Abstract
In reliability theory any coherent system can be represented as either a series-parallel or a parallel-series system. Its lifetime can thus be written as the minimum of maxima or the maximum of minima. For large-scale coherent systems it is sensible to assume that the number of system components goes to infinity. Then, the possible non-degenerate extreme value laws either for maxima or for minima are eligible candidates for the system reliability or at least for the finding of adequate lower and upper bounds for the reliability. The identification of the possible limit laws for the system reliability of homogeneous series-parallel (or parallel-series) systems has already been done under different frameworks. However, it is well-known that in most situations such non-degenerate limit laws are better approximated by an adequate penultimate distribution. Dealing with regular and homogeneous parallel-series systems, we assess both theoretically and through Monte-Carlo simulations the gain in accuracy when a penultimate approximation is used instead of the ultimate one.
Similar content being viewed by others
References
Anderson CW (1971) Contributions to the asymptotic theory of extreme values. PhD thesis, University of London
Barlow AA, Proschan F (1975) Statistical theory of reliability and life testing: probability models. Holt, Rinehart and Winston, Inc., USA
Beirlant J, Caeiro F, Gomes MI (2012) An overview and open research topics in statistics of univariate extremes. Revstat 10:1–31
Diebolt J, Guillou A (2005) Asymptotic behaviour of regular estimators. Revstat 3:19–44
Dietrich D, de Haan L, Hüsler J (2002) Testing extreme value conditions. Extremes 5:71–85
Drees H, de Haan L, Li D, Pereira TT (2006) Approximations to the tail empirical distribution function with applications to testing extreme value conditions. J Stat Plan Inference 136:3498–3538
Fisher RA, Tippett LHC (1928) Limiting forms of the frequency distributions of the largest or smallest member of a sample. Proc Camb Philos Soc 24:180–190
Gnedenko BV (1943) Sur la distribution limite du terme maximum d’une série aléatoire. Ann Math 44:423–453
Gnedenko BV, Belyayev YuK, Solovyev AD (1969) Mathematical methods of reliability theory. Academic Press, USA
Gomes MI (1984) Penultimate limiting forms in extreme value theory. Ann Inst Stat Math 36:71–85
Gomes MI (1994) Penultimate behaviour of the extremes. In: Galambos J, et al (eds) Extreme value theory and applications. Kluwer Academic Publishers, pp 403–418
Gomes MI, de Haan L (1999) Approximations by penultimate extreme value distributions. Extremes 2:71–85
Gomes MI, Pestana DD (1987) Nonstandard domains of attraction and rates of convergence. In: Puri L, et al (eds) New perspectives in theoretical and applied statistics. Wiley, pp 467–477
Hüsler J, Li D (2006) On testing extreme value conditions. Extremes 9:69–86
Jenkinson AF (1955) The frequency distribution of the annual maximum (or minimum) values of meteorological elements. Q J R Meteorol Soc 81:158–171
Kaufmann E (2000) Penultimate approximations in extreme value theory. Extremes 3:39–55
Laio F (2004) Cramer-von Mises and Anderson–Darling goodness of fit tests for extreme value distributions with unknown parameters. Water Resour Res 40:W09308
Raoult JP, Worms R (2003) Rate of convergence for the generalized Pareto approximation of the excesses. Adv Appl Probab 35:1007–1027
Reis P, Canto e Castro L (2009) Limit model for the reliability of a regular and homogeneous series-parallel system. Revstat 7:227–243
Samaniego F (1985) On closure of the IFR class under formation of coherent systems. IEEE Trans Reliab 34:69–72
von Mises R (1936) La distribution de la plus grande de n valeurs. Revue Math. Union Interbalcanique 1:141–160. Reprinted in Selected Papers of Richard von Mises. Amer Math Soc (1964) 2:271–294
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Reis, P., Canto e Castro, L., Dias, S. et al. Penultimate Approximations in Statistics of Extremes and Reliability of Large Coherent Systems. Methodol Comput Appl Probab 17, 189–206 (2015). https://doi.org/10.1007/s11009-013-9338-7
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11009-013-9338-7