On Likelihood Ratio Ordering of Parallel Systems with Two Generalized Exponential Components

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HAIDARI A.; SATTARI M.; Barmalzan GH.

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Journal: JOURNAL OF STATISTICAL SCIENCES Year:2022 | Volume:16 | Issue:1 Page(s): 109-126

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Title

On Likelihood Ratio Ordering of Parallel Systems with Two Generalized Exponential Components

Author(s)

HAIDARI A. | SATTARI M. | Barmalzan GH. | Issue Writer Certificate

Keywords

Parallel Systems

Likelihood Ratio Order

Generalized Exponential Distribution

Exponentiated Weibull Distribution

Abstract

Introduction The exponential distribution, because of its memoryless, has a central rule in reliability theory and survival analysis. But, this distribution is not a suitable model to fit the data sets in practical situations due to its constant hazard rate function. For this reason, some generalizations of the exponential distribution exist in the literature. The Generalized Exponential Distribution is introduced by adding a shape parameter to the exponential distribution via the exponentiated method. This distribution admits both increasing and decreasing hazard rate function. For more information on the Generalized Exponential Distribution and its applications, one can refer to Gupta and Kundu (2007) and Nadarajah (2011). Comparisons of Parallel Systems with two independent heterogeneous exponential components are studied extensively in the literature. Boland et al. (١, ٩, ٩, 4) proved that the hazard rate order between two Parallel Systems holds under the majorization order between the vectors of the hazard rate parameters. This result is extended to the Likelihood Ratio Order by Dykstra et al. (1997). In this direction, Zhao and Balakrishnan (2012) obtained some characterization results concerning the hazard rate and Likelihood Ratio Orders using the p-larger and weak majorization orders between the hazard rate parameters vectors. Yan et al. (2012) established sufficient conditions to compare two Parallel Systems in the hazard rate and Likelihood Ratio Orders. The present work provides a sufficient condition to compare Parallel Systems comprising two independent heterogeneous generalized exponential components in the Likelihood Ratio Order. Material and Methods The comparison of essential characteristics associated with lifetimes of technical systems is an exciting topic in reliability theory since it usually enables us to approximate complex systems with simpler systems and subsequently obtain various bounds for important ageing characteristics of the complex system. A convenient tool for this purpose is the theory of stochastic orderings. Results and Discussion Consider two Parallel Systems with their component lifetimes following a Generalized Exponential Distribution. In this paper, based on existing shape and scale parameters included in the distribution of one of the systems, we introduce a region such that if the vector of scale parameters of another parallel system lies in that region, then the Likelihood Ratio Ordering between the two systems hold. An extension of this result is also presented for the case when the lifetimes of components follow Exponentiated Weibull Distribution. Conclusion In this paper, based on the shape and scale vectors of parameters involved in the lifetime distribution of a parallel system consisting of two independent heterogeneous generalized exponential components, a region is obtained such that if the scale vector of parameters of another parallel system lies in this region, then the Likelihood Ratio Order between systems holds.

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APA: Copy

HAIDARI, A., SATTARI, M., & Barmalzan, GH.. (2022). On Likelihood Ratio Ordering of Parallel Systems with Two Generalized Exponential Components. JOURNAL OF STATISTICAL SCIENCES, 16(1 ), 109-126. SID. https://sid.ir/paper/1021444/en

Vancouver: Copy

HAIDARI A., SATTARI M., Barmalzan GH.. On Likelihood Ratio Ordering of Parallel Systems with Two Generalized Exponential Components. JOURNAL OF STATISTICAL SCIENCES[Internet]. 2022;16(1 ):109-126. Available from: https://sid.ir/paper/1021444/en

IEEE: Copy

A. HAIDARI, M. SATTARI, and GH. Barmalzan, “On Likelihood Ratio Ordering of Parallel Systems with Two Generalized Exponential Components,” JOURNAL OF STATISTICAL SCIENCES, vol. 16, no. 1 , pp. 109–126, 2022, [Online]. Available: https://sid.ir/paper/1021444/en

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