Volume-11 , Issue-4 , Aug 2024, ISSN 2348-4519 (Online)

• Open Access   Article

  Statistical Inference for Rayleigh Distribution Using Ranked Set Sampling: A Bayesian Approach

  Fahad M. Al Subhi

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.11 , Issue.4 , pp.1-13, Aug-2024

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  Abstract
  Working on symmetrical or asymmetrical data is complicated since each requires a different probability density function. Many statistical distributions can be used for these data types, where choosing one should be satisfied with the correct data type. So, we apply the ranked set sampling technique, which is essential in gaining data when dealing with units in a population is expensive. However, their classification is simple according to the variable of interest. The Rayleigh distribution has recently played a crucial role in analyzing symmetrical or asymmetrical complex data sets specifically in modeling claim and risk data used in actuarial and financial studies, and its density can take different symmetric and asymmetric possible shapes. It is proposed in various areas, such as reliability, survival, economics, actuarial science, and insurance. In this paper, we provide Bayesian estimation for a parameter of the Rayleigh distribution based on a simple random sample (SRS) and ranked set sampling (RSS) using two loss functions; the squared error loss function, and the linex loss function. The results of the simulation study showed that the Bayes estimates based on RSS are more efficient than the estimates based on SRS for different sample sizes of (n) and different values of the parameter ?.

  Key-Words / Index Term
  Bayesian analysis; Linex loss function; Ranked set sampling; Rayleigh distribution; Squared error loss function; Simulation Study

  References
  [1] L. Rayleigh, “On the resultant of a large number of vibrations of the same pitch and of arbitrary phase”, Phil. Mag., Series 5, 43(261), pp.259-272, 1880.
  [2] A.R. Kamel, A. A. Alqarni, M. A. Ahmed, “On the Performance Robustness of Artificial Neural Network Approaches and Gumbel Extreme Value Distribution for Prediction of Wind Speed”, Int. J. Sci. Res. in Mathematical and Statistical Sciences, Vol,9, Issue.4,? 2022.
  [3] A.B. Çolak, T.N. Sindhu, S. A. Lone, A. Shafiq, T.A. Abushal, “ Reliability study of generalized Rayleigh distribution based on inverse power law using artificial neural network with Bayesian regularization”, Tribology International, 185, 108544., 2023.
  [4] S. Dey, “Comparison of Bayes estimators of the parameter and reliability function for Rayleigh distribution under different loss functions”, Malays J. Math. Sci. 3, pp.247–264, 2009.
  [5] S. Dey, M. K. Das, “A note on prediction interval for a Rayleigh distribution: Bayesian approach”, Am. J. Math. Manag. Sci. 1&2, pp.43–48, 2007.
  [6] S. Noor, O. Tajik, J. Golzar, “Simple random sampling”, International Journal of Education & Language Studies, Vol.1, Issue.2, pp.78-82, 2022.
  [7] G. A. McIntyre, “A method for unbiased selective sampling, using ranked sets”, Australian journal of agricultural research, Vol.3, Issue.4, pp.385-390, 1952.
  [8] A. I. Al-Omari, C. N. Bouza, “Review of ranked set sampling: modifications and applications”, Revista Investigación Operacional, Vol.3, Issue.35, pp.215-240, 2014.
  [9] R. D. Thompson, A. P. Basu, “Asymmetric loss functions for estimating system reliability”, In Bayesian analysis in statistics and econometrics: essays in honor of Arnold Zellner. pp.471-482, 1996.
  [10] A. Zellner, “Bayesian estimation and prediction using asymmetric loss functions”, Journal of the American Statistical Association, Vol.81, Issue.394, pp.446-451, 1986.
  [11] K. Takahasi, K. Wakimoto, “On unbiased estimates of the population mean based on the sample stratified by means of ordering”, Annals of the institute of statistical mathematics, Vol.20, Issue.1, pp.1-31, 1968.
  [12] F. Noor, A. Sajid, M. Ghazal, I. Khan, M. Zaman, I. Baig, “Bayesian estimation of Rayleigh distribution in the presence of outliers using progressive censoring”, Hacettepe Journal of Mathematics and Statistics, Vol.49, Issue.6, pp.2119-2133, 2020.
  [13] A.R. Kamel, “Handling outliers in seemingly unrelated regression equations model”, MSc thesis, Faculty of graduate studies for statistical research (FGSSR), Cairo University, Egypt, 2021.
  [14] A. R, Kamel, M. R. Abonazel, “Simple Introduction to Regression Modeling using R” Computational Journal of Mathematical and Statistical Sciences, Vol.2, Issue.1, pp.52-79, 2023.
  [15] A. A. Alharbi, A. R. Kamel, S. A. Atia, “A New Robust Molding of Heat and Mass Transfer Process in MHD Based on Adaptive-Network-Based Fuzzy Inference System”, WSEAS Transactions on Heat and Mass Transfer, 17, pp.80-96, 2022.
  [16] A. H. Youssef, M. R. Abonazel, A. R. Kamel, “Efficiency comparisons of robust and non-robust estimators for seemingly unrelated regressions model”, WSEAS Transactions on Mathematics, 21, pp.218-244, 2022.
  [17] A. H. Youssef, A. R. Kamel, M. R. Abonazel, “Robust SURE estimates of profitability in the Egyptian insurance market”, Statistical journal of the IAOS, Vol.37, Issue.4, pp.1275-1287, 2021.
  [18] M. R. Abonazel, A. R. Kamel, “The impact of using robust estimations in regression models: An application on the Egyptian economy”, Journal of Advanced Research in Applied Mathematics and Statistics, Vol.4, Issue.2, pp.8-16, 2019.
  [19] F. M. Alghamdi, A. R. Kamel, M.S. Mustafa, M.M. Bahloul, M.M. Alsolmi, M. R. Abonazel, “A statistical study for the impact of REMS and nuclear energy on carbon dioxide emissions reductions in G20 countries”, Journal of Radiation Research and Applied Sciences, 17(3), 100993, 2024.

  Citation
  Fahad M. Al Subhi, "Statistical Inference for Rayleigh Distribution Using Ranked Set Sampling: A Bayesian Approach," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.11, Issue.4, pp.1-13, 2024

• Open Access   Article

  Analysis of Commercial Aircrafts Based on Maneuverability Condition

  Raja Munusamy, Sudhir Kumar Chaturvedi, Kokila Vasudevan

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.11 , Issue.4 , pp.14-19, Aug-2024

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  Performance estimation of an aircraft is required to make sure of the ability of the aircraft to perform conventional and mission specific maneuvers. An aircraft`s maneuverability is defined by its capability to alter attitude and speed along its three axes: longitudinal, lateral, and vertical. To acquire the performance of the aircraft for the establishing the operational envelope, initial operation certification and gradual expansion of the flight envelope, an aircraft system is studied directly by conducting some experiments with it or by using its mathematical model. Nowadays, mathematical modelling is more preferred over real flight testing as rehearsal of several situations cannot be practiced in real life. Further, drastically reduced training costs and increased efficiency of the required results. The conceptual model of the specific static maneuvers of the aircrafts can be converted to computer model via programming in a flight simulator such as Matlab, which then simulates the reality. The maneuvering capabilities of an aircraft are decided according to the mission to be performed. Therefore, the commercial passenger aircrafts are more stable and less maneuverable compared to the commercial transport and the military aircrafts, designed so keeping in mind the passenger safety and security as the foremost aim. Hence, it becomes very important to estimate the responses of a commercial aircraft for the maneuvers it performs under its operational flight envelope. Such estimation firstly requires a study of these maneuvers. Secondly, converting them into a matrix model and then simulating them in the Matrix Laboratory to obtain the required results for further analysis to arrive at conclusions. The conclusions required from the flight simulator are the coefficient of lift, drag; load factor, velocity and time estimation for a particular maneuver that an aircraft is capable to perform.

  Key-Words / Index Term
  Maneuverability, Longitudnal, Lateral, Load factor, Operational envelope

  References
  [1] John D. Anderson, Jr., “The Wright Brothers: The First True Aeronautical Engineers,” in The Wright flyer : an engineering perspective , edited by Howard S.Wolko, National Air and Space Museum, Washington, ,pp. 1-17, 1987
  [2] Michael V.Cook, “flight dynamic principles,”: a linear systems approach to aircraft stability and control, Maneuverability, , pp. 210-215,
  2007
  [3] Sibo Ding. “A New Uncertain Programming Model for Grain Supply Chain Design,” An International Interdisciplinary Journal, Vol. 16. No 2 (A). p. 1069-1076, 2013.
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  [5] Wilson, D. J., Riley, D. R., and Citurs, K. D., “Aircraft Maneuvers for the Evaluation of Flying Qualities and Agility Maneuver Development Process and Initial Maneuver Set,” Tech. Rep. Flight Dynamics Directorate, Wright Laboratory, Wright-Patterson AFB, OH, WL-TR-93-3081, Vol 1, 1993.
  [6] V., Raol J.R. Girija G., Parameswaram,“Determination of aircraft performance from flight data,” journal of aeronautical society of India, 43(4). pp. 285-292, 1990.
  [7] L.J. Clancy, “Aerodynamics: Elementary mechanics of flight”, Wiley, ISBN, 0470158379 pp, 406-409,2007
  [8] Bruns M., Paredisa C., Ferson S. “Computational Methods for Decision Making based on Imprecise Information,”. Proceedings of the Reliable Engineering Computing Workshop, p. 22–24, 2006.
  [9] Paul Jackson, “All the World’s Aircraft 2004-2005,” Janes Information Group; Subsequent edition, ISBN 978-0710626141, 1814.
  [10] John D. Anderson, Jr., “Introduction to flight, ”, McGraw—Hill, pp. 467-474, 2000.
  [11] Mazza, C. J., “Agility: A Rational Development of Fundamental Metrics and their Relationship to Flying Qualities,” AGARD Conference Proceedings Flying Qualities, AGARD, No. 508: 1990.
  [12] R Nigam, R Jain, A. Sehgal, R. Maheswori, P.S Mehra, “Estimation of Household Consumption: A Survey,” World Academic Journal of Engineering Science, Vol.6, Issue 1, pp.55-63, 2019
  [13] Nishan Khadka, Sulav Parajuli, Aayush Bhattarai, Sanjaya Neupane, “A Study of Optimization of Aircraft Separation using the Time Based Separation at Tribhuvan International Airport,” World Academics Journal of Engineering Sciences, Vol.7, Issue.2, pp.1-7, 2020.

  Citation
  Raja Munusamy, Sudhir Kumar Chaturvedi, Kokila Vasudevan, "Analysis of Commercial Aircrafts Based on Maneuverability Condition," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.11, Issue.4, pp.14-19, 2024

• Open Access   Article

  Analysis of the Cholera Disease and Control Using the SIRD Model

  S. Munira, M.S. Atureta, K.E. Lasisi

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.11 , Issue.4 , pp.20-23, Aug-2024

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  Abstract
  This Study is centered on modifying SIRD mathematical model for cholera dynamics incorporating four control strategies namely: therapeutic treatment, Surveillance, vaccination and water sanitation. This study is limited to only cholera disease. Data were collected from the Gombe state Hospital for infectious disease and leprosy control in Zambuk. According to this study, total human population has been classified into four sub population such as susceptible humans, recovered humans, Infected Humans and the mortality rate of humans(Death). The susceptible class,Infected class and the recovered and the death category is being related with the homotopy perturbation method and the runge kutta method to observe the most efficient method to use to control the disease better.

  Key-Words / Index Term
  SIRD model, Susceptible, Infected, Recovered, Death, Runge kutta(order 4and order 5) method, Homotopy perturbation method, Vibrio Cholerae

  References
  [1] A. Macelroy and P. townsend, ‘’ Medical arthropology in ecological perspective’’ 2nd Edition, Boulder,co- westview, USA, pp 375, 2009.
  [2] A.P. Lemos paiao, C.J. Silva, D.F. Torres, ‘’ An epidemic model for Cholera with optimal control treatment’’ A journal of computational and applied mathematics, Vol.318, No1, pp 1-7, 2016.
  [3] A.J. Robert and S.Basu, ‘’ Transmission dynamics and control of Cholera in Haiti’’ A journal on epidemic model Lancet. Vol.377, No 3, pp 1248- 1255, 2011.
  [4] C. Virginia and S.L. Paveri- Fontana, ‘’ A mathematical model for the Cholera epidemic in the European Mediterrenean region’’ proceedings of the Revue depimoligie et de Sante publique, France Vol.27 pp 121-132, 1979.
  [5] C. T. Codeco ‘’ Epidemic dynamics of Cholera the role of acquatic reservoir’’ proceedings of the BMC infectious diseases, Berlin pp1-9, 2001.
  [6] C. Denny, M.E. Halloran,I.M. Longini Jr, ‘’ Vaccination strategies for epidemic Cholera in Haiti with implications for the developing world’’ proceedings of the national acdemy of science USA,pp 7081- 7085, 2011.
  [7] E. Berttuzo, R. Casagrandi, M. Gatto, C.R. Blokes, ‘’ prediction of the spatial evolution and effects of control measures for the unfolding Haiti Cholera outbreak’’ proceedings of the geophysical research letters, Haiti. Vol. 38, pp 106403, 2011.
  [8] H.Hethcote,’’The mathematics of infectious disease’’ A journal of mathematical biology, Vol.42, No2, pp599, 2000.
  [9] J. Tien, A.R. Tuite,D.N. Fisman ‘’Using transmission models to explain spatial spread of disease and identify optimal control interventions’’ An international journal of medicine, Vol.154, No 1 pp593-601,2010.
  [10 ]S.L. Paveri-Fontana ‘’A mathematical model for the 1973 cholera epidemic in the European mediterenean region’’ proceedings of the Revue Depidemioligue et de Sante publique, France, pp121-132.1973.
  [11] Z. Mukanvire,S. liao, J. Wang,H.Gaff, D.L. Smith, ‘’Estimating the reproductive numbers for the 2008-2009 cholera outbreaks’’ Proceeding of the National academy of science, Zimbabwe, pp 8767-8772, 2011.

  Citation
  S. Munira, M.S. Atureta, K.E. Lasisi, "Analysis of the Cholera Disease and Control Using the SIRD Model," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.11, Issue.4, pp.20-23, 2024

• Open Access   Article

  A Novel Area-Biased Distribution With Essential Statistical Properties

  Asgar Ali, Aafaq A. Rather, Berihan R. Elemary

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.11 , Issue.4 , pp.24-28, Aug-2024

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  Abstract
  In this paper, we introduce a novel model called the area biased Transmuted Mukherjee-Islam (ABTMI) distribution, which generalizes the Transmuted Mukherjee-Islam distribution through an area biased transformation approach. We thoroughly explore the probability density function (PDF) and the cumulative distribution function (CDF) of the ATMI distribution. Additionally, we investigate the distinctive structural properties of the proposed model, including the survival function, conditional survival function, hazard function, cumulative hazard function, mean residual life, moments, moment generating function (MGF), characteristic function (CF), cumulant generating function (CGF), entropy measures, and Bonferroni and Lorenz curves. The model parameters are estimated using the maximum likelihood estimation method.

  Key-Words / Index Term
  Transmuted Mukherjee-Islam distribution, Area biased distribution, Reliability analysis, Maximum likelihood estimator

  References
  [1] R.A. Fisher, “The Effects of Methods of Ascertainment Upon the Estimation of Frequencies,” Annals of Eugenics, Issue 6, pp.13- 25, 1934.
  [2] C.R. Rao, “On Discrete Distributions Arising Out of Method of Ascertainment, In Classical and Contagious Discrete,” G.P. Patiled; Pergamum Press and Statistical Publishing Society, Calcutta, pp.320-332, 1965.
  [3] A.A. Rather, G. Ozel, “The Weighted Power Lindley Distribution with Applications on the Life Time Data,” Pakistan Journal of Statistics and Operation Research, Vol.16, Issue.2, pp. 225-237, 2020.
  [4] A. Ahmad, A.A. Rather, Y.A. Tashkandy, M.E. Bakr, M.M.M. El-Din, A.M. Gemeay, E.M. Almetwally, M. Salem, “Deriving the New Cotangent Frechet Distribution With Real Data Analysis,” Alexandria Engineering Journal, Vol.105, Issue.10, pp.12-24, 2024. https://doi.org/10.1016/j.aej.2024.06.038
  [5] D. Qayoom, A. A. Rather, “Weighted Transmuted Mukherjee-Islam Distribution with Statistical Properties,” Reliability: Theory and Applications, Vol.78, Issue.2, pp. 124-137, 2024.
  [6] A. Ahmad, A.A. Rather, A.M. Gemeay, M. Nagy, L.P. Sapkota, A.H. Mansi, “Novel Sin-G Class of Distributions with an Illustration of Lomax Distribution: Properties and Data Analysis,” AIP Advances, Vol.14, pp.1-17, 2024.
  [7] A.A. Rather, C. Subramanian, “Transmuted Mukherjee-Islam Failure Model,” Journal of Statistics Applications & Probability, Vol.7, Issue.2, pp.343-347, 2018.
  [8] W.T. Shaw, I.R.C. Buckley, “The Alchemy of Probability Distributions: Beyond Gram-Charlier Expansions and A Skew-Kurtotic-Normal Distribution from A Rank Transmutation Map,” Research report, 2007.
  [9] R. Alfred, “On Measures of Information and Entropy,” Proceedings of the fourth Berkeley Symposium on Mathematics, Statistics and Probability 1960. pp.547–561, 1961.
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  [12] M.O. Lorenz, “Methods of measuring the concentration of wealth,” Publications of the American Statistical Association, Vol.9, pp.209–219, 1905.

  Citation
  Asgar Ali, Aafaq A. Rather, Berihan R. Elemary, "A Novel Area-Biased Distribution With Essential Statistical Properties," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.11, Issue.4, pp.24-28, 2024

• Open Access   Article

  Design and Formulation for Multi-Party Computation in Elliptic Curve Cryptography

  Domven Lohcwat, D.G. Yakubu, M.I. Bello

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.11 , Issue.4 , pp.29-33, Aug-2024

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  Abstract
  This article discusses the use of elliptic curve cryptography (ECC) in creating secure communication schemes that do not require a pre-shared secret key. ECC, a type of public key cryptography, supports applications such as key agreement and digital signatures. Traditionally, secure communication necessitates a secret key exchange beforehand, but the paper demonstrates a method where multiple parties can publicly select a key without others being able to determine it. The proposed protocol leverages elliptic curves and secure multiparty computation (MPC) to allow secure communication over insecure channels, even in the presence of adversaries. MPC protocols enable parties to jointly compute functions of their private inputs while only revealing the output, with potential applications in privacy-preserving auctions, private DNA comparisons, private machine learning, and threshold cryptography.

  Key-Words / Index Term
  Multiparty Computation, Elliptic Curve, Elliptic Curve Cryptography and Cryptography

  References
  [1] N. Koblitz, "Elliptic Curve Cryptosystems," Mathematics of Computation, vol. 48, pp. 203-209, 1987. [Online]. V. Miller, "Use of elliptic curves in cryptography," in Advances in Cryptology—CRYPTO `85, vol. 218, pp. 417-426, 1986.
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  [12] Y. G. Peng, L. Wang, et al., "LS-RQ: A lightweight and forward-secure range query on geographically encrypted data," IEEE Trans. Dependable Secur. Comput., vol. 19, no. 1, pp. 388-401, 2022.
  [13] Y. G. Guan, R. X. Lu, et al., "Toward oblivious location-based k-nearest neighbor query in smart cities," IEEE Internet Things J., vol. 8, no. 18, pp. 14219-14231, 2021.
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  Citation
  Domven Lohcwat, D.G. Yakubu, M.I. Bello, "Design and Formulation for Multi-Party Computation in Elliptic Curve Cryptography," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.11, Issue.4, pp.29-33, 2024

• Open Access   Article

  On The Applications of Stiefel-Whitney Classes of Real Bott Manifolds

  Daniel Danladi, Domven Lohcwat, Isah Abdullahi

  Research Paper | Journal-Paper (IJSRMSS)

  Vol.11 , Issue.4 , pp.34-37, Aug-2024

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  Abstract
  Real Bott manifolds are a type of compact, connected Riemannian manifolds without boundaries, notable as a unique and intriguing category of real toric varieties. This study focuses on identifying a set of topological invariants known as Stiefel-Whitney classes, which are crucial characteristic classes in the context of the set of integers modulo two. We computed the Stiefel-Whitney classes of real Bott manifolds. We applied the method to calculate the Stiefel-Whitney classes of the manifolds. The formula for determining the number of terms in each class was also given.

  Key-Words / Index Term
  Manifolds, Real Bott manifolds, Stiffel-Whiteney Class

  References
  [1] L. S. Charlap, Bieberbach Groups and Flat Manifolds. Springer-Verlag, 1986.
  [2] A. Szczepa?ski, Geometry of Crystallographic Groups. World Scientific, Algebra and Discrete Mathematics, vol. 4, 2012.
  [3] R. Lee and R. Szczarba, "On the integral Pontrjagin classes of a Riemannian flat manifold," Geometriae Dedicata, vol. 3, no. 1, pp. 1-9, 1974.
  [4] Y. Kamishima and M. Masuda, "Cohomological rigidity of real Bott manifolds," Algebraic & Geometric Topology, vol. 9, pp. 2479-2502, 2009.
  [5] A. Gasior, "Note about Stiefel-Whitney classes on Real Bott Manifolds," arXiv preprint arXiv:1808.0803v1, 2018
  [6] A. Gasior, "Note about Stiefel-Whitney classes on Real Bott Manifolds," arXiv:1808.0803v1 [math.GT], 2018.
  [7] H. B. Lawson and M.-L. Michelsohn, Spin Geometry (PMS-38), Volume 38, vol. 84, Princeton, NJ: Princeton University Press, 2016.
  [8] S. Choi, M. Masuda, and S.-i. Oum, "Classification of real bott manifolds and acyclic digraphs," Trans. Amer. Math. Soc., vol. 369, no. 4, pp. 2987-3011, 2016.
  [9] S. Choi and S.-i. Oum, "Real bott manifolds and acyclic digraphs," arXiv preprint, arXiv:1002.4704, 2010.
  [10] M.Grossberg and Y. Karshon, "Bott towers, complete integrability, and the extended character of representations," 1994.
  

  Citation
  Daniel Danladi, Domven Lohcwat, Isah Abdullahi, "On The Applications of Stiefel-Whitney Classes of Real Bott Manifolds," International Journal of Scientific Research in Mathematical and Statistical Sciences, Vol.11, Issue.4, pp.34-37, 2024