Estimation of Parameters of Life for an Inverted Nadarajah–Haghighi Distribution from Type-II Progressively Censored Samples

Abstract

In this paper, the estimating problems of the model parameters, reliability and hazard functions of an inverted Nadarajah–Haghighi distribution when sample is available from Type-II progressive censoring scheme have been considered. The maximum likelihood and maximum product of spacings estimators have been obtained for any function of the model parameters. The normality property of the classical estimators is used to construct the approximate confidence intervals for the unknown parameters and some related functions of them such as the reliability characteristics. Using independent gamma informative priors, the Bayes estimators of the unknown parameters are derived under squared error loss function. Since the Bayes estimators are obtained in a complex form, so we have been used two approximation techniques, namely: Tierney–Kadane approximation method and Metropolis–Hastings algorithm to carry out the Bayes estimates and also to construct the associate highest posterior density credible intervals. To evaluate the performance of the proposed methods, a Monte Carlo simulation study is carried out. To suggest the optimum censoring scheme among different competing censoring plans, four optimality criteria have been considered. One real-life data set is analyzed to discuss how the applicability of the proposed methods in real phenomenon.

Appendices

Appendix 1: Fisher’s Elements of the MLEs

Differentiating (8) with respect to $\alpha$ and $\lambda$, hence, the elements of the observed Fisher information matrix (24) are becomes

$$\begin{aligned} {\mathcal {L}}_{\alpha \alpha }&=\frac{{\partial ^2 \ell }}{{\partial \alpha ^2 }} = - \frac{m}{{\alpha ^2 }} - \mathop \sum \nolimits _{i = 1}^m {\psi ^{''}_\alpha \left( {x_i ;\alpha ,\lambda } \right) } + \mathop \sum \nolimits _{i = 1}^m {R_i \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) } \\&\times \left[ {1 - \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) } \right] ^{ - 2} \left\{ {\left[ {1 - \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) } \right] } \right. \\&\left. \times \left\{ {\psi ^{''}_\alpha \left( {x_i ;\alpha ,\lambda } \right) - \left( {\psi ^{'}_\alpha \left( {x_i ;\alpha ,\lambda } \right) } \right) ^2 } \right\} - \left( {\psi ^{'}_\alpha \left( {x_i ;\alpha ,\lambda } \right) } \right) ^2 \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) \right\} , \\ {\mathcal {L}}_{\lambda \lambda }&=\frac{{\partial ^2 \ell }}{{\partial \lambda ^2 }} = - \frac{m}{{\lambda ^2 }} - (\alpha - 1)\mathop \sum \nolimits _{i = 1}^m {x_i^{ - 2} \left( {1 + \lambda x_i^{ - 1} } \right) ^{ - 2} } - \mathop \sum \nolimits _{i = 1}^m {\psi ^{''}_\lambda \left( {x_i ;\alpha ,\lambda } \right) } \\&+ \mathop \sum \nolimits _{i = 1}^m {R_i \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) \left[ {1 - \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) } \right] ^{ - 2} } \\&\times \left\{ {\left[ {1 - \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) } \right] \left[ {\psi ^{''}_\lambda \left( {x_i ;\alpha ,\lambda } \right) - \left( {\psi ^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) } \right) ^2 } \right] } \right. \\&\left. - \left( {\psi ^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) } \right) ^2 \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) \right\} ,\\ {\mathcal {L}}_{\alpha \lambda }&=\frac{{\partial ^2 \ell }}{{\partial \alpha \partial \lambda }} = \mathop \sum \nolimits _{i = 1}^m {x_i^{ - 1} \left( {1 + \lambda x_i^{ - 1} } \right) ^{ - 1} } - \mathop \sum \nolimits _{i = 1}^m {\psi ^{''}_{\alpha \lambda } \left( {x_i ;\alpha ,\lambda } \right) } \\&+ \mathop \sum \nolimits _{i = 1}^m {R_i } \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) \left[ {1 - \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) } \right] ^{ - 2} \\&\times \left\{ {\left[ {1 - \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) } \right] \left[ {\psi ^{''}_{\alpha \lambda } \left( {x_i ;\alpha ,\lambda } \right) - \psi ^{'}_\alpha \left( {x_i ;\alpha ,\lambda } \right) \psi ^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) } \right] } \right. \\&\left. - \psi ^{'}_\alpha \left( {x_i ;\alpha ,\lambda } \right) \psi ^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) \right\} , \end{aligned}$$

where $\psi ^{''}_\theta ( \cdot )$ is the second-partial derivative with respect to the parameter $\theta$ such as

$$\begin{aligned} \psi ^{''}_\alpha \left( {x_i ;\alpha ,\lambda } \right)&= \psi \left( {x_i ;\alpha ,\lambda } \right) \left[ {\log \left( {1 + \lambda x_i^{ - 1} } \right) } \right] ^2 ,\mathrm{{ }}i = 1,2,\dots ,m,\\ \psi ^{''}_\lambda \left( {x_i ;\alpha ,\lambda } \right)&= \alpha \left( {\alpha - 1} \right) x_i^{ - 2} \left( {1 + \lambda x_i^{ - 1} } \right) ^{\alpha - 2} ,\mathrm{{ }}i = 1,2,\dots ,m,\\ \end{aligned}$$

and

$$\begin{aligned} \psi ^{''}_{\alpha \lambda } \left( {x_i ;\alpha ,\lambda } \right) = \psi ^{''}_{\lambda \alpha } \left( {x_i ;\alpha ,\lambda } \right) = \psi ^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) \left[ {\alpha ^{ - 1} + \log \left( {1 + \lambda x_i^{ - 1} } \right) } \right] ,\mathrm{{ }}i = 1,2,\dots ,m. \end{aligned}$$

Appendix 2: Fisher’s Elements of the MPSEs

Differentiating (12) with respect to $\alpha$ and $\lambda$, hence, the elements of the observed Fisher information matrix (25) are becomes

$$\begin{aligned} H^{''}_\alpha&= \frac{{\partial ^2 H}}{{\partial \alpha ^2 }} = \sum \nolimits _{i = 1}^{m + 1} \{ {[ {F^{''}_\alpha \left( {x_{i - 1} ;\alpha ,\lambda } \right) - \left( {F^{'}_\alpha \left( {x_{i - 1} ;\alpha ,\lambda } \right) } \right) ^2 } ]F\left( {x_{i - 1} ;\alpha ,\lambda } \right) } \\&- [ {F^{''}_\alpha \left( {x_i ;\alpha ,\lambda } \right) - \left( {F^{'}_\alpha \left( {x_i ;\alpha ,\lambda } \right) } \right) ^2 } ]F\left( {x_i ;\alpha ,\lambda } \right) \} \\&\times \left( {F\left( {x_i ;\alpha ,\lambda } \right) - F\left( {x_{i - 1} ;\alpha ,\lambda } \right) } \right) ^{ - 1} - \left( {F\left( {x_i ;\alpha ,\lambda } \right) - F\left( {x_{i - 1} ;\alpha ,\lambda } \right) } \right) ^{ - 2} \\&\times \left[ {F^{'}_\alpha \left( {x_{i - 1} ;\alpha ,\lambda } \right) F\left( {x_{i - 1} ;\alpha ,\lambda } \right) - F^{'}_\alpha \left( {x_i ;\alpha ,\lambda } \right) F\left( {x_i ;\alpha ,\lambda } \right) } \right] ^2 , \\ H^{''}_\lambda&= \frac{{\partial ^2 H}}{{\partial \lambda ^2 }} = \sum \nolimits _{i = 1}^{m + 1} \{ {[ {F^{''}_\lambda \left( {x_{i - 1} ;\alpha ,\lambda } \right) - \left( {F^{'}_\lambda \left( {x_{i - 1} ;\alpha ,\lambda } \right) } \right) ^2 } ]F\left( {x_{i - 1} ;\alpha ,\lambda } \right) } \\&- [ {F^{''}_\lambda \left( {x_i ;\alpha ,\lambda } \right) - \left( {F^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) } \right) ^2 } ]F\left( {x_i ;\alpha ,\lambda } \right) \}\\&\times \left( {F\left( {x_i ;\alpha ,\lambda } \right) - F\left( {x_{i - 1} ;\alpha ,\lambda } \right) } \right) ^{ - 1} - \left( {F\left( {x_i ;\alpha ,\lambda } \right) - F\left( {x_{i - 1} ;\alpha ,\lambda } \right) } \right) ^{ - 2} \\&\times \left[ {F^{'}_\lambda \left( {x_{i - 1} ;\alpha ,\lambda } \right) F\left( {x_{i - 1} ;\alpha ,\lambda } \right) - F^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) F\left( {x_i ;\alpha ,\lambda } \right) } \right] ^2 , \\ H^{''}_{\alpha \lambda }&= \frac{{\partial ^2 H}}{{\partial \alpha \partial \lambda }} = \sum \nolimits _{i = 1}^{m + 1} \{ {[ {F^{''}_{\alpha \lambda } \left( {x_{i - 1} ;\alpha ,\lambda } \right) - F^{'}_\alpha \left( {x_{i - 1} ;\alpha ,\lambda } \right) F^{'}_\lambda \left( {x_{i - 1} ;\alpha ,\lambda } \right) } ]F\left( {x_{i - 1} ;\alpha ,\lambda } \right) } \\&- [ {F^{''}_{\alpha \lambda } \left( {x_i ;\alpha ,\lambda } \right) - F^{'}_\alpha \left( {x_i ;\alpha ,\lambda } \right) F^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) }]F\left( {x_i ;\alpha ,\lambda } \right) \} \\&\times \left( {F\left( {x_i ;\alpha ,\lambda } \right) - F\left( {x_{i - 1} ;\alpha ,\lambda } \right) } \right) ^{ - 1} - \left( {F\left( {x_i ;\alpha ,\lambda } \right) - F\left( {x_{i - 1} ;\alpha ,\lambda } \right) } \right) ^{ - 2} \\&\times \left\{ {F^{'}_\alpha \left( {x_{i - 1} ;\alpha ,\lambda } \right) F\left( {x_{i - 1} ;\alpha ,\lambda } \right) - F^{'}_\alpha \left( {x_i ;\alpha ,\lambda } \right) F\left( {x_i ;\alpha ,\lambda } \right) } \right\} \\&\times \left\{ {F^{'}_\lambda \left( {x_{i - 1} ;\alpha ,\lambda } \right) F\left( {x_{i - 1} ;\alpha ,\lambda } \right) - F^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) F\left( {x_i ;\alpha ,\lambda } \right) } \right\} , \\ M^{''}_\alpha&= \frac{{\partial ^2 M}}{{\partial \alpha ^2 }} = \sum \nolimits _{i = 1}^m {R_i \{ [{F^{''}_\alpha \left( {x_i ;\alpha ,\lambda } \right) - \left( {F^{'}_\alpha \left( {x_i ;\alpha ,\lambda } \right) } \right) ^2 }]F\left( {x_i ;\alpha ,\lambda } \right) \left( {1 - F\left( {x_i ;\alpha ,\lambda } \right) } \right) ^{ - 1} } \\&- [ {F^{'}_\alpha \left( {x_i ;\alpha ,\lambda } \right) F\left( {x_i ;\alpha ,\lambda } \right) } ]^2 \left( {1 - F\left( {x_i ;\alpha ,\lambda } \right) } \right) ^{ - 2} \}, \\ M^{''}_\lambda&= \frac{{\partial ^2 M}}{{\partial \lambda ^2 }} = \sum \nolimits _{i = 1}^m {R_i } \{ [ {F^{''}_\lambda \left( {x_i ;\alpha ,\lambda } \right) - \left( {F^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) } \right) ^2 } ]F\left( {x_i ;\alpha ,\lambda } \right) \left( {1 - F\left( {x_i ;\alpha ,\lambda } \right) } \right) ^{ - 1} \\&- [ {F^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) F\left( {x_i ;\alpha ,\lambda } \right) } ]^2 \left( {1 - F\left( {x_i ;\alpha ,\lambda } \right) } \right) ^{ - 2} \}, \\ M^{''}_{\alpha \lambda }&= \frac{{\partial ^2 M}}{{\partial \alpha \partial \lambda }} = \sum \nolimits _{i = 1}^m {R_i \{ [ {F^{''}_{\alpha \lambda } \left( {x_i ;\alpha ,\lambda } \right) - F^{'}_\alpha \left( {x_i ;\alpha ,\lambda } \right) F^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) } ]F\left( {x_i ;\alpha ,\lambda } \right) \left( {1 - F\left( {x_i ;\alpha ,\lambda } \right) } \right) ^{ - 1} } \\&- F^{'}_\lambda \left( {x_i ;\alpha ,\lambda } \right) F^{'}_\alpha \left( {x_i ;\alpha ,\lambda } \right) \left( {F\left( {x_i ;\alpha ,\lambda } \right) } \right) ^2 \left( {1 - F\left( {x_i ;\alpha ,\lambda } \right) } \right) ^{ - 2} \}, \end{aligned}$$

where

$$\begin{aligned} F\left( {x_i ;\alpha ,\lambda } \right)&= \exp \left( {1 - \psi \left( {x_i ;\alpha ,\lambda } \right) } \right) ;\\ F\left( {x_{i - 1} ;\alpha ,\lambda } \right)&= \exp \left( {1 - \psi \left( {x_{i - 1} ;\alpha ,\lambda } \right) } \right) ,\\ F^{''}_\alpha \left( {x_i ;\alpha ,\lambda } \right)&= \left( {\log \left( {1 + \lambda x_i^{ - 1} } \right) } \right) ^2 \psi \left( {x_i ;\alpha ,\lambda } \right) ,\\ F^{''}_\alpha \left( {x_{i - 1} ;\alpha ,\lambda } \right)&= \left( {\log \left( {1 + \lambda x_{i - 1}^{ - 1} } \right) } \right) ^2 \psi \left( {x_{i - 1} ;\alpha ,\lambda } \right) ,\\ F^{''}_\lambda \left( {x_i ;\alpha ,\lambda } \right)&= \alpha \left( {\alpha - 1} \right) x_i^{ - 2} \left( {1 + \lambda x_i^{ - 1} } \right) ^{\alpha - 2} ,\\ F^{''}_\lambda \left( {x_{i - 1} ;\alpha ,\lambda } \right)&= \alpha \left( {\alpha - 1} \right) x_{i - 1}^{ - 2} \left( {1 + \lambda x_{i - 1}^{ - 1} } \right) ^{\alpha - 2} , \end{aligned}$$

and

$$\begin{aligned} F^{''}_{\alpha \lambda } \left( {x_i ;\alpha ,\lambda } \right) = x_i^{ - 1} \left( {1 + \lambda x_i^{ - 1} } \right) ^{ - 1} \psi \left( {x_i ;\alpha ,\lambda } \right) \left[ {\alpha \log \left( {1 + \lambda x_i^{ - 1} } \right) + 1} \right] . \end{aligned}$$