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Statistical analysis using multiple censoring scheme under partially accelerated life tests for the power Lindley distribution

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Abstract

If the items are very reliable, assessing the lifetime of them in regular usage requires more time and money than in accelerated conditions. As a result, life testing studies benefit greatly from accelerated life testing (ALT) since it saves time and money. In ALT, the components are subjected to higher levels of stress than usual in order to detect early failures in a short period of time and so decrease the expenses associated with component testing while maintaining quality. In this paper, a problem-based constant stress partially accelerated life test for multiple censored data is investigated. Under normal and accelerated settings, failure data from test objects is supposed to follow the Power Lindley distribution. The maximum likelihood estimates (MLEs) are obtained by solving the likelihood equations numerically. The approximate confidence intervals for the parameters are calculated using a normal approximation to the asymptotic distribution of MLEs. A simulation study is then used to test and validate the estimators' performance.

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Authors and Affiliations

  1. School of Engineering and Technology, Maharishi University of Information Technology, Noida, India

    Intekhab Alam, Trapty Agarwal & Awakash Mishra

  2. Department of Basic Sciences, College of Science and Theoretical Studies, Saudi Electronic University, Dammam, Saudi Arabia

    Mustafa Kamal

  3. Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh, India

    Ahmadur Rahman

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  1. Intekhab Alam
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  2. Mustafa Kamal
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  3. Ahmadur Rahman
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  4. Trapty Agarwal
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  5. Awakash Mishra
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Correspondence to Intekhab Alam.

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Appendix

Appendix

$$\frac{{\partial }^{2}\mathit{ln}L}{\partial {\theta }^{2}}=\left[-\frac{2}{\theta }+\frac{1}{(\theta +1{)}^{2}}\right]{n}_{f}+{\sum\limits\limits }_{i=1}^{n}{\delta }_{i},1.c\frac{{{t}_{i}}^{\lambda }}{(\theta +1{)}^{2}\left(1+\frac{\theta }{\theta +1}{{t}_{i}}^{\lambda }\right)}\left[\frac{2}{\theta +1}-\frac{{{t}_{i}}^{\lambda }\left(\theta (\theta +1{)}^{-2}-(\theta +1{)}^{-1}\right)}{1+\frac{\theta }{\theta +1}{{t}_{i}}^{\lambda }}\right]+{\sum\limits\limits }_{i=1}^{n}{\delta }_{i,2,c}\frac{(\beta {x}_{i}{)}^{\lambda }}{(\theta +1{)}^{2}\left(1+\frac{\theta }{\theta +1}(\beta {x}_{i}{)}^{\lambda }\right)}\left[\frac{2}{\theta +1}-\frac{(\beta {x}_{i}{)}^{\lambda }\left(\theta (\theta +1{)}^{-2}-(\theta +1{)}^{-1}\right)}{1+\frac{\theta }{\theta +1}(\beta {x}_{i}{)}^{\lambda }}\right]$$
$$\frac{{\partial }^{2}\mathit{ln}L}{\partial {\lambda }^{2}}=-\frac{{n}_{f}}{{\lambda }^{2}}+{\sum\limits\limits }_{i=1}^{n}{\delta }_{i,1,f}\left[\frac{\lambda {{t}_{i}}^{\lambda -1}\left((1+{{t}_{i}}^{\lambda })-\lambda {{t}_{i}}^{2\lambda -1}\right)\mathit{ln}({t}_{i})}{(1+{{t}_{i}}^{\lambda }{)}^{2}}-\theta {{t}_{i}}^{2\lambda }{\mathit{ln}}^{2}({t}_{i})\right]+{\sum\limits\limits }_{i=1}^{n}{\delta }_{i},1.c\left[\frac{\theta \mathit{ln}({t}_{i})\left(1+\frac{\theta }{\theta +1}{{t}_{i}}^{\lambda }\right)\lambda {{t}_{i}}^{\lambda -1}-\frac{\lambda \theta {{t}_{i}}^{2\lambda -1}}{\theta +1}}{(\theta +1){\left(1+\frac{\theta }{\theta +1}{{t}_{i}}^{\lambda }\right)}^{2}}-\theta {{t}_{i}}^{2\lambda }{\mathit{ln}}^{2}({t}_{i})\right]+{\sum\limits\limits }_{i=1}^{n}{\delta }_{i,2,f}\left[\frac{\lambda (\beta {x}_{i}{)}^{\lambda -1}\left((1+(\beta {x}_{i}{)}^{\lambda })-\lambda (\beta {x}_{i}{)}^{2\lambda -1}\right)\mathit{ln}({t}_{i})}{(1+(\beta {x}_{i}{)}^{\lambda }{)}^{2}}-\theta {{t}_{i}}^{2\lambda }{\mathit{ln}}^{2}(\beta {x}_{i})\right]+{\sum\limits\limits }_{i=1}^{n}{\delta }_{i,2,c}\left[\frac{\theta \mathit{ln}(\beta {x}_{i})\left(1+\frac{\theta }{\theta +1}(\beta {x}_{i}{)}^{\lambda }\right)\lambda (\beta {x}_{i}{)}^{\lambda -1}-\frac{\lambda \theta (\beta {x}_{i}{)}^{2\lambda -1}}{\theta +1}}{(\theta +1){\left(1+\frac{\theta }{\theta +1}(\beta {x}_{i}{)}^{\lambda }\right)}^{2}}-\theta (\beta {x}_{i}{)}^{2\lambda }{\mathit{ln}}^{2}(\beta {x}_{i})\right]$$
$$\frac{{\partial }^{2}\mathit{ln}L}{\partial {\beta }^{2}}=\sum\limits_{i=1}^{n}{\delta }_{i,2,f}\left[-\left(\frac{\lambda -1}{{\beta }^{2}}\right)+\frac{\lambda {x}_{i}(\beta {x}_{i}{)}^{\lambda -1}}{1+(\beta {x}_{i}{)}^{\lambda }}\left(\frac{\lambda -1}{\beta }-\frac{\lambda {x}_{i}(\beta {x}_{i}{)}^{\lambda -1}}{1+(\beta {x}_{i}{)}^{\lambda }}\right)-{x}_{i}^{2}\theta \lambda (\lambda -1)(\beta {x}_{i}{)}^{\lambda -2}\right]$$
$$+\sum\limits_{i=1}^{n}{\delta }_{i,2,c}\left[\frac{\theta \lambda {x}_{i}(\beta {x}_{i}{)}^{\lambda -1}}{(\theta +1)\left(1+\frac{\theta }{\theta +1}(\beta {x}_{i}{)}^{\lambda }\right)}\left(\frac{\lambda -1}{\beta }-\frac{\frac{\theta }{\theta +1}\lambda {x}_{i}(\beta {x}_{i}{)}^{\lambda -1}}{1+\frac{\theta }{\theta +1}(\beta {x}_{i}{)}^{\lambda }}\right)-\theta {x}_{i}^{2}\lambda (\lambda -1)(\beta {x}_{i}{)}^{\lambda -2}\right]-\frac{{n}_{2f}}{{\beta }^{2}}$$
$$\frac{{\partial }^{2}lnL}{\partial \theta \partial \lambda }=-\left({t}_{i}^{\lambda -1}-(\beta {x}_{i}{)}^{\lambda -1}\right)\lambda {n}_{f}-\left[\sum\limits_{i=1}^{n}{\delta }_{i,1,f}{t}_{i}+\sum\limits_{i=1}^{n}{\delta }_{i,2,f}(\beta {x}_{i}{)}^{\lambda }\right]-\sum\limits_{i=1}^{n}{\delta }_{i},1.c\frac{{{t}_{i}}^{\lambda }}{(\theta +1{)}^{2}\left(1+\frac{\theta }{\theta +1}{{t}_{i}}^{\lambda }\right)}\left[\mathit{ln}({t}_{i})-\frac{\lambda {{t}_{i}}^{\lambda -1}\theta }{(\theta +1)\left(1+\frac{\theta }{\theta +1}{{t}_{i}}^{\lambda }\right)}\right]-\sum\limits_{i=1}^{n}{\delta }_{i,2,c}\frac{(\beta {x}_{i}{)}^{\lambda }}{(\theta +1{)}^{2}\left(1+\frac{\theta }{\theta +1}(\beta {x}_{i}{)}^{\lambda }\right)}\left[\mathit{ln}({t}_{i})-\frac{\lambda (\beta {x}_{i}{)}^{\lambda -1}\theta }{(\theta +1)\left(1+\frac{\theta }{\theta +1}(\beta {x}_{i}{)}^{\lambda }\right)}\right]$$
$$\frac{{\partial }^{2}\mathit{ln}L}{\partial \beta \partial \theta }=\sum\limits_{i=1}^{n}{\delta }_{i,2,f}\left[-{x}_{i}\lambda (\beta {x}_{i}{)}^{\lambda -1}\right]+\sum\limits_{i=1}^{n}{\delta }_{i,2,c}\left[\frac{\theta \lambda {x}_{i}(\beta {x}_{i}{)}^{\lambda -1}}{(\theta +1)\left(1+\frac{\theta }{\theta +1}(\beta {x}_{i}{)}^{\lambda }\right)}\left(\frac{1}{\theta }-\frac{1}{\theta +1}-\frac{(\beta {x}_{i}{)}^{\lambda }((\theta +1{)}^{-1}-\theta (\theta +1{)}^{-2})}{1+\frac{\theta }{\theta +1}(\beta {x}_{i}{)}^{\lambda }}\right)-\lambda {x}_{i}(\beta {x}_{i}{)}^{\lambda -1}\right]$$
$$\frac{{\partial }^{2}\mathit{ln}L}{\partial \beta \partial \lambda }=\sum\limits_{i=1}^{n}{\delta }_{i,2,f}\left[\begin{array}{c}\frac{1}{\beta }+\frac{\lambda {x}_{i}(\beta {x}_{i}{)}^{\lambda -1}}{1+(\beta {x}_{i}{)}^{\lambda }}\left(\frac{\lambda (\lambda -1)(\beta {x}_{i}{)}^{\lambda -2}+(\beta {x}_{i}{)}^{\lambda -1}}{\lambda (\beta {x}_{i}{)}^{\lambda -1}}-\frac{\lambda (\beta {x}_{i}{)}^{\lambda -1}}{1+(\beta {x}_{i}{)}^{\lambda }}\right)-\\ {x}_{i}\theta \left(\frac{1}{\lambda }+\mathit{ln}(\beta {x}_{i})\right)\lambda (\beta {x}_{i}{)}^{\lambda -1}\end{array}\right]+\sum\limits_{i=1}^{n}{\delta }_{i,2,c}\left[\begin{array}{c}\frac{\theta \lambda {x}_{i}(\beta {x}_{i}{)}^{\lambda -1}}{(\theta +1)\left(1+\frac{\theta }{\theta +1}(\beta {x}_{i}{)}^{\lambda }\right)}\left(\frac{\lambda (\lambda -1)(\beta {x}_{i}{)}^{\lambda -2}+(\beta {x}_{i}{)}^{\lambda -1}}{\lambda (\beta {x}_{i}{)}^{\lambda -1}}-\frac{\frac{\theta }{\theta +1}\lambda (\beta {x}_{i}{)}^{\lambda -1}}{1+\frac{\theta }{\theta +1}(\beta {x}_{i}{)}^{\lambda }}\right)\\ -\theta \lambda {x}_{i}(\beta {x}_{i}{)}^{\lambda -1}\left(\frac{1}{\lambda }+\mathit{ln}(\beta {x}_{i})\right)\end{array}\right]$$

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Alam, I., Kamal, M., Rahman, A. et al. Statistical analysis using multiple censoring scheme under partially accelerated life tests for the power Lindley distribution. Int J Syst Assur Eng Manag (2024). https://doi.org/10.1007/s13198-024-02350-7

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