Estimation of $ P(X &lt; Y) $ for Fréchet, reversed Weibull and Weibull distributions: Analytical expressions, simulations and applications

Tiago A. da Fonseca; Felipe S. Quintino; Luan C. S. M. Ozelim; Pushpa N. Rathie; Tiago A. da Fonseca; Felipe S. Quintino; Luan C. S. M. Ozelim; Pushpa N. Rathie

doi:10.3934/nhm.2024061

Networks and Heterogeneous Media

2024, Volume 19, Issue 4: 1424-1447. doi: 10.3934/nhm.2024061

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Estimation of $ P(X < Y) $ for Fréchet, reversed Weibull and Weibull distributions: Analytical expressions, simulations and applications

1.
Gama Engineering College, University of Brasília, Brasília 72.444-240, Brazil
2.
Department of Statistics, University of Brasília, Brasília 70.910-900, Brazil
3.
Department of Civil and Environmental Engineering, University of Brasília, Brasília 70.910-900, Brazil

Received: 22 August 2024 Revised: 29 October 2024 Accepted: 04 November 2024 Published: 12 December 2024

This work aimed to derive new analytical formulas for the stress–strength reliability of the type $ P(X < Y) $ when both $ X $ and $ Y $ follow Fréchet, reversed Weibull or Weibull distributions. The new expressions were given in terms of extreme value $ \mathbb{H} $-functions and have been obtained under fewer parameter restrictions while compared to similar results in the literature of these distributions. The performance of the maximum likelihood estimator was evaluated through Monte-Carlo simulations and the results were compared with a nonparametric estimator. Three real dataset applications were carried out. First, we analyzed the statistical behavior of financial assets' returns, showing how $ P(X < Y) $ can be used to build an interesting approach to perform asset selection. Second, minimum monthly flows of water were analyzed. Finally, we compared failure voltage levels of two types of electrical cable insulation. For all the real case applications, confidence intervals for $ P(X < Y) $ were obtained by Bootstrap methods.
- stress–strength reliability,
- extreme value $ \mathbb{H} $-function,
- extreme distributions,
- Monte Carlo simulations,
- statistical modeling
Citation: Tiago A. da Fonseca, Felipe S. Quintino, Luan C. S. M. Ozelim, Pushpa N. Rathie. Estimation of $ P(X < Y) $ for Fréchet, reversed Weibull and Weibull distributions: Analytical expressions, simulations and applications[J]. Networks and Heterogeneous Media, 2024, 19(4): 1424-1447. doi: 10.3934/nhm.2024061

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Abstract

This work aimed to derive new analytical formulas for the stress–strength reliability of the type $ P(X < Y) $ when both $ X $ and $ Y $ follow Fréchet, reversed Weibull or Weibull distributions. The new expressions were given in terms of extreme value $ \mathbb{H} $-functions and have been obtained under fewer parameter restrictions while compared to similar results in the literature of these distributions. The performance of the maximum likelihood estimator was evaluated through Monte-Carlo simulations and the results were compared with a nonparametric estimator. Three real dataset applications were carried out. First, we analyzed the statistical behavior of financial assets' returns, showing how $ P(X < Y) $ can be used to build an interesting approach to perform asset selection. Second, minimum monthly flows of water were analyzed. Finally, we compared failure voltage levels of two types of electrical cable insulation. For all the real case applications, confidence intervals for $ P(X < Y) $ were obtained by Bootstrap methods.

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