Abstract
The \(\alpha \)-mixtures is a new, flexible family of distributions that includes many mixture models as special cases. This paper is mainly focused on relevant stochastic comparisons and ageing properties of \(\alpha \)-mixtures of survival functions. In particular, we prove that ageing properties of \(\alpha \)-mixtures for additive and multiplicative hazards models depend on the properties of the baseline failure rate functions and the corresponding conditional moments of mixing distributions. Partial orderings of the finite \(\alpha \)-mixtures in the sense of the usual stochastic order and the hazard rate order are discussed. Finally, we extend some results on the shape of the mixture failure rate obtained in the literature for usual mixtures to the case of \(\alpha \)-mixtures.
Similar content being viewed by others
Notes
We say that a function \(h : R^{m} \rightarrow R \) is increasing (decreasing) if \(h(x_{1},..., x_{m}) \le (\ge )h(y_{1},..., y_{m})\) for all \( x_{i} \le y_{i}, i = 1,..., m\).
For two vectors u and v, we say that \({u}\le {v}\) if for all i, \(i = 2,..., n\), \( u_{i} \le v_{i}\).
References
Amini-Seresht E, Zhang Y (2017) Stochastic comparisons on two finite mixture models. Operations Research Letters 45(5):475–480
Asadi M, Ebrahimi N, Soofi ES (2019) The alpha-mixture of survival functions. Journal of Applied Probability 56(4):1151–1167
Badia FG, Cha JH (2017) On bending (down and up) property of reliability measures in mixtures. Metrika 80(4):455–482
Badia FG, Lee H (2020) On stochastic comparisons and ageing properties of multivariate proportional hazard rate mixtures. Metrika 83(3):355–375
Badia FG, Berrade MD, Campos CA (2002) Aging properties of the additive and proportional hazard mixing models. Reliability Engineering and System Safety 78(2):165–172
Barlow RE, Proschan F (1981) Statistical Theory of Reliability and Life Testing: Probability models. To Begin With, Silver Spring, MD
Block HW, Joe H (1997) Tail behavior of the failure rate functions of mixtures. Lifetime Data Analysis 3(3):269–288
Block HW, Savits TH (1976) The IFRA closure problem. The Annals of Probability 4:1030–1032
Block HW, Mi J, Savits TH (1993) Burn-in and mixed population. Journal of Applied probability 30(3):692–702
Block HW, Li Y, Savits TH (2003) Preservation of properties under mixture. Probability in the Engineering and Informational Sciences 17(2):205–212
Cha JH (2011) Comparison of combined stochastic risk processes and its applications. European journal of operational research 215(2):404–410
Cha JH, Badia FG (2016) An information based burn-in procedure for minimally repaired items from mixed population. Applied Stochastic Models in Business and Industry 32(4):511–525
Cuadras CM (2002) On the covariance between functions. Journal of Multivariate Analysis 81(1):19–27
Finkelstein, M. (2008). Failure Rate Modelling for Reliability and Risk. Springer Science & Business Media
Finkelstein M, Esaulova V (2001) Modeling a failure rate for a mixture of distribution function. Probability in the Engineering and Informational Sciences 15(3):383–400
Finkelstein M, Esaulova V (2006) On mixture failure rates ordering. Communications in Statistics - Theory and Methods 35(11):1943–1955
Hazra NK, Finkelstein M (2018) On stochastic comparisons of finite mixtures for some semiparametric families of distributions. Test 27(4):988–1006
Hazra NK, Finkelstein M, Cha JH (2017) On optimal grouping and stochastic comparisons for heterogeneous items. Journal of Multivariate Analysis 160:146–156
Joag-dev K, Kochar S, Proschan F (1995) A general composition theorem and its applications to certain partial orderings of distributions. Statistics & Probability Letters 22:111–119
Lynch JD (1999) On conditions for mixtures of increasing failure rate distributions to have an increasing failure rate. Probability in the Engineering and Informational Sciences 13(1):33–36
Navarro J (2008) Likelihood ratio ordering of order statistics, mixtures and systems. Journal of Statistical Planning and Inference 138(5):1242–1257
Navarro J (2016) Stochastic comparisons of generalized mixtures and coherent systems. Test 25(1):150–169
Navarro J, del Aguila Y (2017) Stochastic comparisons of distorted distributions, coherent systems and mixtures with ordered components. Metrika 80(6–8):627–648
Navarro J, Hernandez PJ (2004) How to obtain bathtub-shaped failure rate models from normal mixtures. Probability in the Engineering and Informational Sciences 18(4):511–531
Navarro J, Hernandez PJ (2008) Mean residual life functions of finite mixtures, order statistics and coherent systems. Metrika 67(3):277–298
Navarro J, Guillamon A, Ruiz MDC (2009) Generalized mixtures in reliability modelling: applications to the construction of bathtub shaped hazard models and the study of systems. Applied Stochastic Models in Business and Industry 25(3):323–337
Savits TH (1985) A multivariate IFR class. Journal of Applied Probability 22(1):197–204
Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer, Berlin
Shaked M, Spizzichino F (2001) Mixtures and monotonicity of failure rate functions. Advances in Reliability. Springer, Berlin, pp 185–198
Acknowledgements
Asadi’s research work was performed in IPM Isfahan branch and was in part supported by a grant from IPM (No. 99620213).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest:
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Shojaee, O., Asadi, M. & Finkelstein, M. On Some Properties of \(\alpha \)-Mixtures. Metrika 84, 1213–1240 (2021). https://doi.org/10.1007/s00184-021-00818-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00184-021-00818-1