**3. Skewness Coefficient**

As discussed, the skewness coefficient is simple way to verify whether the approximation to normality is adequate. The model presented in (2) does not has a closed-form for this coefficient. The alternative is to apply the [3] result. These authors derived an approximation of order <sup>O</sup>(*n*<sup>−</sup><sup>2</sup>) for the third cumulant of the MLE of the *a*-th regressor, i.e.,

$$\kappa\_3(\mathfrak{f}\_a) = \mathbb{E}\left\{ \left[ \mathfrak{f}\_a - \mathbb{E}(\mathfrak{f}\_a) \right]^3 \right\},$$

*a* = 1, . . . , *p*, which can be expressed as

$$\kappa\_3(\hat{\beta}\_a) = \sum' \kappa^{a,b} \kappa^{a,c} \kappa^{a,d} m\_{bc}^{(d)} \, , \tag{3}$$

where *m*(*d*) *bc* = <sup>5</sup>*κ*(*d*) *bc* − (*κ*(*b*) *cd* + *κ*(*c*) *bd* + *<sup>κ</sup>bcd*), *a* = 1, ... , *p*. Here, ∑ represents the summation over all combinations of parameters and over all the observations. From (3), after some algebra, we can express the third cumulant of the distribution of *β* ˆ for the Weibull censored data as

$$\kappa\_3(\hat{\boldsymbol{\beta}}) = -\sigma^{-3} \mathbf{P}^{(3)} \left( \mathbf{W} + 3\sigma \mathbf{W}' \right) \mathbf{1},\tag{4}$$

where *W* = diag(*<sup>w</sup>* 1, ... , *<sup>w</sup> n*), *w i* = −*σ*<sup>−</sup>1*L*1/*<sup>σ</sup> i* exp{−*L*1/*<sup>σ</sup> i* exp(−*μi*/*σ*) − *μi*/*σ*}, *P* = *K*−<sup>1</sup> *ββX* = *σ*<sup>2</sup> *XW X*−<sup>1</sup> *<sup>X</sup>*, *P*(3) = *P P P*, represents a direct product of matrices and **1** is a *n*-dimensional vector of ones. Finally, by (4) and the Fisher information matrix, the asymmetry coefficient of the distribution of *β* to order *n*<sup>−</sup>1/2 is given by

$$\gamma\_1(\hat{\boldsymbol{\beta}}) = -\boldsymbol{\sigma}^{-3} \mathbf{P}^{(3)} \left( \boldsymbol{\mathcal{W}} + 3 \boldsymbol{\sigma} \mathbf{W}' \right) \mathbf{1} \odot \left\{ \text{diag} \left( \boldsymbol{\mathcal{K}}\_{\boldsymbol{\beta}\boldsymbol{\beta}}^{-1} \right) \mathbf{1} \right\}^{-3/2},\tag{5}$$

for type II censoring, *W* = **0**, then (5) reduces to *<sup>γ</sup>*1(*β*<sup>ˆ</sup>) = −*σ*<sup>−</sup>3*P*(3)*W***<sup>1</sup>** )diag *<sup>K</sup>*−<sup>1</sup> *ββ* **1**\*−3/2. More details about the involved expressions are presented in Appendix A. The study of asymptotic properties of the Weibull censored data was the goal of many papers. Refs. [6,7] derived the Bartlett and the Bartlett-type correction factors for likelihood ratio and score tests, respectively, for the exponential censored data. Ref. [8] generalized these previous for the Weibull censored data and also derived the Bartlett-type correction factors for the gradient test. Ref. [9] presented the asymptotic expansions up to order *n*<sup>−</sup>1/2 of the non null distribution functions of the likelihood ratio, Wald, Rao score and gradient statistics also for the censored exponential data. The result in expression (5) can be incorporated in this gallery.
