Bayesian stress-strength inference for the Moran-Downton bivariate exponential distribution under modified Type-Ⅱ censoring

Yu-Jau Lin; Tzong-Ru Tsai; Yuhlong Lio; Hon Keung Tony Ng; Liang Wang; Yu-Jau Lin; Tzong-Ru Tsai; Yuhlong Lio; Hon Keung Tony Ng; Liang Wang

doi:10.3934/math.20251271

AIMS Mathematics

2025, Volume 10, Issue 12: 28878-28907. doi: 10.3934/math.20251271

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Bayesian stress-strength inference for the Moran-Downton bivariate exponential distribution under modified Type-Ⅱ censoring

1.
Department of Applied Mathematics, Chung Yuan Christian University, Zhongli District, Taoyuan City 320314, Taiwan
2.
Department of Statistics and Data Science, Tamkang University, Tamsui District, New Taipei City 251301, Taiwan
3.
Department of Mathematical Sciences, University of South Dakota, Vermillion, SD 57069, USA
4.
Department of Mathematical Sciences, Bentley University, Waltham, MA 02452, USA
5.
School of Mathematics, Yunnan Normal University, Kunming, 650092, China

Received: 26 August 2025 Revised: 23 November 2025 Accepted: 25 November 2025 Published: 10 December 2025
MSC : 62F10, 62N05, 62P30

The stress-strength reliability $ \delta = \Pr(X < Y) $, where $ (X, Y) $ is Moran-Downton bivariate exponential distributed, is investigated under a modified bivariate Type-Ⅱ censoring scheme. Likelihood contributions for the four censoring configurations are derived via series representations of the modified Bessel function and evaluated with finite partial-sum approximations that admit geometric error bounds, enabling direct likelihood computation. Bayesian inference is developed using two Markov chain Monte Carlo strategies: (ⅰ) Metropolis-Hastings within Gibbs sampler using the likelihood based on censored data; and (ⅱ) a data-augmentation scheme that imputes censored observations. Extensive simulations across different censoring proportions and parameter settings show a decreasing mean squared error with increasing sample size and a slightly superior correlation estimate between $ X $ and $ Y $ for the approach using a likelihood based on censored data, while both methods perform comparably for $ \delta $. Finally, the methodology is illustrated with a simulated numerical example and a real bivariate dataset, highlighting implementation details and the benefits of the proposed censoring design for estimating the model parameters and stress-strength reliability.
- dependence,
- incomplete data,
- maximum likelihood,
- reliability
Citation: Yu-Jau Lin, Tzong-Ru Tsai, Yuhlong Lio, Hon Keung Tony Ng, Liang Wang. Bayesian stress-strength inference for the Moran-Downton bivariate exponential distribution under modified Type-Ⅱ censoring[J]. AIMS Mathematics, 2025, 10(12): 28878-28907. doi: 10.3934/math.20251271

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Abstract

The stress-strength reliability $ \delta = \Pr(X < Y) $, where $ (X, Y) $ is Moran-Downton bivariate exponential distributed, is investigated under a modified bivariate Type-Ⅱ censoring scheme. Likelihood contributions for the four censoring configurations are derived via series representations of the modified Bessel function and evaluated with finite partial-sum approximations that admit geometric error bounds, enabling direct likelihood computation. Bayesian inference is developed using two Markov chain Monte Carlo strategies: (ⅰ) Metropolis-Hastings within Gibbs sampler using the likelihood based on censored data; and (ⅱ) a data-augmentation scheme that imputes censored observations. Extensive simulations across different censoring proportions and parameter settings show a decreasing mean squared error with increasing sample size and a slightly superior correlation estimate between $ X $ and $ Y $ for the approach using a likelihood based on censored data, while both methods perform comparably for $ \delta $. Finally, the methodology is illustrated with a simulated numerical example and a real bivariate dataset, highlighting implementation details and the benefits of the proposed censoring design for estimating the model parameters and stress-strength reliability.

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