**Appendix A**

In this Section we provided some additional details related to the computation of the manuscript.

*Appendix A.1. W's Quantities*

In order to compute *wi* = E exp *yi*−*μ<sup>i</sup> σ* , *i* = 1, ... , *n*, we first study the case type I censoring. Note that

$$\exp\left(\frac{y\_i - \mu\_i}{\sigma}\right) = \begin{cases} \exp\left(\frac{y\_i - \mu\_i}{\sigma}\right), & \text{if } y\_i \le \log L\_i \\\exp\left(\frac{\log L\_i - \mu\_i}{\sigma}\right), & \text{otherwise} \end{cases}$$

Therefore,

$$\begin{split} w\_{i} &= \int\_{-\infty}^{\log L\_{i}} \frac{1}{\sigma} \exp\left(\frac{2(y\_{i} - \mu\_{i})}{\sigma} - \exp\left(\frac{y\_{i} - \mu\_{i}}{\sigma}\right)\right) dy\_{i} + \exp\left(\frac{\log L\_{i} - \mu\_{i}}{\sigma}\right) \mathbb{P}(T\_{i} > L\_{i}) \\ &= 1 - \exp\left(-L\_{i}^{1/\sigma} \varepsilon^{-\mu\_{i}/\sigma}\right) \left(1 + L\_{i}^{1/\sigma} \varepsilon^{-\mu\_{i}/\sigma}\right) + L\_{i}^{1/\sigma} \varepsilon^{-\mu\_{i}/\sigma} \exp\left(-L\_{i}^{1/\sigma} \varepsilon^{-\mu\_{i}/\sigma}\right) \\ &= 1 - \exp\left(-L\_{i}^{1/\sigma} \varepsilon^{-\mu\_{i}/\sigma}\right). \end{split}$$

Direct computation also shows that

$$\begin{split} v\_{i} &= \mathbb{E}\left[\exp\left(\frac{2(y\_{i}-\mu\_{i})}{\sigma}\right)\right] \\ &= \int\_{-\infty}^{\log L\_{i}} \frac{1}{\sigma} \exp\left(\frac{3(y\_{i}-\mu\_{i})}{\sigma} - \exp\left(\frac{y\_{i}-\mu\_{i}}{\sigma}\right)\right) dy\_{i} + \exp\left(\frac{2(\log L\_{i}-\mu\_{i})}{\sigma}\right) \mathbb{P}(T\_{i} > L\_{i}) \\ &= 2 - \exp\left(-L\_{i}^{1/\sigma}e^{-\mu\_{i}/\sigma}\right) \left[2 + 2L\_{i}^{1/\sigma}e^{-\mu\_{i}/\sigma} + L\_{i}^{2/\sigma}e^{-2\mu\_{i}/\sigma}\right] + L\_{i}^{2/\sigma}e^{-2\mu\_{i}/\sigma} \exp\left(-L\_{i}^{1/\sigma}e^{-\mu\_{i}/\sigma}\right) \\ &= 2 \left\{1 - \exp\left(-L\_{i}^{1/\sigma}e^{-\mu\_{i}/\sigma}\right) \left[1 + L\_{i}^{1/\sigma}e^{-\mu\_{i}/\sigma}\right]\right\} \\ &= 2\left\{w\_{i} + \sigma w\_{i}^{\prime}\right\}. \end{split}$$

On the other hand, for the type II censoring note that *Wi* = exp *yi*−*μ<sup>i</sup> σ* ∼ *<sup>E</sup>*(1). By [12], we have that

$$\mathcal{W}\_{(i)} \stackrel{\mathcal{D}}{=} \sum\_{j=1}^{i} \frac{Z\_j}{n - j + 1},\tag{A1}$$

where *<sup>W</sup>*(*i*) is the *i*th order statistic from *W*1, ... , *Wn*, D= denotes "equal in distribution" and *Z*1, ... , *Zn* are independent and identically distributed *E*(1) random variables. Therefore, <sup>E</sup>(*<sup>W</sup>*(*j*)) = Var(*<sup>W</sup>*(*j*)) = <sup>∑</sup>*jk*=<sup>1</sup>(*<sup>n</sup>* − *k* + <sup>1</sup>)−<sup>1</sup> and

$$\mathcal{W}\_{\mathbb{I}} = \begin{cases} \mathcal{W}\_{(1)'} & \text{with probability } 1/n \\ \vdots & \\ \mathcal{W}\_{(r-1)'} & \text{with probability } 1/n \\ \mathcal{W}\_{(r)'} & \text{with probability } (n-r+1)/n \end{cases}$$

Therefore,

$$w\_i = \mathbb{E}(W\_i) = \frac{1}{n} \sum\_{j=1}^{r-1} \sum\_{k=1}^j (n - k + 1)^{-1} + \frac{(n - r + 1)}{n} \sum\_{k=1}^r (n - k + 1)^{-1}.$$

With some manipulations, we obtain that E(*Wi*) = *<sup>r</sup>*/*<sup>n</sup>*. Also, we have that <sup>E</sup>(*W*<sup>2</sup>(*j*)) = <sup>E</sup>(*<sup>W</sup>*(*j*)) + <sup>E</sup><sup>2</sup>(*<sup>W</sup>*(*j*)) and

$$V\_i = \begin{cases} \begin{array}{c} \mathcal{W}^2\_{(1)'} & \text{with probability } 1/n \\ \vdots & \\ \mathcal{W}^2\_{(r-1)'} & \text{with probability } 1/n \\ \mathcal{W}^2\_{(r)'} & \text{with probability } (n-r+1)/n \end{array} \end{cases}$$

Therefore,

$$w\_i = w\_i + \frac{1}{n} \sum\_{j=1}^{r-1} \left[ \sum\_{k=1}^j (n - k + 1)^{-1} \right]^2 + \frac{(n - r + 1)}{n} \left[ \sum\_{k=1}^r (n - k + 1)^{-1} \right]^2.$$

Algebraic manipulations shows that

$$v\_i = \frac{1}{n} \left[ r + \sum\_{k=1}^{r} \frac{2(r-k)+1}{n-k+1} \right].$$

Finally, as the hybrid scheme can be seen as a mixture between type I and II censoring, we obtain directly that

$$\begin{aligned} w\_i &= q \times \left( 1 - \exp\left( -L\_i^{1/\sigma} e^{-\mu\_i/\sigma} \right) \right) + (1 - q) \times (r/n), \\ v\_i &= q \times 2 \left\{ 1 - \exp\left( -L\_i^{1/\sigma} e^{-\mu\_i/\sigma} \right) \left[ 1 + L\_i^{1/\sigma} e^{-\mu\_i/\sigma} \right] \right\} + (1 - q) \times \frac{1}{n} \left[ r + \sum\_{k=1}^r \frac{2(r - k) + 1}{n - k + 1} \right], \end{aligned}$$

where *q* is the mixing probability given by *q* = <sup>P</sup>(*<sup>W</sup>*(*r*) ≤ log *<sup>L</sup>*). By (A1), *<sup>W</sup>*(*r*) has hypoexponential [13] distribution with vector of parameters *λ* = (*<sup>λ</sup>*1,..., *<sup>λ</sup>n*), where *λj* = (*n* − *j* + <sup>1</sup>)−1. Therefore,

$$q = 1 - \sum\_{j=1}^{n} \frac{L^{-\lambda\_j}}{P\_j} \gamma$$

where *Pj* = ∏*nk*=1,*<sup>k</sup>* =*j*(*<sup>k</sup>* − *j*)/(*n* − *j* + <sup>1</sup>). *Symmetry* **2019**, *11*, 1351

### *Appendix A.2. Derivatives and Cumulants*

Let *Y*1, ... ,*Yn* a random sample from Weibull censored data, the logarithm of the likelihood function is given by

$$\ell(\mathcal{B}) = \sum\_{i=1}^{n} \left\{ \delta\_i \left[ -n \log \sigma + \frac{y\_i - \mu\_i}{\sigma} \right] - \exp \left( \frac{y\_i - \mu\_i}{\sigma} \right) \right\}. \tag{A2}$$

The first four derivatives of (A2) can be expressed, respectively, for

$$\begin{split} \frac{\partial}{\partial \beta\_{r}} \ell(\boldsymbol{\mathcal{B}}) &= \frac{1}{\sigma} \sum\_{i=1}^{n} \left\{-\delta\_{i} + \exp\left(\frac{y\_{i} - \mu\_{i}}{\sigma}\right)\right\} \mathbf{x}\_{i\bar{i}};\\ \frac{\partial^{2}}{\partial \beta\_{r} \partial \beta\_{s}} \ell(\boldsymbol{\mathcal{B}}) &= -\frac{1}{\sigma^{2}} \sum\_{i=1}^{n} \exp\left(\frac{y\_{i} - \mu\_{i}}{\sigma}\right) \mathbf{x}\_{i\bar{i}} \mathbf{x}\_{i\bar{i}};\\ \frac{\partial^{3}}{\partial \beta\_{r} \partial \beta\_{s} \partial \beta\_{t}} \ell(\boldsymbol{\mathcal{B}}) &= \frac{1}{\sigma^{3}} \sum\_{i=1}^{n} \exp\left(\frac{y\_{i} - \mu\_{i}}{\sigma}\right) \mathbf{x}\_{i\bar{i}} \mathbf{x}\_{i\bar{i}} \mathbf{x}\_{i\bar{i}}. \end{split}$$

The second- to third-order cumulants are

$$\kappa\_{rs} = -\frac{1}{\sigma^2} \sum\_{i=1}^n w\_i \mathbf{x}\_{ri} \mathbf{x}\_{si}; \kappa\_{r,s} = -\kappa\_{rs} = \frac{1}{\sigma^2} \sum\_{i=1}^n w\_i \mathbf{x}\_{ri} \mathbf{x}\_{si} \mathbf{x}\_{si}$$

$$\kappa\_{rst} = \frac{1}{\sigma^3} \sum\_{i=1}^n w\_i \mathbf{x}\_{ri} \mathbf{x}\_{si} \mathbf{x}\_{ti}; \; \kappa\_{rs}^{(t)} = -\frac{1}{\sigma^2} \sum\_{i=1}^n w\_i' \mathbf{x}\_{ri} \mathbf{x}\_{si} \mathbf{x}\_{ti};$$

where *wi* = E )exp *yi*−*μ<sup>i</sup> σ* \*,

$$\begin{aligned} w\_i &= 1 - \exp\left\{-L\_i^{1/\sigma} \exp(-\mu\_i/\sigma)\right\}, \\ w\_i' &= -\frac{1}{\sigma} L\_i^{1/\sigma} \exp\{-L\_i^{1/\sigma} \exp(-\mu\_i/\sigma) - \mu\_i/\sigma\}. \end{aligned}$$

It can be observed that *w i* = 0 for type II censoring.
