Gajjar

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.10 , Issue.5 , pp.22-25, Oct-2023

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  Abstract
  Let G = (V(G),E(G)) be a graph and let g? V(G) ? {0,1} be a mapping from the set of vertices to {0,1} and for each edge uv ? E assign the label | g(u)- g(v)|. If the number of vertices labeled with 0 and the number of vertices labeled with 1 differ by at most 1 and the number of edges labeled with 0 and the number of edges labeled with 1 differ by at most 1, then f is called a cordial labeling. We discuss cordial labeling of graphs obtained from duplication of certain graph elements in quadrilateral snake graphs.

  Key-Words / Index Term
  Graph Labeling, Cordial Labeling, Cordial Graph, Quadrilateral Snake Graph.

  References
  [1] V. Ramachandran and C. Sekar, “One modulo N gracefulness of n-polygonal-snakes, C_n^((t)) and P_(a,b)”, Internat. J. Engineering Res. and tech., (IJERT), Vol.2, issue.10, pp.3514-3529.
  [2]. S. K. Vaidya and C. M. Barasara, “Product Cordial graph in context of some graph operations”, J. Math. Comput. Sci. Vol.1, issue.2, pp.1-6, 2011.
  [3] S. K. Vaidya and N. A. Dani, “Cordial and 3-Equitable Graphs Induced by Duplication of Edge”, Mathematics Today, 27, pp.71-82, 2011.
  [4] I. Cahit, “Cordial Graphs: A weaker version of graceful and harmonious Graphs”, Ars Combinatoria, 23, pp.201-207, 1987.
  [5] D. B. West, “Introduction to Graph Theory”, Prentice-Hall of India, New Delhi (2001).
  [6] J.A. Gallian, “A Dynamic Survey of Graph Labeling”, The Electronic Journal of Combinatorics 25, 2022.
  [7] J. Gross and J. Yellen, “Graph Theory and Its Applications”, CRC Press, Boca Raton, 1999.
  [8]. U.M. Prajapati and R. M. Gajjar, “Cordial Labeling of Complement of some Graph”, Mathematics Today, 30, pp.99-118, 2015.
  [9]. U.M. Prajapati and R. M. Gajjar, “Cordial Labeling in the Context of Duplication of some Graph Elements”, International Journal of Mathematics and Soft Computing, Vol.6, No.2, pp.65 – 73, 2016.

  Citation
  U. M. Prajapati, Rohoullah Asad, R. M. Gajjar, "Cordiality in the Context of Duplication in Quadrilateral Snake Graph," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.10, Issue.5, pp.22-25, 2023

• Open Access   Article

  On the Estimation of Current Population Median in Two-Occasion Rotation Sampling

  Mandu Otto Inyang, Etebong Clement

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.10 , Issue.5 , pp.26-33, Oct-2023

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  Abstract
  It has been observed that most authors addressed their estimation problems in successive sampling based on ideal condition that variable of interest is normally distributed; an assumption that may not always be the case in practical situations. Supposed the normality condition does not hold; and that the variable of interest follows a distribution that is highly skewed; this study is a proposal in this direction. This paper deals with problem of estimating the current population median in two-occasion rotation sampling; using the convex combination of two estimators; one based on u units drawn afresh at the current occasion and the other based on m units retained from the previous occasion. The bias and Mean Square Error (MSE) as well as Optimal Replacement Policy are formulated for the proposed estimator. The empirical study conducted to validate the efficacy of the new proposal proved its dominances over existing estimators under review.

  Key-Words / Index Term
  convex combination, efficiency, measure of central tendency, normality assumption, optimal replacement policy, outliers, successive sampling

  References
  [1] Singh, H. P. and Vishwakarma, G. K. ‘Modified exponential ratio and product estimators for finite population mean in double sampling.”Austrian Journal of Statistics, Vol.36, Issue.3, pp.217-225, 2007.
  [2] Clement, E. P. ‘An Improved Ratio Estimator for Population Mean in Stratified Random Sampling’ European Journal of Statistics and Probability’ Vol.4, Issue.4, pp.12-17, 2016.
  [3] Clement, E. P. ‘Efficient Exponential Estimators of Population Mean in Survey Sampling’. International Journal of Mathematics and Computation Vol.28 Issue.3 pp.94-106, 2017.
  [4] Clement, E. P. and Enang, E. I. ‘On the Efficiency of Ratio Estimator over the Regression Estimator’, Communication in Statistics: Theory and Methods, Vol.46, Issue.11, pp.5357-5367, 2017.
  [5] Kamlesh Kumar and Mukesh Kumar ”Two-phase sampling exponential type estimator for ratio and product of two population means in the presence of non-response” International Journal of Scientific Research in Mathematical and Statistical Sciences,Vol.4, Issue.6, pp.26-34, 2017.
  [6] Izunobi, C. H. and Onyeka, A, C. “Logarithmic ratio and product-type Estimators of population mean”. International Journal of Advanced Statistics and Probability.,Vol.7, Issue 2, pp.47-55. 2019.
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  [8] Clement, E. P. and Inyang, E. J.‘A Class of Modified Calibration Ratio Estimators of Population Mean with Known Coefficient of Kurtosis in Stratified Double Sampling’ Journal of modern Mathematics and Statistics Vol.14 Issue 4, pp.55-60, 2020.
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  Citation
  Mandu Otto Inyang, Etebong Clement, "On the Estimation of Current Population Median in Two-Occasion Rotation Sampling," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.10, Issue.5, pp.26-33, 2023