Bayesian survival analysis of logistic exponential distribution for adaptive progressive Type-II censored data

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.

Appendix 1

1.1 Proof of Theorem 2.1

Equation (2.3) can be written as

$$\begin{aligned} l(\alpha , \lambda |\textbf{x})&= m \log \alpha + m \log \lambda + \lambda \sum _{i=1}^{m} x_i + (\alpha -1) \sum _{i=1}^{m} \log (e^{\lambda x_i}-1) \\&\quad -2 \sum _{i=1}^{m} \log \bigg [1+ (e^{\lambda x_i}-1)^{\alpha }\bigg ] - \sum _{i=1}^{J} R_i \log \bigg [1+ (e^{\lambda x_i}-1)^{\alpha }\bigg ] \\& \quad - R_m^* \log \bigg [1+ (e^{\lambda x_m}-1)^{\alpha }\bigg ]. \end{aligned}$$

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 $\lambda$,

$$\begin{aligned} \lim \limits _{\alpha \rightarrow 0} l(\alpha ,\lambda |{\bf x})= -\infty \quad \text{ and } \quad \lim \limits _{\alpha \rightarrow \infty } l(\alpha ,\lambda |{\bf x})= -\infty , \end{aligned}$$

and for fixed $\alpha$,

$$\begin{aligned} \lim \limits _{\lambda \rightarrow 0} l(\alpha ,\lambda |{\bf x})= -\infty \quad \text{ and } \quad \lim \limits _{\lambda \rightarrow \infty } l(\alpha ,\lambda |{\bf x})= -\infty . \end{aligned}$$

Therefore, for fixed $\lambda (\alpha )$, $l(\alpha ,\lambda |{\bf x})$ is unimodal function with respect to $\alpha (\lambda )$. Furthermore,

$$\begin{aligned} \lim \limits _{\alpha \rightarrow 0,\lambda \rightarrow 0} l(\alpha ,\lambda |{\bf x})&= -\infty ,~ \lim \limits _{\alpha \rightarrow \infty ,\lambda \rightarrow 0} l(\alpha ,\lambda |{\bf x})= -\infty ,\\ \lim \limits _{\alpha \rightarrow 0,\lambda \rightarrow \infty } l(\alpha ,\lambda |{\bf x})&= -\infty ,~ \lim \limits _{\alpha \rightarrow \infty ,\lambda \rightarrow \infty } l(\alpha ,\lambda |{\bf x})= -\infty . \end{aligned}$$

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. $\square$