**1. Introduction**

In its first appearance, the Weibull distribution [1] claimed its wide applicability. Survival analysis, reliability engineering, and extreme value theory are some of its applicability. To amplify the relevance of the Weibull, a regression structure is added to one of the parameters, i.e., the behavior of the distribution may be explained from covariates (explanatory variables) and unknown parameters to be estimated from observable data.

In statistical inference, it is often desirable to test if there are regression parameters statistically significant and the Wald test is commonly performed. Under standard regularity conditions, the null distribution of the Wald statistic is asymptotically chi-squared, a consequence of the maximum likelihood estimators (MLE) distribution. Therefore, the Wald test must be avoided if the sample size is not large enough, because the distribution of the MLE will be poorly approximated by the normal distribution.

Preventing the complexity of the statistical tests, the skewness coefficient (say *γ*) of the distribution of the MLE is an easy way to verify if the approximation to normality is adequate. A value of *γ* far from zero indicates a departure from the normal distribution. Pearson's standardized third cumulant defined by *γ* = *κ*3/*κ*3/2 2 , where *κr* is the *r*th cumulant of the distribution, is the most well-known measure of skewness. When *γ* > 0 (*γ* < 0) the distribution is positively (negatively) skewed and will have a longer (shorter) right tail and a shorter (longer) left tail. If the distribution is symmetrical, *γ* equals zero. However, there are in [2] (Exercise 3.26) asymmetrical distributions with as many zero-odd order central moments as desired, so, the value of *γ* must be interpreted with some caution.

In the statistical literature, there is not a closed-form for the skewness coefficient of *γ* of the MLE in several regression models. Ref. [3] obtained a general *n*<sup>−</sup>1/2 *γ* expression (say *γ*1) for the distribution of the MLE, where *n* is the sample size. Following [3], several works have been developed in order to obtain the *γ*1 coefficient. In the first, Ref. [4] determined its expression for the class of generalized linear models and, the last one, Ref. [5] defined the *γ*1 for the varying dispersion beta regression model and showed that this coefficient for the distribution of the MLE of the precision parameter is relatively large in small to moderate sample sizes. This paper is the first focused on a censored model.

In this work, we derive the *γ*1 coefficient of the distribution of the MLE of the linear parameters in the Weibull censored data, assuming *σ* known, as *σ* = 1/2 and 1, the Rayleigh and exponential models, respectively. We discuss the situation when *σ* is unknown, however, it can be replaced by a consistent estimator, and then we can turn back to the original situation. This type of procedure was performed, for instance, by [4].

The remainder of the paper is organized as follows. Section 2 defines the Weibull censored data. In Section 3, we obtain a simple matrix expression, of order *n*<sup>−</sup>1/2, for the skewness coefficients of the distributions of the MLEs of the linear regression parameters. In Section 4, some Monte Carlo simulations are performed. Two applications are presented in Section 5. Concluding remarks are offered in Section 6.

### **2. The Weibull Censored Data**

We say that a continuous random variable *T* has Weibull distribution with scale parameter *θ* and shape parameter *σ*, or *T* ∼ WE(*θ*, *σ*), if its probability density function (pdf) is given by

$$f(t; \theta, \sigma) = \frac{1}{\sigma \theta^{1/\sigma}} t^{1/\sigma - 1} \exp\left\{-\left(t/\theta\right)^{1/\sigma}\right\},\tag{1}$$

with *t* > 0, *σ* > 0 and *θ* > 0. From (1), we can observe two particular distributions: the exponential and the Rayleigh, where *σ* = 1 and *σ* = 1/2, respectively. In lifetime data, there is the censoring restriction, i.e, if *T*1, ... , *Tn* are a random sample from (1), instead of *Ti*, we observe, under right censoring, *ti* = min(*Ti*, *Li*), where *Li* is the censoring time, independent of *Ti*, *i* = 1, ... , *n*. In this work, we consider an hybrid censoring scheme, where the study is finalized when a pre-fixed number, *r* ≤ *n*, out of *n* observations have failed, as well as when a prefixed time, say *L*1 = ... = *Ln* = *L*, has been reached. The type I censoring is a particular case for *r* = *n* and the type II censoring appears when *L*1, ... , *Ln* = + ∞. Additionally, we add the non-informative censoring assumption, i.e., the random variables *Li* does not depend on *θ*. Under this setup, the log-likelihood function has the form

$$L(\theta, \sigma) \;= \left(\sigma \theta^{1/\sigma}\right)^{-r} \exp\left\{ \left(\frac{1}{\sigma} - 1\right) A\_1 - \frac{1}{\theta^{1/\sigma}} A\_2 \right\},$$

where *r* = ∑*n i*=1 *δi*, *A*1 = ∑*n i*=1 *δi* log *ti*, *A*2 = ∑*n i*=1 *t*1/*<sup>σ</sup> i* , *δi* = 1, if *Ti* ≤ *Li* and *δi* = 0, otherwise. Usually, the regression modeling considers the distribution of *Yi* = log(*Ti*) instead of *Ti*. The distribution of *Yi* is of the extreme value form with pdf given by

$$f(y\_i; \mathbf{x}\_i) = \frac{1}{\sigma} \exp\left\{ \frac{y\_i - \mu\_i}{\sigma} - \exp\left(\frac{y\_i - \mu\_i}{\sigma}\right) \right\}, \quad -\infty < y\_i < \infty \tag{2}$$

.

where *μi* = log *θi*. The regression structure can be incorporated in (2) by making *θi* = exp *x i β* , where *β* is a p-vector of unknown parameters and *xi* is a vector of regressors related to the *i*th observation. From this moment, we assume that *σ* is known, then, the log-likelihood function derived from (2) is given by

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

The total score function and the total Fisher information matrix for *β* are, respectively, *<sup>U</sup>β* = *σ*<sup>−</sup>1*XW*1/2*v* and *<sup>K</sup>ββ* = *σ*<sup>−</sup>2*XW X*, where *X* = (*<sup>x</sup>*1, ... , *<sup>x</sup>n*), the model matrix, assuming rank (*X*) = *p*, *W* = diag(*<sup>w</sup>*1, ... , *wn*), *wi* = E exp *yi*−*μ<sup>i</sup> σ* and *v* = (*<sup>v</sup>*1, ... , *vn*), *vi* = )−*δ<sup>i</sup>* + exp *yi*−*μ<sup>i</sup> σ* \* *w*<sup>−</sup>1/2 *i* . It can observed that the value of *wi* depends on the mechanism of censoring. That means *wi* = *q* × 1 − exp )−*L*1/*<sup>σ</sup> i* exp(−*μi*/*σ*)\* + (1 − *q*) × (*r*/*n*), where *<sup>W</sup>*(*r*) denotes the *r*th order statistic from *W*1, ... , *Wn* and *q* = P *<sup>W</sup>*(*r*) ≤ log *Li*. Note that *q* = 1 and *q* = 0 for types I and II censoring, respectively. The proof is presented in the Appendix A. The MLE of *β*, *β* , is the solution of *<sup>U</sup>β* = **0**. The *β* can not be expressed in closed-form. It is typically obtained by numerically maximizing the log-likelihood function using a Newton or quasi-Newton nonlinear optimization algorithm. Under mild regularity conditions and in large samples,

$$
\widehat{\boldsymbol{\theta}} \sim \operatorname{N}\_p \left( \boldsymbol{\beta}, \boldsymbol{\mathsf{K}}\_{\boldsymbol{\theta}\boldsymbol{\beta}}^{-1} \right),
$$

 ,

approximately.
