Abstract
To reduce total test time and increase the efficiency of statistical analysis of a life-testing experiment adaptive progressive Type-II censoring scheme has been proposed. This paper addresses the statistical inference of the unknown parameters, reliability, and hazard rate functions of logistic exponential distribution under adaptive progressive Type-II censored samples. Maximum likelihood estimates (MLEs) and maximum product spacing estimates (MPSEs) for the model parameters, reliability, and hazard rate functions can not be obtained explicitly, hence these are derived numerically using the Newton–Raphson method. Bayes estimates for the unknown parameters and reliability and hazard rate functions are computed under squared error loss function (SELF) and linear exponential loss function (LLF). It has been observed that the Bayes estimates are not in explicit forms, hence an approximation method such as Markov Chain Monte Carlo (MCMC) method is employed. Further, asymptotic confidence intervals (ACIs) and highest posterior density (HPD) credible intervals for the unknown parameters, reliability, and hazard rate functions are constructed. Besides, point and interval Bayesian predictions have been derived for future samples. A Monte Carlo simulation study has been carried out to compare the performance of the proposed estimates. Furthermore, three different optimality criteria have been considered to obtain the optimal censoring plan. Two real-life data sets, one from electronic industry and other one from COVID-19 data set containing the daily death rate from France are re-analyzed to demonstrate the proposed methodology.
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Abbreviations
- AB:
-
Average bias
- ACI:
-
Asymptotic confidence interval
- AL:
-
Average length
- APC-II:
-
Adaptive progressive Type-II censoring
- ALT:
-
Accelerated life testing
- BCI:
-
Bayesian credible interval
- BPI:
-
Bayesian predictive interval
- CDF:
-
Cumulative distribution function
- CP:
-
Coverage probability
- HPD:
-
Highest posterior density
- K–S:
-
Kolmogorov–Smirnov
- LE:
-
Logistic exponential
- LF:
-
Likelihood function
- LLF:
-
Linear exponential loss function
- MCMC:
-
Markov chain Monte Carlo
- MLE:
-
Maximum likelihood estimation
- MPS:
-
Maximum product spacing
- MPSE:
-
Maximum product spacing estimation
- MH:
-
Metropolis–Hastings
- MSE:
-
Mean squared error
- N–R:
-
Newton–Raphson
- PDF:
-
Probability density function
- P–P:
-
Probability–probability
- PSF:
-
Product spacing function
- Q–Q:
-
Quantile–quantile
- SELF:
-
Squared error loss function
- VCM:
-
Variance–covariance matrix
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Acknowledgements
The authors would like to thank the Editor in Chief, an Associate Editor and two anonymous reviewers for their positive remarks and useful comments. The author S. Dutta, thanks the Council of Scientific and Industrial Research (C.S.I.R. Grant No. 09/983(0038)/2019-EMR-I), India, for the financial assistantship received to carry out this research work. The authors thank the research facilities received from the Department of Mathematics, National Institute of Technology Rourkela, India.
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Appendix 1
Appendix 1
1.1 Proof of Theorem 2.1
Equation (2.3) can be written as
We will show that for \((\alpha ,\lambda ) \in (0,\infty ) \times (0,\infty )\), the maximum of \(l(\alpha ,\lambda )\) exists and it is unique.
From Sect. 2.3, it has been observed that \(\frac{\partial ^2l}{\partial \alpha ^2}<0\) and \(\frac{\partial ^2l}{\partial \lambda ^2}<0\). Hence, for fixed \(\lambda (\alpha )\), \(l(\alpha ,\lambda |{\bf x})\) is strictly concave function of \(\alpha (\lambda )\). Moreover, for fixed \(\lambda\),
and for fixed \(\alpha\),
Therefore, for fixed \(\lambda (\alpha )\), \(l(\alpha ,\lambda |{\bf x})\) is unimodal function with respect to \(\alpha (\lambda )\). Furthermore,
Now, the rest of the proof follows from that of Theorem 1 of Dey et al. (2016) (see pp. 453–54). Thus, it is omitted. \(\square\)
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Dutta, S., Dey, S. & Kayal, S. Bayesian survival analysis of logistic exponential distribution for adaptive progressive Type-II censored data. Comput Stat 39, 2109–2155 (2024). https://doi.org/10.1007/s00180-023-01376-y
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DOI: https://doi.org/10.1007/s00180-023-01376-y