**4. Parameters Estimation**

*4.1. Maximum Likelihood Estimation (MLE) Based on Complete Samples.*

In this section, we derive the MLE of the unknown parameters *λ*, *β*, and *α* of the OEHLIEx model based on complete samples. Consider a random sample *X*1, *X*2, ... , *Xn* from the OEHLIEx model; then, the log-likelihood (*LL*) function can be expressed as

$$L = \sum\_{i=1}^{n} \log f(x\_i; \lambda\_\prime \beta\_\prime \alpha). \tag{17}$$

By substituting from Equation (3) into Equation (17), the MLEs of the OEHLIEx parameters can be obtained by maximizing

$$\begin{split} LL &= n \log 2a\lambda \beta - 2\sum\_{i=1}^{n} \log x\_i - \beta \sum\_{i=1}^{n} \frac{1}{x\_i} - \lambda \sum\_{i=1}^{n} \left(e^{\frac{\beta}{x\_i}} - 1\right)^{-1} - 2\sum\_{i=1}^{n} \log \left(1 - e^{-\frac{\beta}{x\_i}}\right) + \\ & (a-1) \sum\_{i=1}^{n} \log \left(1 - e^{-\lambda(e^{\frac{\beta}{\Gamma}} - 1)}\right) - (a+1) \sum\_{i=1}^{n} \log \left(1 + e^{-\lambda(e^{\frac{\beta}{\Gamma}} - 1)}\right), \end{split} \tag{18}$$

with respect to *λ*, *β*, and *α*. We used **R** software to obtain the parameters' values. The (1 − *δ*) 100% confidence intervals (CIs) of the model parameters can be calculated using the following relations:

$$
\hat{\lambda} \pm Z\_{\frac{\delta}{2}} \sqrt{var(\hat{\lambda})}, \ \hat{\beta} \pm Z\_{\frac{\delta}{2}} \sqrt{var(\hat{\beta})} \text{ and } \hat{\kappa} \pm Z\_{\frac{\delta}{2}} \sqrt{var(\hat{\kappa})}.
$$

where *Zδ* 2is the upper *<sup>δ</sup>*2*th* percentile of the standard normal distribution.

### *4.2. MLE Based on Type-II Censored Samples*

The likelihood function for a type-II censored sample can be reported as

$$I = \frac{n!}{(n-k)!} (R(\mathbf{x}\_k))^{n-k} \prod\_{i=1}^k f(\mathbf{x}\_i)\_{\prime} \tag{19}$$

where *n* represents the number of components, and the experiment is stopped when *k* items failed. If *X*1, *X*2, ... , *Xn* represent an independent and identically distributed random sample from the OEHLIEx distribution and *X*1, *X*2, ... , *Xk, k* ≤ *n* represent an ordered sample obtained from a type-II right censoring sample, then the log-likelihood (*LL*∗) function is

$$LL^\* = \ln \frac{n!}{(n-k)!} + (n-k)\ln R(\mathbf{x}\_k; \boldsymbol{\lambda}, \boldsymbol{\beta}, \boldsymbol{\alpha}) + \sum\_{i=1}^k \log f(\mathbf{x}\_i; \boldsymbol{\lambda}, \boldsymbol{\beta}, \boldsymbol{\alpha}). \tag{20}$$

By substituting from Equations (3) and (5) into Equation (20), the MLEs of the OEHLIEx parameters can be obtained by maximizing results of the equation with respect to *λ*, *β*, and *α*.
