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2023, Volume 19, Issue 8: 6286-6301. Doi: 10.3934/jimo.2022216

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Stochastic comparison of the second-largest order statistics from heterogeneous models

Sangita Das^1,2, and
Suchandan Kayal^2, ,

1.
Department of Mathematics, SRM University - AP, Guntur-522240, India
2.
Department of Mathematics, NIT Rourkela, Rourkela-769008, India

^*Corresponding author: Suchandan Kayal
^*Corresponding author: Suchandan Kayal

Received: December 2021

Revised: October 2022

Early access: November 2022

Published: August 2023

S. Das is supported by Ministry of Education (formerly MHRD), Govt. of India and S. Kayal is supported by SERB, India (Grant no. MTR/2018/000350)

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Abstract

The second-largest order statistic is of special importance in reliability theory since it represents the time to failure of a $ 2 $-out-of-$ n $ system. Consider two $ 2 $-out-of-$ n $ systems with heterogeneous random component lifetimes, following a general family of exponentiated location-scale distributions. In this article, usual stochastic order between the lifetimes of systems has been established when the components are interdependent. For the independent heterogeneous distributions, sufficient conditions under which reversed hazard rate order between the second-largest order statistics holds are investigated. To illustrate theoretical findings, some examples are considered.

Keywords:

Mathematics Subject Classification: Primary: 60E15; Secondary: 90B25.

Citation:

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Figure 1. (a) Plots of $ {F}_{X_{2:3}}(x) $ and $ {F}_{Y_{2:3}}(x) $ as in Example 3.1(i). (b) Plots of $ {F}_{X_{2:3}}(x) $ and $ {F}_{Y_{2:3}}(x) $ as in Example 3.1(ii)

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Figure 2. (a) Plots of $ {F}_{X_{2:3}}(x) $ and $ {F}_{Y_{2:3}}(x) $ as in Counterexample 3.1. (b) Plots of $ {F}_{X_{2:3}}(x) $ and $ {F}_{Y_{2:3}}(x) $ as in Counterexample 3.2

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Figure 3. (a) Plots of $ {F}_{X_{2:3}}(x) $ and $ {F}_{Y_{2:3}}(x) $ as in Example 3.2(i). (b) Plots of $ {F}_{X_{2:3}}(x) $ and $ {F}_{Y_{2:3}}(x) $ as in Example 3.2(ii)

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Figure 4. (a) Plot of the difference $ {F}_{X_{2:3}}(x)-{F}_{Y_{2:3}}(x) $ as in Counterexample 3.3. (b) Plot of $ \tilde{r}_{X_{2:3}}(x)-\tilde{r}_{Y_{2:3}}(x) $ as in Example 3.3

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Figure 5. (a) Plot of $ \frac{\tilde{r}_{Y_{2:3}}(x)}{\tilde{r}_{X_{2:3}}(x)} $ as in Counterexample 3.4. (b) Plot of $ \frac{f_{Y_{2:3}}(x)}{f_{X_{2:3}}(x)} $ as in Counterexample 3.4

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