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Navarro, Jorge; Gomis, M. Carmen

Comparisons in the mean residual life order of coherent systems with identically distributed components. (English) Zbl 1411.90127

Appl. Stoch. Models Bus. Ind. 32, No. 1, 33-47 (2016).
Summary: The comparisons of the performance of coherent systems (under different stochastic criteria) is an important task in the reliability theory. Several results have been obtained in the literature for the stochastic, hazard rate and likelihood ratio orders. In this paper, we obtain comparison results for the mean residual life order of coherent systems with identically distributed (ID) component lifetimes. These results can be applied not only to the usual case of systems with independent and identically distributed components but also to the case of systems with exchangeable components and to the more general case of just ID components. The results obtained are based on the representation of the system distribution as a distorted distribution of the common components’ distribution. Some specific comparison results are given to illustrate the theoretical results. The comparison results for distorted distributions given here can also be applied to other statistical concepts such as order statistics, generalized order statistics or record values.

Cited in 25 Documents

MSC:

90B25 Reliability, availability, maintenance, inspection in operations research
62K10 Statistical block designs
60E15 Inequalities; stochastic orderings

Keywords:

stochastic orders; mean residual life order; coherent systems; distorted distributions; copulas
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References:

[1] KocharS, MukerjeeH, SamaniegoFJ. The signature of a coherent system and its application to comparison among systems. Naval Research Logistics1999; 46:507-523. · Zbl 0948.90067
[2] NavarroJ, SamaniegoFJ, BalakrishnanN, BhattacharyaD. On the application and extension of system signatures in engineering reliability. Naval Research Logistics2008; 55:313-327. · Zbl 1153.90386
[3] NavarroJ, RubioR. A note on necessary and sufficient conditions for ordering properties of coherent systems with exchangeable components. Naval Research Logistics2011; 58:478-489. · Zbl 1218.90068
[4] NavarroJ, delAguilaY, SordoMA, Suárez‐LlorensA. Stochastic ordering properties for systems with dependent identically distributed components. Applied Stochastic Models in Business and Industry2013; 29:264-278. · Zbl 1291.60042
[5] NavarroJ, delAguilaY, SordoMA, Suárez‐LlorensA.. Preservation of reliability classes under the formation of coherent systems. Applied Stochastic Models in Business and Industry2014; 30:444-454. · Zbl 07880609
[6] NavarroJ, RychlikT. Comparisons and bounds for expected lifetimes of reliability systems. European Journal of Operational Research2010; 207:309-317. · Zbl 1217.62155
[7] NavarroJ, SpizzichinoF. Comparisons of series and parallel systems with components sharing the same copula. Applied Stochastic Models in Business and Industry2010; 26:775-791. · Zbl 1226.90031
[8] NavarroJ, delAguilaY, SordoMA, Suárez‐LlorensA. Applications to coherent systems and other concepts, Preservation of stochastic orders under the formation of generalized distorted distributions, Submitted.
[9] NavarroJ, PellereyF, Di CrescenzoA. Orderings of coherent systems with randomized dependent components. To appear in European Journal of Operational Research2015; 240:127-139. · Zbl 1339.60015
[10] YouY, LiX. On allocating redundancies to k‐out‐of‐n reliability systems. Applied Stochastic Models in Business and Industry2014; 30:361-371. · Zbl 07879467
[11] BelzunceF, Martinez‐RiquelmeC, RuizJM. On sufficient conditions for mean residual life and related orders. Computational Statistics and Data Analysis2013; 61:199-210. · Zbl 1349.62497
[12] QuigginJ. A theory of anticipated utility. Journal of Economic Behavior and Organization1982; 3:323-343.
[13] YaariME. The dual theory of choice under risk. Econometrica1987; 55:95-115. · Zbl 0616.90005
[14] WangS. Premium calculation by transforming the layer premium density. ASTIN Bulletin1996; 26:71-92.
[15] WangS, YoungVR. Ordering risks: expected utility theory versus Yaari’s dual theory of risk. Insurance: Mathematics and Economics1998; 22:145-161. · Zbl 0907.90102
[16] BelzunceF, FrancoM, RuizJM, RuizMC. On partial orderings between coherent systems with different structures. Probability in the Engineering and Informational Sciences2001; 15:273-293. · Zbl 0984.60026
[17] HürlimannW.. Distortion risk measures and economic capital. North American Actuarial Journal2004; 8:86-95. · Zbl 1085.91526
[18] KhalediBE, ShakedM. Stochastic comparisons of multivariate mixtures. Journal of Multivariate Analysis2010; 101:2486-2498. · Zbl 1201.62111
[19] SordoMA, Suarez‐LlorensA. Stochastic comparisons of distorted variability measures. Insurance: Mathematics and Economics2011; 49:11-17. · Zbl 1218.91095
[20] GuptaN, KumarS. Stochastic comparisons of component and system redundancies with dependent components. Operations Research Letters2014; 42:284-289. · Zbl 1408.90110
[21] ShakedM, ShanthikumarJG. Stochastic Orders. Springer‐Verlag: New York, 2007. · Zbl 1111.62016
[22] BarlowRE, ProschanF. International Series in Decision Processes, Statistical Theory of Reliability and Life Testing. Holt Rinehart and Winston, Inc.: New York, 1975. · Zbl 0379.62080
[23] SatyanarayanaA, PrabhakarA. New topological formula and rapid algorithm for reliability analysis of complex networks. IEEE Transactions on Reliability1978; R‐27:82-97. · Zbl 0409.90039
[24] NavarroJ, RuizJM, SandovalCJ. Properties of coherent systems with dependent components. Communications in Statistics Theory and Methods2007; 36:175-191. · Zbl 1121.60015
[25] NavarroJ, RubioR. Computation of signatures of coherent systems with five components. Communications in Statistics Simulation and Computation2010; 39:68-84. · Zbl 1183.62178
[26] NavarroJ, EryilmazS.. Mean residual lifetimes of consecutive k‐out‐of‐n systems. Journal of Applied Probability2007; 44:82-98. · Zbl 1135.62084
[27] Rodríguez‐LallenaJA, Ú‐FloresM.. Multivariate copulas with quadratic sections in one variable. Metrika2010; 72:331-349. · Zbl 1197.62051
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