Bayesian Estimation of the Stress–Strength Parameter for Bivariate Normal Distribution Under an Updated Type-II Hybrid Censoring
To save time and cost for a parameter inference, the type-II hybrid censoring scheme has been broadly applied to collect one-component samples. In the current study, one of the essential parameters for comparing two distributions, that is, the stress–strength probability <inline-formula><ma...
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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-02-01
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| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/5/792 |
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| Summary: | To save time and cost for a parameter inference, the type-II hybrid censoring scheme has been broadly applied to collect one-component samples. In the current study, one of the essential parameters for comparing two distributions, that is, the stress–strength probability <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>δ</mi><mo>=</mo><mo form="prefix">Pr</mo><mo>(</mo><mi>X</mi><mo><</mo><mi>Y</mi><mo>)</mo></mrow></semantics></math></inline-formula>, is investigated under a new proposed type-II hybrid censoring scheme that generates the type-II hybrid censored two-component sample from the bivariate normal distribution. The difficult issues occurred from extending the one-component type-II hybrid censored sample to a two-component type-II hybrid censored sample are keeping useful information from both components and the establishment of the corresponding likelihood function. To conquer these two drawbacks, the proposed type-II hybrid censoring scheme is addressed as follows. The observed values of the first component, X, of data pairs <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></semantics></math></inline-formula> are recorded up to a random time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>=</mo><mo movablelimits="true" form="prefix">max</mo><mo>{</mo><msub><mi>X</mi><mrow><mi>r</mi><mo>:</mo><mi>n</mi></mrow></msub><mo>,</mo><mi>T</mi><mo>}</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mrow><mi>r</mi><mo>:</mo><mi>n</mi></mrow></msub></semantics></math></inline-formula> is the rth ordered statistic among n items with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo><</mo><mi>n</mi></mrow></semantics></math></inline-formula> as two pre-specified positive integers and T is a pre-determined experimental time. The observed value from the other component variable Y is recorded only if it is the counterpart of X and also observed before time <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>; otherwise, it is denoted as occurred or not at <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>τ</mi></semantics></math></inline-formula>. Under the new proposed scheme, the likelihood function of the new bivariate censored data is derived to include the factors of double improper integrals to cover all possible cases without the loss of data information where any component is unobserved. A Monte Carlo Markov chain (MCMC) method is applied to find the Bayesian estimate of the bivariate distribution model parameters and the stress–strength probability, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>δ</mi></semantics></math></inline-formula>. An extensive simulation study is conducted to demonstrate the performance of the developed methods. Finally, the proposed methodologies are applied to a type-II hybrid censored sample generated from a bivariate normal distribution. |
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| ISSN: | 2227-7390 |