A distribution function \(F\) is a generalized mixture of the distribution functions \(F_1,\ldots ,F_k\) if \(F=w_1 F_1+\ldots +w_k F_k\), where \(w_1,\ldots ,w_k\) are some real numbers (weights) which should satisfy \(w_1+\ldots +w_k=1\). If all the weights are positive, then we have a classical finite mixture. If some weights are negative, then we have a negative mixture. Negative mixtures appear in different applied probability models (order statistics, estimators, coherent systems, etc.). The conditions to obtain stochastic comparisons of classical (positive) mixtures are well known in the literature. However, for negative mixtures, there are only results for the usual stochastic order. In this paper, conditions for hazard rate and likelihood ratio comparisons of generalized mixtures are obtained. These theoretical results are applied in this paper to study distribution-free comparisons of coherent systems using their representations as generalized mixtures. They can also be applied to other probability models in which the generalized mixtures appear.
Generalized mixtures Stochastic comparisons Coherent systems Hazard rate
Mathematics Subject Classification
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I would like to thank the editors and the anonymous reviewers for several helpful suggestions. This research was supported in part by Ministerio de Economía y Competitividad under Grant MTM2012-34023-FEDER.
Balakrishnan N, Cramer E (2014) The art of progressive censoring: applications to reliability and quality. Birkhäuser, New YorkCrossRefGoogle Scholar
Balakrishnan N, Xie Q (2007a) Exact inference for a simple step-stress model with type-I hybrid censored data from the exponential distribution. J Stat Plan Inference 137:3268–3290CrossRefMathSciNetzbMATHGoogle Scholar
Balakrishnan N, Xie Q (2007b) Exact inference for a simple step-stress model with type-II hybrid censored data from the exponential distribution. J Stat Plan Inference 137:2543–2563CrossRefMathSciNetzbMATHGoogle Scholar
Balakrishnan N, Xie Q, Kundu D (2009) Exact inference for a simple step-stress model from the exponential distribution under time constraint. Ann Inst Stat Math 61:251–274CrossRefMathSciNetzbMATHGoogle Scholar
Barlow RE, Proschan F (1975) Statistical theory of reliability and life testing. holt, rinehart and winston, New YorkzbMATHGoogle Scholar
Borgonovo E (2010) The reliability importance of components and prime implicants in coherent and non-coherent systems including total-order interactions. Eur J Oper Res 204:485–495CrossRefzbMATHGoogle Scholar
Navarro J, Guillamon A, Ruiz MC (2009) Generalized mixtures in reliability modelling: applications to the construction of bathtub shaped hazard models and the study of systems. Appl Stoch Models Bus Ind 25:323–337CrossRefMathSciNetzbMATHGoogle Scholar
Navarro J, Pellerey F, Di Crescenzo A (2015) Orderings of coherent systems with randomized dependent components. Eur J Oper Res 240:127–139CrossRefGoogle Scholar
Navarro J, Rubio R (2011) A note on necessary and sufficient conditions for ordering properties of coherent systems with exchangeable components. Naval Res Logist 58:478–489CrossRefMathSciNetzbMATHGoogle Scholar
Navarro J, Samaniego FJ, Balakrishnan N, Bhattacharya D (2008b) On the application and extension of system signatures in engineering reliability. Naval Res Logist 55:313–327CrossRefMathSciNetzbMATHGoogle Scholar
Wu JW (2001) Characterizations of generalized mixtures of geometric and exponential distributions based on upper record values. Stat Papers 42:123–133CrossRefzbMATHGoogle Scholar
Wu JW, Lee WC (1998) Characterization of generalized mixtures of exponential distribution based on conditional expectation of order statistics. J Jpn Stat Soc 28:39–44CrossRefMathSciNetzbMATHGoogle Scholar