Volume-5 , Issue-6 , Dec 2018, ISSN 2348-4519 (Online) Go Back

• Open Access   Article

  Harmonious and Pell Labeling for Some Extended Duplicate Graph

  R. Avudainayaki, B. Selvam

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.152-156, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.152156

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  Abstract
  A Graph G is said to be harmonious if there exist an injection f : V(G)→ Zq such that the induced function f *: E(G)→ Zq defined by f *(uv) = (f(u) + f(v)) (mod q) is a bijection. Let G be a (p, q) graph. If there exists a mapping f: V(G) →{ 0,1, …., p-1} such that the induced function f* : E(G) → N given by f*(uv) = f(u) +2 f(v) for every uv E(G) are all distinct where u,v ≥ 0, then the function f is called a Pell labeling. In this paper, we proved that the extended duplicate graph of arrow graph and splitting graph of path admits harmonious and pell labeling.

  Key-Words / Index Term
  Arrow graph, Splitting graph of path, Duplicate graph, Harmonious , Pell labeling

  References
  [1] Gallian J.A, “A Dynamic Survey of graph labeling”, the Electronic Journal of combinatories, 19, # DS6 (2015).Pp. 189-198.
  [2] R. L. Graham and N.J.A. Sloane, on additive bases and harmonious graphs, SIAM, J. Alg. Disc. Meth. 1, (1980), 382 – 404.
  [3] Rosa A, On certain Valuations of the vertices of a graph, Theory of graphs (Internat. Symposium, Rome, July 1966), Gordon and Breech, N.Y. and Dunod paris, 1967.pp. 349- 355.
  [4] E.Sampath kumar, “On duplicate graphs”, Journal of the Indian Math. Soc. 37 (1973), 285 – 293.
  [5] J.Shiama, Pell labeling for some graphs, Asian Journal of Current Engineering and Maths, 2(4) (2013).
  [6] Thirusangu,K, Selvam.B. and Ulaganathan.P.P, Cordial labelings in extended duplicate twig graphs ,International Journal of computer, mathematical sciences and applications, Vol.4, Nos 3-4, (2010), pp.319-328.
  

  Citation
  R. Avudainayaki, B. Selvam, "Harmonious and Pell Labeling for Some Extended Duplicate Graph," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.152-156, 2018

• Open Access   Article

  Total Neighborhood Prime Labeling of Some Graphs

  N P Shrimali, Parul B Pandya

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.157-163, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.157163

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  Let G=(V,E) be a graph with p vertices and q edges. A bijection f:V(G)∪E(G) → {1,2,3,…,p + q} is said to be total neighborhood prime labeling if it satisfies that, for each vertex of degree at least two, the gcd of labeling on its neighborhood vertices is one and for each vertex of degree at least two, the gcd of labeling on the induced edges is one. In this paper we investigate total neighborhood prime labeling for comb, disjoint union of paths, disjoint union of sunlet graphs, disjoint union of wheel graphs, graph obtained by one copy of path P_n (which has n vertices) and n copies of k_(1,m) and joining i^th vertex of P_n with an edge to fix vertex in the i^th copy of of k_(1,m) , corona product of cycle and star and subdivision of bistar.

  Key-Words / Index Term
  Neighborhood prime labeling, Total Neighborhood prime labeling, disjoint union of graphs, corona product of graphs, subdivision of graph

  References
  [1] A. Tout, A. N. Dabboucy, K. Howalla, “Prime labeling of graphs”, Nat. Acad. Sci. Letters, 11 (1982), 365-368.
  [2] S. K. Patel, N. P. Shrimali,” Neighborhood-prime labeling” ,
  International Journal of Mathematics and Soft Computing, 5, No. 2.
  (2015), 135-143.
  [3] Rajesh Kumar T.J., Mathew Varkey T.K., “A note on Total neighbourhood prime labeling”, International journal of Mathematics Combinatorics, 4 (2018).
  [4] J. Gross, J. Yellen, “Graph theory and its applications”, CRC press,
  (1999).
  [5] J.A.Gallian, “A dynamic survey of graph labeling”, The Electronic
  Journal of Combinatorics (2017), # DS6.
  [6] David M. Burton, Elementary Number Theory, Sixth Edition, Tata
  MacGraw Hill, India, 2012
  [7] Parul B. Pandya, N P Shrimali, “Vertex edge neighborhood prime labeling of some graphs”, International Journal of Scientific Research and Review Volume 7, Issue 10, 2018
  [8] S. K. Patel, N. P. Shrimali, “Neighborhood-prime labeling of some union graphs”, International Journal of Mathematics and Soft Computing, 6, - No. 1 (2016), 39-47.
  

  Citation
  N P Shrimali, Parul B Pandya, "Total Neighborhood Prime Labeling of Some Graphs," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.157-163, 2018

• Open Access   Article

  Parametric Generalizations of ‘Useful’ R-Norm Fuzzy Information Measures

  S.M. Sofi, S. Peerzada, M.A.K. Baig

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.164-169, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.164169

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  In this paper, we introduce a family of ‘useful’ R-norm fuzzy information measures. These measures include no parameter, one parameter and two parameters. We show that these proposed measures are valid fuzzy information measures and are the generalizations of various ‘useful’ R-norm fuzzy information measures. Further, to establish its validity we take a numerical data and obtain the results by using R-software. In addition, we study the monotonic behaviour of the proposed measures with respect to the parameters.

  Key-Words / Index Term
  Fuzzy Set, Fuzzy Information Measure, ‘Useful’ Information Measure, R-Norm Fuzzy Information Measure

  References
  [1] C. E. Shannon, “A Mathematical Theory of Communication”, Bell System Technical Journal, Vol. 27, pp. 379-423, 623-659, 1948.
  [2] A. Renyi. “On Measure of Entropy and Information”, Proceeding Fourth Berkley Symposium on Mathematical Statistics and Probability, University of California, Press 1, pp. 546-561, 1961.
  [3] B. D. Sharma and I. J. Taneja, “Entropies of Type α, β and Other Generalized Measures of Information Theory”, Mathematika, Vol. 22, pp. 202-215, 1975.
  [4] S. Peerzada, S. M. Sofi and R. Nisa, “A New Generalized Fuzzy Information Measure and its Properties” International Journal of Advance Research in Science and Engineering, Vol. 6, No. 12, pp. 1647-1654, 2017.
  [5] S. M. Sofi, S. Peerzada and A. H. Bhat, “Coding Theorems on New ‘Useful’ Fuzzy Information Measure of Order Alpha and Type Beta”, International Journal of Advance Research in Science and Engineering, Vol. 7, No. 1, pp. 112-120, 2018.
  [6] L. A. Zadeh, “Fuzzy Sets”, Information and Control, Vol. 8, pp. 338-353. 1965.
  [7] De Luca and S. Termini, “A Definition of Non-Probabilistic Entropy in the Setting of Fuzzy Set Theory”, Information and Control, Vol. 20, pp.301-312, 1972.
  [8] D. E. Boekee and J. C. A. Van der Lubbe, “The R-norm Information Measure”, Information Control 45, pp. 136-145, 1980.
  [9] D. S. Hooda and A. Ram, “Characterization Of The Generalized R-Norm Entropy” Caribbean Journal of Mathematical & Computer Science, Vol. 8, 1998.
  [10] V. P. Tomar and A. Ohlan, “Two New Parametric Generalized R-norm Fuzzy Information Measures”, International journal of computer applications, Vol. 93, No. 13, pp. 22-27, 2014.
  [11] S. Peerzada, S. M. Sofi and M. A. K. Baig, “Properties of New Generalized R-Norm Information Measure and Its Bounds” (communicated).
  [12] D. S. Hooda, “On Generalized Measures of Fuzzy Entropy”, Mathematica Slovaca, Vol. 54, pp. 315-325, 2004.
  [13] M. Belis and S. Guiasu, “A Quantitative-Qualitative Measure of Information in Cybernetic System”, IEEE Transaction on Information Theory, Vol. 14, pp. 593-594, 1968.
  [14] G. Longo, “Quantitative-Qualitative Measure of Information, Springer-Verlag”, New York, 1972.
  [15] R. P. Singh, R. Kumar and R. K. Tuteja, “Application of Holder’s Inequality in Information Theory”, Information Sciences, 152, pp. 145–154, 2003.
  [16] S. Kumar, “Some More Results on a Generalized ‘Useful’ R-norm Information Measure”, Tamkang Journal of Mathematics, Vol. 40, No. 2, pp. 211-216, 2009.
  [17] D. S. Hooda., K. Upadhyay and D. K. Sharma, On Parametric Generalization of ‘Useful’ R-Norm Information Measure, British Journal Of Mathematics & Computer Science, 8, Vol. 1, pp. 1-15, 2015.
  [18] S. M. Sofi, S. Peerzada and A. H. Bhat, “Two Parametric Generalized ‘Useful’ R-Norm Information Measure & Its Coding Theorems.” (accepted for publication in PJS).
  

  Citation
  S.M. Sofi, S. Peerzada, M.A.K. Baig, "Parametric Generalizations of ‘Useful’ R-Norm Fuzzy Information Measures," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.164-169, 2018

• Open Access   Article

  Correlated Gamma Frailty Model Based on Logistic Exponential Baseline Distribution

  A.Pandey, Lalpawimawha, S.Bhushan, P.K. Misra

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.170-176, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.170176

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  Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g. matched pairs experiments, twin or family data), the shared frailty models were suggested. Shared frailty models are used despite their limitations. To overcome their disadvantages correlated frailty models may be used. In this paper, we propose correlated gamma frailty model with logistic exponential distribution as baseline distribution to analyze real-life bivariate survival dataset of McGilchrist and Aisbett [9] related to kidney infection. The Bayesian approach of Markov Chain Monte Carlo was employed to estimate the parameters involved in the models and modles comparison was done by using Bayesian comparison techniques such as Akaike information criteria (AIC), Bayesian information criteria (BIC), Deviance information criteria (DIC) and Bayes factor. Simulation study also carried out to compare the true values of parameters and estimated values of the parameters. A better model suggested for the data.

  Key-Words / Index Term
  Bayesian model comparison, Correlated gamma frailty,Logistic exponential distribution, MCMC

  References
  [1].D.Clayton, J.Cuzick, “ Multivariate Generalizations of the Proportional Hazards Model”, Royal Statistical Society A, Vol.48, pp.82-117,1985.
  [2] A.Gelman, D.B.Rubin,” A single series from the Gibbs sampler provides a false sense of security. In Bayesian Statistics 4 “(J. M. Bernardo, J. 0.Berger, A. P. Dawid and A. F. M. Smith, eds.).Oxford Univ. Press, Oxford, pp.625-632, 1992.
  [3] J.Geweke, "Evaluating the Accuracy of Sampling-Based Approaches to the Calculation of Posterior Moments." In Bayesian Statistics 4 (eds. J.M. Bernardo, J. Berger, A.P. Dawid and A.F.M. Smith), Oxford University Press, Oxford, pp. 169-193, 1992.
  [4] P.Hougaard,”Survival models for heterogeneous populations derived from stable distributions”, Biometrika, Vol.73, pp.387-396, 1986.
  [5] J.G.Ibrahim , C.Ming-Hui, D.Sinha,”Bayesian Survival Analysis”, Springer, Verlag, 2001.
  [6] R.E.Kass, A.E.Raftery,”Bayes Factor”. Journal of the American Statistical Association, Vol.90,Issue. 430,pp. 773-795,1995.
  [7] S.Kheiri, A.Kimber, M.R.Meshkani,“Bayesian analysis of an inverse Gaussian correlated frailty model”. Computational Statistics and Data Analysis, Vo.51, pp.5317-5326,2007.
  [8] Y.Lan, L.M.Leemis,“The logisticexponential survival distribution”,Naval Research Logistics (NRL), Vol.55, Issue.3, pp.252-264,2008.
  [9] C.A.McGilchrist,C.W.Aisbett,“Regression with frailty in survival analysis”,Biometrics, Vol.47, pp.461-466, 1991.
  [10] J.H.Petersen, “An additive frailty model for correlated lifetimes”, Biometrics, Vol.54,pp. 646,1998.
  [11] C.A.Santos, J.A.Achcar, “A Bayesian analysis for multivariate survival data in the presence of covariates”,Journal of Statistical Theory and Applications, Vol.9, pp.233-253,2010.
  [12] S.K.Sahu, D.K.Dey, H.Aslanidou, D.Sinha,”A Weibull regression model with gamma frailties for multivariate survival data”,Life Time Data Analysis, Vol.3, pp.123-137,1997.
  [13] K. Sree Divya, P.Bhargavi, S.Jyothi,“Machine Learning Algorithms in Big data Analytics”. International Journal of Computer Sciences and Engineering, Vol 6, Issue 1, pp.63-70, 2018.
  [14]A.Wienke,”Frailty Models in Survival Analysis”,Chapman & Hall/CRC, 2011.
  [15] A.I.Yashin, I.A.Iachine, “Environment determines 50% of variability in individual frailty: Results from Danish twin study, Research Report, Population Studies of Aging, 10, Odense University, Odense, 1994.
  

  Citation
  A.Pandey, Lalpawimawha, S.Bhushan, P.K. Misra, "Correlated Gamma Frailty Model Based on Logistic Exponential Baseline Distribution," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.170-176, 2018

• Open Access   Article

  On Asymptotic Testing of Equality of Parameters of K Exponential Family of Distributions

  Patel S. R.

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.177-182, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.177182

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  In this paper attempt has been made to obtain the asymptotic tests for testing the equality of parameters involved in several exponential family of distributions. Any exponential family has been reduced to naturalized form if it is not so. The naturalized exponential family is asymptotically reduced to multivariate normal distribution with the use of well-known multivariate central limit theorem and the test for equality of components of multivariate normal gives equivalently the test for equality of parameters of the family, which turns out as Uniformly Most Powerful (UMP) Invariant (UMPI) test. This general result has been applied to some specific distributions such as Binomial, Poisson, Exponential, Gamma distributions as particular cases of exponential family.

  Key-Words / Index Term
  Exponential family of distributions, asymptotic distribution, Asymptotic UMPI test

  References
  [1] Bayoud HA, Kittaneh OA, “Testing the equality of two exponential Distributions”. Communications in Statistics-Simulation and Computation, Vol. 45, issue.7, pp.2249–56, 2016
  [2] Engelhardt M, Bain LJ, “Uniformly most powerful unbiased tests on the scale parameter of a gamma distribution with a nuisance shape parameter”, Technometrics, Vol. 2, pp77–81, 1977
  [3] Kambo NS, Awad AM, “Testing equality of location parameters of K exponential distributions”, Comm Statist Theory Methods Vol. 14, pp.567–85, 1985
  [4] Keating JP, Glaser,RE, Ketchum NS, “Testing hypotheses about the shape parameter of a gamma distribution”, Technometrics. Vol.32, pp.67–82, 1990
  [5] Lehmann EL,” Testing Statistical Hypotheses”, Springer, New York, 2005
  [6] Muirhead RJ, “On the Distribution of the Likelihood Ratio Test of Equality of Normal Populations”, The Canadian Journal of Statistics, Vol.10, pp.59–62, 1982
  [7] Patel, S.R., “Asymptotic tests for truncated parameters of several one-parameter truncation family of distributions”, Research and Reviews: Journal of Statistics, Vol. 7, issue.2, pp.35-41, 2018
  

  Citation
  Patel S. R., "On Asymptotic Testing of Equality of Parameters of K Exponential Family of Distributions," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.177-182, 2018

• Open Access   Article

  Application of Kernel Density Estimation in Chain Ladder Method for Claim Reserving

  K.M. Sakthivel, C. S. Rajitha

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.183-190, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.183190

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  Abstract
  The process of estimating the accurate reserves for incurred but not reported (IBNR) claims is the important task performed by the actuaries in non-life insurance business. Chain ladder (CL) method is probably the most commonly used method in loss reserving. But this method is not derived from any fundamental theory about the way the claims occur. So that theoretical justification of this method is rather difficult. To overcome this, we introduced a modified method using a well defined non parametric model called Kernel Density Estimation (KDE) for estimating the IBNR claims reserves with the help of percentiles. In this paper, we developed a procedure for the modification of the CL predictors (outstanding claims estimates) for the future cumulative claims (lower triangle values) with the use of KDE and showed that our procedure provides better results compare to the existing CL method for finding the IBNR claims reserves in terms of the Mack’s standard error.

  Key-Words / Index Term
  Chain-ladder method,Kernel density estimation; Mack’s standard error

  References
  [ 1] S. Christofides, “Regression Models Based on Log Incremental Payments”, Institute of Actuaries, Claims Reserving Manual, Vol.2, Section D5, pp.1- 53, 1990.
  [ 2] P.D. England, R.J. Verrall, “Stochastic Claims Reserving in General Insurance”, British Actuarial Journal, Vol.8, pp. 443-544, 2002
  [ 3] P.D. England, R.J. Verrall, “Predictive distributions of outstanding liabilities in general insurance”, Annals of Acturial Science, Vol. 1 , pp. 221-270, 2006.
  [ 4] M.C. Jones, J.S. Marron, S.J. Sheather, “A Brief Survey of Bandwidth Selection for Density Estimation”, Journal of American Statistical Association Vol. 91, pp. 401- 407, 1996
  [ 5] T. Mack, “A Simple Parametric Model for Rating Automobile Insurance or Estimating IBNR Claims Reserves”, Astin Bulletin, Vol.39, pp. 35-60, 1991.
  [ 6] T. Mack, “Distribution-free Calculation of the Standard Error of Chain Ladder Reserve Estimates”, Astin Bulletin,Vol. 23, pp. 213-225, 1993.
  [ 7] T. Mack, “The Standard Error of Chain Ladder Reserve Estimates: Recursive Calculation and Inclusion of a Tail Factor”, Astin Bulletin, Vol. 29, pp. 361-366, 1999.
  [ 8] K. Peremans, P. Segaert, S. Van Aelst, T. Verdonck, “Robust Bootstrap Procedures for the Chain-Ladder Method”, Scandinavian Actuarial Journal, Vol.1238, pp. 1-28, 2017
  [ 9] P. Martin, K. Morgan, H. Pickt, “Historical Loss Development”, Reinsurance Association of America (RAA), pp.96,1991
  [ 10] A.E. Renshaw, R.J. Verrall, “A Stochastic Model Underlying the Chain-Ladder Technique”, British Actuarial Journal, Vol.4, pp. 903-923,1998
  [ 11] K. M. Sakthivel, C. S. Rajitha, “Kernel Density Estimation for Claim Size Distributions using Shifted Power Transformation”, International Journal of Science and Research Vol. 5, pp. 2025–2028, 2016
  [ 12] R. Schnieper, “Separating True IBNR and IBNER Claims”, Astin Bulletin Vol.21, pp. 111-127, 1991
  [ 13] S.J. Sheather, M.C. Jones, “A Reliable Data-based Bandwidth Selection Method for Kernel Density Estimation”, Journal of the Royal statistical Society, Serial B, Vol. 53, pp. 683-690, 1991
  [ 14] B.W. Silverman, “Density Estimation for Statistics and Data Analysis”, Chapman & Hall, London, 1986.
  [ 15] G.C. Taylor, and F.R. Ashe, “Second Moments of Estimates of Outstanding Claims”, Journal of Econometrics Vol. 23, pp. 37-61,1983.
  [ 16] T. Verdonck, M. V. Wouwe, J. Dhaene, “A Robustification of the Chain-Ladder Method”, North American Actuarial Journal, Vol.13, pp. 280–298, 2009.
  [ 17] R.J. Verrall, “Bayes and Empirical Bayes Estimation for the Chain Ladder Model”, Astin Bulletin, Vol. 20, pp. 217-243, 1990.
  [ 18] R.J. Verrall, “An Investigation in to Stochastic Claims Reserving Models and the Chain- Ladder Technique”, Insurance: Mathematics and Economics, Vol. 26, pp. 91-99, 2000.
  [ 19] M.P. Wand, M.C. Jones, “Kernel Smoothing”, Chapman & Hall, London, 1995.
  [ 20] M. Wuthrich, M. Merz, “Stochastic Claims Reserving Methods in Insurance”, Wiley, England, 2008.
  [ 21] A. Zambom, R. Dias, “A review of Kernel Density Estimation with Applications to Econometrics”, International Econometric Review (IER), Econometric Research Association, Vol. 5, No. 1, pp. 20-42, 2013.

  Citation
  K.M. Sakthivel, C. S. Rajitha, "Application of Kernel Density Estimation in Chain Ladder Method for Claim Reserving," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.183-190, 2018

• Open Access   Article

  Magicness of Circular Human Chain Graph

  K.Anitha, B. Selvam

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.191-195, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.191195

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  Abstract
  In this paper, we introduce a new graph called circular human chain graph and investigate the existence of Z3- vertex magic total, Z3- edge magic total, Z4-bi magic and n-edge magic labeling of circular human chain graph.

  Key-Words / Index Term
  Circular human chain,Magic labeling,Vertex –magic labeling,Edge –magic labeling

  References
  [1] K.Anitha, B.Selvam ,Human chain graph, International journal of Engineering, Science and Mathematics, Vol.7 Issue 8, August 2018.
  [2] K.Anitha, B.Selvam, Magic labeling on Human chain graph, International journal of Engineering, Science and Mathematics, Vol.7 Issue 11, November 2018.
  [3] K.Anitha, B.Selvam, Cordial labelings on Human chain graph, Journal of Computer and Mathematical Sciences Vol.9, Issue 11, November 2018.
  [4] Cahit, On cordial and 3-equitable labelings of graphs, Util. Math., Vol. 37 (1990) pp.189-198
  [5] Gallian J.A, “A Dynamic Survey of graph labeling”, the Electronic Journal of combinatories, 19, # DS6 (2012).
  [6] B.Selvam, K.Thirusangu and P.P.Ulaganathan, “Z3- vertex magic total labeling and Z3- edge magic total labeling in extended duplicate graph of twig Tm” 2012.
  

  Citation
  K.Anitha, B. Selvam, "Magicness of Circular Human Chain Graph," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.191-195, 2018

• Open Access   Article

  Double Calibration Approach Based Estimator for Population Mean in Two-Stage Stratified Random Sampling When Auxiliary Information Is Available At Element Level

  Dhirendra Singh, B.V.S. Sisodia

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.196-207, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.196207

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  Abstract
  Singh et al. (2017) have developed calibration estimators of population mean in two-stage stratified random sampling by calibrating integrated sampling design weight when the auxiliary information is available at element (second stage unit) level for the entire population. Obviously, if the auxiliary information is available at the element level for entire population, then population mean/total of the auxiliary variable is also known. In the present paper, double (two-steps) calibration estimators of the population mean have been developed by calibrating integrated sampling design weight at first step and calibrating stratum weight at the second step using known population total/mean of the auxiliary variable. A limited simulation study with real data has been conducted to examine the relative performance of the calibration estimators over the usual estimator of the population mean without using auxiliary information in two-stage stratified random sampling. It has been found from the results of simulation study that double (two-steps) calibration estimator has brought considerable improvement in the precision of the estimate of population mean.

  Key-Words / Index Term
  Calibration Estimator, Auxiliary Information, Two-Stage Sampling, Stratified Random Sampling.

  References
  [1]. Aditya, Kaustav., Sud, U.C., and Chandra, H. (2016). Calibration based regression type estimator of the population total under two-stage sampling design. Journal of the Indian Society of Agricultural Statistics, 70 (1), 19-24.
  [2]. Deville, J.C. and Särndal, C.E. (1992). Calibration estimators in survey sampling. Journal of the American Statistical Association, 87, 376–382.
  [3]. Estevao, V.M. and Särndal, C.E. (2009). A new face on two-phase sampling with calibration estimators. Survey Methodology, 35(1), 3-4.
  [4]. Horvitz, D.G. and Thompson, D.J. (1952). A generalization of sampling without replacement from a finite universe. Journal of the American Statistical Association, 47, 663-685.
  [5]. Kim, J.M., Sunger, E.A. and Heo, T.Y. (2007). Calibration approach estimators in stratified sampling. Statistics and Probability Letters, 77, 99-103.
  [6]. Kim, J.K. and Park, M. (2010). Calibration estimation in survey sampling. International Statistical Review, 78, 21-39.
  [7]. Mourya, K.K., Sisodia, B.V.S. and Chandra, H. (2016). Calibration approach for estimating finite population parameter in two-stage sampling, Journal of Statistical Theory and Practice, (Online, 19 May 2016).
  [8]. Mourya, K.K., Sisodia, B.V.S., Singh, Amar and Rai, V.N. (2016). Calibration estimators in cluster sampling when cluster are of unequal size. Int. J. Agricult. and Statistical Sci., 12(2), 351-358.
  [9]. Särndal, C.E., Swensson, B. and Wretman, J. (1992). Model-assisted Survey Sampling. New York: Springer-Verlag.
  [10]. Särndal, C.E. (2007). The calibration approach in survey theory and practice. Survey Methodology, 33(2), 99-119.
  [11]. Singh, Dhirendra., Sisodia, B.V.S., Kumar, Sunil., and Kumar, Sandeep. (2017). Some calibration estimators for finite population mean in two-stage stratified random sampling. Communication in Statistics- Theory and Methods, (submitted).
  [12]. Singh, Dhirendra., Sisodia, B.V.S., Rai, V.N., and Kumar, Sandeep. (2017). A calibration approach based regression and ratio type estimators of finite population mean in two-stage stratified random sampling. Journal of the Indian Society of Agricultural Statistics, (submitted).
  [13]. Singh, S., Horn, S. and Yu, F. (1998). Estimation of variance of the general regression estimators: Higher level calibration approach. Survey methodology, 24(1), 41-50.
  [14]. Sinha, N., Sisodia, B.V.S., Singh, S. and K. Singh, Sanjay. (2016). Calibration approach estimation of mean in stratified sampling and stratified double sampling. Communication in Statistics- Theory and Methods, (Online, 27 May, 2016).
  [15]. Singh, S. and Arnab, R. (2011). On calibration of design weights. Metron, 69(2), 185-205.
  [16]. Tracy, D.S., Singh, S. and Arnab, R. (2003). Note on calibration estimators in stratified and double sampling. Survey Methodology, 29, 99-106.
  

  Citation
  Dhirendra Singh, B.V.S. Sisodia, "Double Calibration Approach Based Estimator for Population Mean in Two-Stage Stratified Random Sampling When Auxiliary Information Is Available At Element Level," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.196-207, 2018

• Open Access   Article

  Dynamics of a Discrete Duffing Equation with Fractional Order

  A.George Maria Selvam, D. Vignesh

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.208-211, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.208211

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  Abstract
  This Paper considers a discrete fractional order 2-Dimensional system for a damped and driven Duffing Oscillator with linear and cubic term. Stability of the system is analyzed using the fixed points of the system. Time plot and Phase portraits are provided for different values of the parameters. Chaotic behavior of the system is studied with Bifurcation diagram. Numerical simulations are performed supporting the theoretical analysis.

  Key-Words / Index Term
  Duffing equation, Fixed Points, Stability, Bifurcation

  References
  [1] Saber Elaydi, An Introduction to Difference Equations, Third Edition, Springer International Edition, First Indian Reprint, 2008.
  [2] Xiaoli Liu, Dongmei Xiao, Complex dynamic behaviors of a discrete-time predator - prey system, Chaos, Solitons and Fractals, 32 (2007) 80 - 94.
  [3] The Duffing Equation: Nonlinear Oscillators and their Behaviour, First Edition(2011), Edited by I. Kovacic and M. J. Brennan. Publisher John Wiley & Sons, Ltd. ISBN: 978-0-470-71549-9
  [4] Zheng B.D., Liang L.J., Zhang C.R., “Extended Jury criterion”, Science China Math., 53(4), (2010), 1133 – 1150
  [5] G.C. Layek, An Introduction to Dynamical Systems and Chaos(2015), Springer India, ISBN 978-81-322-2555-3, ISBN 978-81-322-2556-0 (eBook),DOI 10.1007/978-81-322-2556-0.
  [6] El-Sayed et al., Discretization of forced Duffing system with fractional-order damping. Advances in Difference Equations 2014, 2014:66
  [7] Ravi P Agarwal, Ahmed MA El Sayed and Sanaa M Salman, Fractional-order Chua.s system: discretization,bifurcation and chaos, Advances in Difference Equations 2013, 2013:320.
  
  

  Citation
  A.George Maria Selvam, D. Vignesh, "Dynamics of a Discrete Duffing Equation with Fractional Order," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.208-211, 2018

• Open Access   Article

  A Class of Super-Efficient Estimators of the Normal Variance: A Study on Sample Size Preference

  K. Sivasakthi, Martin L. William

  Research Paper | Isroset-Journal (IJSRMSS)

  Vol.5 , Issue.6 , pp.212-221, Dec-2018

  CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v5i6.212221

  Abstract

  Full-Text HTML

  References

  Citation

  Abstract
  A class of super-efficient estimators of the variance of a normal population with known mean has been recently constructed by Sivasakthi, Durairajan, and William (2017) through the ‘Delta Method’. The preference for a super-efficient estimator over the asymptotically efficient estimator (say, maximum likelihood estimator) is for ‘large samples’. In this paper, we address the super-efficient estimation of the normal variance when the population mean is known. The issue that is taken up in this paper is on the sample size required for a super-efficient estimator to be preferred over the maximum likelihood estimator and is addressed through a numerical study. The answer is sought for a subset of the class of super-efficient estimators of the normal variance. Mathematics Subject Classification: 62F10, 62F12

  Key-Words / Index Term
  Maximum Likelihood Estimators, Super-efficiency

  References
  [1] Bahadur, R. R. (1983). Hodges Super-Efficiency. In: Encyclopedia of Statistical Sciences. 3, John-Wiley, 645-646
  [2] Basu, D. (1952). Unpublished Thesis
  [3] Durairajan, T.M. (2012).Sub-score and Super- Efficient Estimator. In: Navin Chandra and Gopal, G., eds. Applications of Reliability Theory and Survival Analysis, Bonfring, 120-129
  [4] Hodges, J. L. (1951). Unpublished
  [5] Le Cam, L. (1953). On some asymptotic properties of maximum likelihood estimates and related Baye’s estimates. University of California Publications in Statistics. 1, 277-330.
  [6] Le Cam, L. (1956). On the asymptotic theory of estimation and testing hypotheses. In: Neyman, J. ed. Proceedings of the Third Berkeley Symposium on Mathematics and Probability. 1, University of California Press, Berkeley, 129-156
  [7] Le Cam, L. (1960). Locally asymptotically normal families of distributions. University of California Publications in Statistics. 3, 37-98
  [8] Le Cam, L. (1972). Limits of experiments. In: Proceedings of the Sixth Berkeley Symposium on Mathematics and Probability. 1, University of California Press, Berkeley, 245-261
  [9] Sethuraman, J. (2004). Are Super- Efficient Estimators Super-powerful?. Communications in Statistics-Theory and Methods. 33(9), 2003-2013
  [10] Sivasakthi, K., Durairajan, T. M., and William, M. L. (2017), Obtaining Super-Efficient Estimators for a Real-Valued Parameter: Delta Method. International Journal of Applied Mathematics and Statistical Sciences, 6(1), 81-88.
  [11] Sivasakthi, K., Sakthivel, R. and William, M. L. (2017). Preference for a class of Super-Efficient Estimators of the normal mean: A study on sample size requirement. International Journal of Statistics and Applied Mathematics, 2(6), 241-249
  [12] Stein, C. (1956). Inadmissibility of the usual estimator for the mean of a normal distribution. In: Neyman, J. ed. Proceedings of the Third Berkeley Symposium on Mathematics and Probability. 1, University of California Press, Berkeley, 197-206
  

  Citation
  K. Sivasakthi, Martin L. William, "A Class of Super-Efficient Estimators of the Normal Variance: A Study on Sample Size Preference," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.5, Issue.6, pp.212-221, 2018