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Journal of Mathematics and Applications

Journal of Mathematics and Applications
10/42, DOI: 10.7862/rf.2019.10

A Minimax Approach to Mapping Partial Interval Uncertainties into Point Estimates

Vadim Romanuke

DOI: 10.7862/rf.2019.10

Abstract

A problem of simultaneously reducing a group of interval uncertainties is considered. The intervals are positively normalized. There is a constraint, by which the sum of any point estimates taken from those intervals is equal to 1. Hence, the last interval is suspended. For mapping the interval uncertainties into point estimates, a minimax decision-making method is suggested. The last interval's point estimate is then tacitly found. Minimax is applied to a maximal disbalance between a real unknown amount and a guessed amount. These amounts are interpreted as aftermaths of the point estimation. According to this model, the decision-maker is granted a pure strategy, whose components are the most appropriate point estimates. Such strategy is always single. Its components are always less than the right endpoints. The best mapping case is when we obtain a totally regular strategy whose components are greater than the left endpoints. The irregular strategy's components admitting many left endpoints are computed by special formulae. The worst strategy exists, whose single component is greater than the corresponding left endpoint. Apart from the point estimation, irregularities in the decision-maker's optimal strategy may serve as an evidence of the intervals' incorrectness. The irregularity of higher ranks is a criterion for correcting the intervals.

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References

  1. E. Alhassan, H. Sjöstrand, P. Helgesson, M. Österlund, S. Pomp, A.J. Koning, D. Rochman, On the use of integral experiments for uncertainty reduction of reactor macroscopic parameters within the TMC methodology, Progress in Nuclear Energy 88 (2016) 43-52.
  2. M.H. Bazerman, D.A. Moore, Judgment in Managerial Decision Making (8th ed.), Wiley, River Street, Hoboken, NJ, 2013.
  3. J.O. Berger, Minimax analysis, in: Statistical Decision Theory and Bayesian Analysis, J.O. Berger (ed.), Springer, New York, NY, 1985, 308-387.
  4. C. Dong, G.H. Huang, Y.P. Cai, Y. Xu, An interval-parameter minimax regret programming approach for power management systems planning under uncertainty, Applied Energy 88 (8) (2011) 2835-2845.
  5. J.P.C. Driessen, H. Peng, G.J. van Houtum, Maintenance optimization under non-constant probabilities of imperfect inspections, Reliability Engineering & System Safety 165 (2017) 115-123.
  6. Y.C. Eldar, Minimax estimation of deterministic parameters in linear models with a random model matrix, IEEE Transactions on Signal Processing 54 (2) (2006) 601-612.
  7. R. Festa, Bayesian point estimation, verisimilitude, and immodesty, in: Optimum Inductive Methods, R. Festa (ed.), Springer, Dordrecht, 1993, 38-47.
  8. C. Fu, X. Ren, Y. Yang, W. Qin, Dynamic response analysis of an overhung rotor with interval uncertainties, Nonlinear Dynamics 89 (3) (2017) 2115-2124.
  9. E. Ghashim, É. Marchand, W.E. Strawderman, On a better lower bound for the frequentist probability of coverage of Bayesian credible intervals in restricted parameter spaces, Statistical Methodology 31 (2016) 43-57.
  10. M. González, C. Minuesa, I. del Puerto, Maximum likelihood estimation and expectation-maximization algorithm for controlled branching processes, Computational Statistics & Data Analysis 93 (2016) 209-227.
  11. G.C. Goodwin, R.L. Payne, Dynamic System Identification: Experiment Design and Data Analysis, Academic Press, New York, NY, 1977.
  12. P. Guo, G.H. Huang, Y.P. Li, Inexact fuzzy-stochastic programming for water resources management under multiple uncertainties, Environmental Modeling & Assessment 15 (2) (2010) 111-124.
  13. P. Guo, H. Tanaka, Decision making with interval probabilities, European Journal of Operational Research 203 (2) (2010) 444-454.
  14. M.A. Howe, B. Rustem, M.J.P. Selby, Multi-period minimax hedging strategies, European Journal of Operational Research 93 (1) (1996) 185-204.
  15. A. Jablonski, T. Barszcz, M. Bielecka, P. Breuhaus, Modeling of probability distribution functions for automatic threshold calculation in condition monitoring systems, Measurement 46 (1) (2013) 727-738.
  16. Y. Kobayashi, K. Okabe, S. Kondo, Y. Togo, Application of minimax principle to design of reactor in-core monitoring system, Journal of Nuclear Science and Technology 10 (12) (1973) 731-738.
  17. E.L. Lehmann, G. Casella, Theory of Point Estimation (2nd ed.), Springer, New York, NY, 1998.
  18. M. Leonelli, J.Q. Smith, Directed expected utility networks, Decision Analysis 14 (2) (2017) 108-125.
  19. Y.P. Li, G.H. Huang, S.L. Nie, A robust interval-based minimax-regret analysis approach for the identification of optimal water-resources-allocation strategies under uncertainty, Resources, Conservation and Recycling 54 (2) (2009) 86-96.
  20. J. Liebowitz, The Handbook of Applied Expert Systems, CRC Press, Boca Raton, FL, 1997.
  21. P. Liu, F. Jin, X. Zhang, Y. Su, M. Wang, Research on the multi-attribute decision-making under risk with interval probability based on prospect theory and the uncertain linguistic variables, Knowledge-Based Systems 24 (4) (2011) 554-561.
  22. Y.K. Liu, The completion of a fuzzy measure and its applications, Fuzzy Sets and Systems 123 (2) (2001) 137-145.
  23. W.A. Lodwick, K.D. Jamison, Interval-valued probability in the analysis of problems containing a mixture of possibilistic, probabilistic, and interval uncertainty, Fuzzy Sets and Systems 159 (21) (2008) 2845-2858.
  24. H. Luss, Minimax resource allocation problems: Optimization and parametric analysis, European Journal of Operational Research 60 (1) (1992) 76-86.
  25. W. Ma, K. McAreavey, W. Liu, X. Luo, Acceptable costs of minimax regret equilibrium: a solution to security games with surveillance-driven probabilistic information, Expert Systems with Applications 108 (2018) 206-222.
  26. M.J. Machina, W.K. Viscusi, Handbook of the Economics of Risk and Uncertainty. Volume 1, North-Holland, Oxford, UK, 2014.
  27. C.A. Martínez, K. Khare, M.A. Elzo, On the Bayesness, minimaxity and admissibility of point estimators of allelic frequencies, Journal of Theoretical Biology 383 (2015) 106-115.
  28. A. Miele, T. Wang, C.Y. Tzeng, W.W. Melvin, Transformation techniques for minimax optimal control problems and their application to optimal fight trajectories in a windshear: optimal abort landing trajectories, IFAC Proceedings Volumes 20 (5/8) (1987) 131-150.
  29. S. Mishra, A. Datta-Gupta, Multivariate data analysis, in: Applied Statistical Modeling and Data Analytics, S. Mishra, A. Datta-Gupta (eds.), Elsevier, 2018, 97-118.
  30. J. Moon, T. Başar, Minimax control over unreliable communication channels, Automatica 59 (2015) 182-193.
  31. D.E. Morris, J.E. Oakley, J.A. Crowe, A web-based tool for eliciting probability distributions from experts, Environmental Modelling & Software 52 (2014) 1-4.
  32. C. Ning, F. You, Adaptive robust optimization with minimax regret criterion: Multiobjective optimization framework and computational algorithm for planning and scheduling under uncertainty, Computers & Chemical Engineering 108 (2018) 425-447.
  33. N. Nisan, T. Roughgarden, É. Tardos, V.V. Vazirani, Algorithmic Game Theory, Cambridge University Press, Cambridge, UK, 2007.
  34. L. Pan, D.N. Politis, Bootstrap prediction intervals for Markov processes, Computational Statistics & Data Analysis 100 (2016) 467-494.
  35. G. Parmigiani, L. Inoue, Decision Theory: Principles and Approaches, Wiley, Chichester, UK, 2009.
  36. M.M. Rajabi, B. Ataie-Ashtiani, Efficient fuzzy Bayesian inference algorithms for incorporating expert knowledge in parameter estimation, Journal of Hydrology 536 (2016) 255-272.
  37. R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
  38. V.V. Romanuke, A generalized model of removing N partial indeterminancies of the probabilistic type as a continuous antagonistic game on (2N - 2)-dimensional parallelepiped by maximal disbalance minimization, Herald of Khmelnytskyi National University. Technical Sciences 3 (2011) 45-60.
  39. V.V. Romanuke, Convergence and estimation of the process of computer implementation of the optimality principle in matrix games with apparent play horizon, Journal of Automation and Information Sciences 45 (10) (2013) 49-56.
  40. V.V. Romanuke, Designer's optimal decisions to fit cross-section squares of the supports of a construction platform in overestimations of uncertainties in the generalized model, Cybernetics and Systems Analysis 50 (3) (2014) 426-438.
  41. V.V. Romanuke, Interval uncertainty reduction via division-by-2 dichotomization based on expert estimations for short-termed observations, Journal of Uncertain Systems 12 (1) (2018) 3-21.
  42. S. Tesfamariam, K. Goda, Handbook of Seismic Risk Analysis and Management of Civil Infrastructure Systems, Woodhead Publishing, Cambridge, UK, 2013.
  43. A.D. Torshizi, M.H.F. Zarandi, Hierarchical collapsing method for direct defuzzification of general type-2 fuzzy sets, Information Sciences 277 (2014) 842-861.
  44. N.N. Vorob'yov, Game Theory Fundamentals. Noncooperative Games, Nauka, Moscow, 1984 (in Russian).
  45. N.N. Vorob'yov, Game Theory for Economists-Cyberneticists, Nauka, Moscow, 1985 (in Russian).
  46. R.E. Walpole, R.H. Myers, S.L. Myers, K. Ye, Probability & Statistics for Engineers & Scientists (9th ed.), Prentice Hall, Boston, MA, 2012.
  47. É. Walter, L. Pronzato, Identification of Parametric Models from Experimental Data, Springer, New York, NY, 1997.
  48. M. Xia, C.S. Cai, F. Pan, Y. Yu, Estimation of extreme structural response distributions for mean recurrence intervals based on short-term monitoring, Engineering Structures 126 (2016) 121-132.
  49. R.C. Yadava, P.K. Rai, Analyzing variety of birth intervals: A stochastic approach, in: Handbook of Statistics. Volume 40, A.S.R.S. Rao, C.R. Rao (eds.), Elsevier, 2019, 195-283.
  50. Y. Zhou, N. Fenton, M. Neil, Bayesian network approach to multinomial parameter learning using data and expert judgments, International Journal of Approximate Reasoning 55 (5) (2014) 1252-1268.
  51. S. Zinodiny, S. Rezaei, S. Nadarajah, Bayes minimax estimation of the multivariate normal mean vector under balanced loss function, Statistics & Probability Letters 93 (2014) 96-101.

About this Article

TITLE:
A Minimax Approach to Mapping Partial Interval Uncertainties into Point Estimates

AUTHORS:
Vadim Romanuke

AUTHORS AFFILIATIONS:
Polish Naval Academy, POLAND

JOURNAL:
Journal of Mathematics and Applications
10/42

KEY WORDS AND PHRASES:
Game theory; Interval uncertainties; Minimax decision-making; Pure strategy; Point estimates.

FULL TEXT:
http://doi.prz.edu.pl/pl/pdf/jma/87

DOI:
10.7862/rf.2019.10

URL:
http://dx.doi.org/10.7862/rf.2019.10

RECEIVED:
2019-01-28

ACCEPTED:
2019-04-19

COPYRIGHT:
Oficyna Wydawnicza Politechniki Rzeszowskiej, al. Powstańców Warszawy 12, 35-959 Rzeszów

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