View More View Less
  • 1 Eötvös University Department of Operations Research Pázmány Péter sétány 1/c Budapest H-1117 Hungary
  • 2 Rutgers Center for Operations Research RUTCOR 640 Bartholomew Rd. Piscataway NJ 08854 USA
Open access

Let A 1,...,A N and B 1,...,B M be two sequences of events and let ν N(A) and ν M(B) be the number of those A i and B j, respectively, that occur. Based on multivariate Lagrange interpolation, we give a method that yields linear bounds in terms of S k,t, k+tm on the distribution of the vector (ν N(A), ν M(B)). For the same value of m, several inequalities can be generated and all of them are best bounds for some values of S k,t. Known bivariate Bonferroni-type inequalities are reconstructed and new inequalities are generated, too.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • Biermann, O., Über näherungsweise Kubaturen, Monatshefte Math. Phys., 14 (1903), 211–225.

    Biermann O. , 'Über näherungsweise Kubaturen ' (1903 ) 14 Monatshefte Math. Phys. : 211 -225.

    • Search Google Scholar
  • Boros, E. and Prékopa, A., Closed Form Two-Sided Bounds for Probabilities That Exactly r and at Least r out of n Events Occur, Mathematics of Operations Research, 14 (1989), 317–342.

    Prékopa A. , 'Closed Form Two-Sided Bounds for Probabilities That Exactly r and at Least r out of n Events Occur ' (1989 ) 14 Mathematics of Operations Research : 317 -342.

    • Search Google Scholar
  • Boros, E., Scozzari, A., Tardella, F. and Veneziani, P., Polynomially Computable Bounds for the Probability of the Union of Events, Mathematics of Operations Research (2014). http://dx.doi.org/10.1287/moor.2014.0657

    Veneziani P. , '', in Mathematics of Operations Research , (2014 ) -.

  • Bukszár, J., Hypermultitrees and sharp Bonferroni inequalities, Math. Inequal. Appl., 6(4) (2003), 727–743.

    Bukszár J. , 'Hypermultitrees and sharp Bonferroni inequalities ' (2003 ) 6 Math. Inequal. Appl. : 727 -743.

    • Search Google Scholar
  • Bukszár, J., Mádi-Nagy, G. and Szántai, T., Computing bounds for the probability of the union of events by different methods, Annals of Operations Research, 201(1) (2012), 63–81.

    Szántai T. , 'Computing bounds for the probability of the union of events by different methods ' (2012 ) 201 Annals of Operations Research : 63 -81.

    • Search Google Scholar
  • Bukszár, J. and Szántai, T., Probability bounds given by hypercherry trees, Optimization Methods and Software, 17 (2002), 409–422.

    Szántai T. , 'Probability bounds given by hypercherry trees ' (2002 ) 17 Optimization Methods and Software : 409 -422.

    • Search Google Scholar
  • Chen, J., Multivariate Bonferroni-Type Inequalities: Theory and Applications, CRC Press (2014).

    Chen J. , '', in Multivariate Bonferroni-Type Inequalities: Theory and Applications , (2014 ) -.

    • Search Google Scholar
  • Chen, T. and Seneta, E., Multivariate Bonferroni-Type Lower Bounds, J. Appl. Probab., 33(3) (1996), 729–740.

    Seneta E. , 'Multivariate Bonferroni-Type Lower Bounds ' (1996 ) 33 J. Appl. Probab. : 729 -740.

    • Search Google Scholar
  • Chen, T. and Seneta, E., A Refinement of Multivariate Bonferroni-Type Inequalities, J. Appl. Probab., 37(1) (2000), 276–282.

    Seneta E. , 'A Refinement of Multivariate Bonferroni-Type Inequalities ' (2000 ) 37 J. Appl. Probab. : 276 -282.

    • Search Google Scholar
  • Dohmen, K. and Tittmann, P., Bonferroni-type inequalities and binomially bounded functions, Discrete Math., 310(6–7) (2007), 1265–1268.

    Tittmann P. , 'Bonferroni-type inequalities and binomially bounded functions ' (2007 ) 310 Discrete Math. : 1265 -1268.

    • Search Google Scholar
  • Galambos, J. and Simonelli, I., Bonferroni-Type Inequalities with Applications, Springer-Verlag, Berlin/New York (1996).

    Simonelli I. , '', in Bonferroni-Type Inequalities with Applications , (1996 ) -.

  • Galambos, J. and Xu, Y., Some Optimal Bivariate Bonferroni-Type Bounds Proc. of the American Mathematical Society, 117(2) (1993), 523–528.

    Xu Y. , 'Some Optimal Bivariate Bonferroni-Type Bounds ' (1993 ) 117 Proc. of the American Mathematical Society : 523 -528.

    • Search Google Scholar
  • Galambos, J. and Xu, Y.,. Bivariate Extension of the Method of Polynomials for Bonferroni-type Inequalities, J. Multivariate Analysis, 52 (1995), 131–139.

    Xu Y. , 'Bivariate Extension of the Method of Polynomials for Bonferroni-type Inequalities ' (1995 ) 52 J. Multivariate Analysis : 131 -139.

    • Search Google Scholar
  • Habib, A. and Szántai, T., New bounds on the reliability of the consecutive k-out-of-r-from-n: F system, Reliability Engineering and System Safety, 68 (2000), 97–106.

    Szántai T. , 'New bounds on the reliability of the consecutive k-out-of-r-from-n: F system ' (2000 ) 68 Reliability Engineering and System Safety : 97 -106.

    • Search Google Scholar
  • Lee, M.-Y., Improved bivariate Bonferroni-type inequalities, Statistics and Probability Letters, 31 (1997), 359–364.

    Lee M.-Y. , 'Improved bivariate Bonferroni-type inequalities ' (1997 ) 31 Statistics and Probability Letters : 359 -364.

    • Search Google Scholar
  • Mádi-Nagy, G., On multivariate discrete moment problems: generalization of the bivariate min algorithm for higher dimensions, SIAM Journal on Optimization, 19(4) (2009), 1781–1806.

    Mádi-Nagy G. , 'On multivariate discrete moment problems: generalization of the bivariate min algorithm for higher dimensions ' (2009 ) 19 SIAM Journal on Optimization : 1781 -1806.

    • Search Google Scholar
  • Mádi-Nagy, G. and Prékopa, A., On Multivariate Discrete Moment Problems and Their Applications to Bounding Expectations and Probabilities, Mathematics of Operations Research, 29(2) (2004), 229–258.

    Prékopa A. , 'On Multivariate Discrete Moment Problems and Their Applications to Bounding Expectations and Probabilities ' (2004 ) 29 Mathematics of Operations Research : 229 -258.

    • Search Google Scholar
  • Prékopa, A., Boole-Bonferroni Inequalities and Linear Programming, Operations Research, 36(1) (1988), 145–162.

    Prékopa A. , 'Boole-Bonferroni Inequalities and Linear Programming ' (1988 ) 36 Operations Research : 145 -162.

    • Search Google Scholar
  • Prékopa, A., Sharp bounds on probabilities using linear programming, Operations Research, 38 (1990a), 227–239.

    Prékopa A. , 'Sharp bounds on probabilities using linear programming ' (1990 ) 38 Operations Research : 227 -239.

    • Search Google Scholar
  • Prékopa, A., The discrete moment problem and linear programming, Discrete Applied Mathematics, 27 (1990b), 235–254.

    Prékopa A. , 'The discrete moment problem and linear programming ' (1990 ) 27 Discrete Applied Mathematics : 235 -254.

    • Search Google Scholar
  • Prékopa, A., Inequalities on Expectations Based on the Knowledge of Multivariate Moments. M. Shaked and Y. L. Tong, eds., Stochastic Inequalities, Institute of Mathematical Statistics, Lecture Notes — Monograph Series, Vol 22 (1992), 309–331.

    Prékopa A. , '', in Stochastic Inequalities , (1992 ) -.

  • Prékopa, A., Stochastic Programming, Kluwer Academic Publishers, Dordrecht, Boston (1995).

    Prékopa A. , '', in Stochastic Programming , (1995 ) -.

  • Prékopa, A., Bounds on Probabilities and Expectations Using Multivariate Moments of Discrete Distributions, Studia Scientiarum Mathematicarum Hungarica, 34 (1998), 349–378.

    Prékopa A. , 'Bounds on Probabilities and Expectations Using Multivariate Moments of Discrete Distributions ' (1998 ) 34 Studia Scientiarum Mathematicarum Hungarica : 349 -378.

    • Search Google Scholar
  • Prékopa, A., On Multivariate Discrete Higher Order Convex Functions and their Applications. RUTCOR Research Report 39-2000 (2000). Also in: Proceedings of the Sixth International Conference on Generalized Convexity and Monotonicity, Karlovasi, Samos, Greece, August 29 - September 2, to appear.

  • Simonelli, I., An Extension of the Bivariate Method of Polynomials and a Reduction Formula for Bonferroni-Type Inequalities, J. Multivariate Analysis, 69 (1999), 1–9.

    Simonelli I. , 'An Extension of the Bivariate Method of Polynomials and a Reduction Formula for Bonferroni-Type Inequalities ' (1999 ) 69 J. Multivariate Analysis : 1 -9.

    • Search Google Scholar
  • Veneziani, P., Upper bounds of degree 3 for the probability of the union of events via linear programming, Discrete Applied Mathematics, 157(4) (2009), 858–863.

    Veneziani P. , 'Upper bounds of degree 3 for the probability of the union of events via linear programming ' (2009 ) 157 Discrete Applied Mathematics : 858 -863.

    • Search Google Scholar
  • Vizvári, B., New upper bounds on the probability of events based on graph structures, Math. Inequal. Appl., 10(1) (2007), 217–228.

    Vizvári B. , 'New upper bounds on the probability of events based on graph structures ' (2007 ) 10 Math. Inequal. Appl. : 217 -228.

    • Search Google Scholar

The author instruction is available in PDF.

Please, download the file from HERE

Manuscript submission: HERE

 

  • Impact Factor (2019): 0.486
  • Scimago Journal Rank (2019): 0.234
  • SJR Hirsch-Index (2019): 23
  • SJR Quartile Score (2019): Q3 Mathematics (miscellaneous)
  • Impact Factor (2018): 0.309
  • Scimago Journal Rank (2018): 0.253
  • SJR Hirsch-Index (2018): 21
  • SJR Quartile Score (2018): Q3 Mathematics (miscellaneous)

Language: English, French, German

Founded in 1966
Publication: One volume of four issues annually
Publication Programme: 2020. Vol. 57.
Indexing and Abstracting Services:

  • CompuMath Citation Index
  • Mathematical Reviews
  • Referativnyi Zhurnal/li>
  • Research Alert
  • Science Citation Index Expanded (SciSearch)/li>
  • SCOPUS
  • The ISI Alerting Services

 

Subscribers can access the electronic version of every printed article.

Senior editors

Editor(s)-in-Chief: Pálfy Péter Pál

Managing Editor(s): Sági, Gábor

Editorial Board

  • Biró, András (Number theory)
  • Csáki, Endre (Probability theory and stochastic processes, Statistics)
  • Domokos, Mátyás (Algebra (Ring theory, Invariant theory))
  • Győri, Ervin (Graph and hypergraph theory, Extremal combinatorics, Designs and configurations)
  • O. H. Katona, Gyula (Combinatorics)
  • Márki, László (Algebra (Semigroup theory, Category theory, Ring theory))
  • Némethi, András (Algebraic geometry, Analytic spaces, Analysis on manifolds)
  • Pach, János (Combinatorics, Discrete and computational geometry)
  • Rásonyi, Miklós (Probability theory and stochastic processes, Financial mathematics)
  • Révész, Szilárd Gy. (Analysis (Approximation theory, Potential theory, Harmonic analysis, Functional analysis))
  • Ruzsa, Imre Z. (Number theory)
  • Soukup, Lajos (General topology, Set theory, Model theory, Algebraic logic, Measure and integration)
  • Stipsicz, András (Low dimensional topology and knot theory, Manifolds and cell complexes, Differential topology)
  • Szász, Domokos (Dynamical systems and ergodic theory, Mechanics of particles and systems)
  • Tóth, Géza (Combinatorial geometry)

STUDIA SCIENTIARUM MATHEMATICARUM HUNGARICA
Gábor Sági
Address: P.O. Box 127, H–1364 Budapest, Hungary
Phone: (36 1) 483 8344 ---- Fax: (36 1) 483 8333
E-mail: smh.studia@renyi.mta.hu