A note on the "saturation" of poisson-exponential cumulative function in Hausdorff sense

Year 2018, Volume: 1 Issue: 1, 39 - 44, 30.09.2018

Anton Iliev , Nikolay Kyurkchiev

https://doi.org/10.33434/cams.444785

Abstract

In this paper we study the important ''saturation'' characteristic for the Poisson--exponential cumulative distribution function in the Hausdorff sense. The results have independent significance in the study of issues related to lifetime analysis, insurance mathematics, biochemical kinetics, population dynamics and debugging theory. Numerical examples, illustrating our results are presented using programming environment Mathematica.

Keywords

Hausdorff distance, Poisson--exponential cumulative distribution function (Pcdf), Upper and lower bounds

References

• [1] V. Cancho, F. Louzada and G. Barriga, The Poisson–exponential lifetime distribution, Comp. Stat. Data Anal., vol. 55, pp. 677–686, 2011.
• [2] G. Rodrigues, F. Louzada and P. Ramos, Poisson–exponential distribution: different methods of estimation, J. of Appl. Stat., vol. 45 (1), pp. 128–144, 2018.
• [3] F. Louzada, P. Ramos and P. Ferreira, Exponential–Poisson distribution: estimation and applications to rainfall and aircraft data with zero occurrence, Communication in Statistics–Simulation and Computation, 2018.
• [4] F. Hausdorff, Set Theory (2 ed.) (Chelsea Publ., New York, (1962 [1957]) (Republished by AMS-Chelsea 2005), ISBN: 978–0–821–83835–8.
• [5] N. Pavlov, A. Iliev, A. Rahnev and N. Kyurkchiev, Some software reliability models: Approximation and modeling aspects, LAP LAMBERT Academic Publishing, 2018, ISBN: 978-613-9-82805-0.
• [6] N. Pavlov, A. Iliev, A. Rahnev and N. Kyurkchiev, Nontrivial Models in Debugging Theory (Part 2), LAP LAMBERT Academic Publishing, 2018, ISBN: 978-613-9-87794-2.
• [7] M. Ramos, A. Percontini, G. Cordeiro and R. Silva, The Burr XII Negative Binomial Distribution with applications to Lifetime Data, Int. J. of Stat. and Prob., vol. 4 (1), pp. 109–124, 2015.
• [8] N. Kyurkchiev and S. Markov, Sigmoid functions: Some Approximation and Modelling Aspects. (LAP LAMBERT Academic Publishing, Saarbrucken, 2015); ISBN 978-3-659-76045-7.
• [9] N. Kyurkchiev and S. Markov, On the Hausdorff distance between the Heaviside step function and Verhulst logistic function. J. Math. Chem., vol. 54 (1), pp. 109–119, DOI:10.1007/S10910-015-0552-0.
• [10] A. Iliev, N. Kyurkchiev and S. Markov, On the Approximation of the step function by some sigmoid functions, Mathematics and Computers in Simulation, 2015, DOI:10.1016/j.matcom.2015.11.005.
• [11] N. Kyurkchiev, A new transmuted cumulative distribution function based on the Verhulst logistic function with application in population dynamics. Biomath Communications, vol. 4 (1), 2017.
• [12] N. Kyurkchiev, A new class of activation function based on the correcting amendments Int. J. for Sci., Res. and Developments, vol. 6 (2), pp. 565–568, 2018.
• [13] N. Kyurkchiev, The new transmuted C.D.F. based on Gompertz function, Biomath Communications, vol. 5 (1), 2018.
• [14] V. Kyurkchiev and N. Kyurkchiev, A Family of Recurrence Generated Functions Based on ”Half-Hyperbolic Tangent Activation Function”. Biomedical Statistics and Informatics, vol. 2 (3), pp. 87–94, 2017.
• [15] A. Iliev, N. Kyurkchiev and S. Markov, A Note on the New Activation Function of Gompertz Type, Biomath Communications, vol. 4 (2), 2017.
• [16] N. Kyurkchiev, A. Iliev, Extension of Gompertz-type Equation in Modern Science: 240 Anniversary of the birth of B. Gompertz, LAP LAMBERT Academic Publishing, 2018, ISBN: 978-613-9-90569-0.
• [17] V. Kyurkchiev, A. Malinova, O. Rahneva, P. Kyurkchiev, On the Burr XII-Weibull Software Reliability Model, Int. J. of Pure and Appl. Math., vol. 119 (4), pp. 639—650, 2018.
• [18] V. Kyurkchiev, A. Malinova, O. Rahneva, P. Kyurkchiev, Some Notes on the Extended Burr XII Software Reliability Model, Int. J. of Pure and Appl. Math., vol. 120 (1), pp. 127–136, 2018.
• [19] N. Pavlov, A. Iliev, A. Rahnev, N. Kyurkchiev, Application of a New Class Cumulative Lifetime Distribution to Software Reliability Analysis, Communications in Applied Analysis, vol. 22 (4), pp. 555-–565, 2018.
• [20] V. Kyurkchiev, H. Kiskinov, O. Rahneva, G. Spasov, A Note on the Exponentiated Exponential-Poisson Software Reliability Model, Neural, Parallel, and Scientific Computations, vol. 26, 2018. (to appear)
• [21] N. Kyurkchiev, A. Iliev and S. Markov, Some Techniques for Recurrence Generating of Activation Functions: Some Modeling and Approximation Aspects, LAP LAMBERT Academic Publishing, 2017, ISBN: 978-3-330-33143-3.