The Weibull distribution considered here has probability density function
$$ f(t;\lambda, \sigma)=\frac{1}{\sigma
\lambda^{1/\sigma}}t^{1/\sigma-1}\exp\left\{-\left(\frac{t}{\lambda}\right)^{1/\sigma}\right\},
\quad t, \sigma, \lambda>0. $$ The regression structure is incorporated as
$$ \log(\lambda_i)={\bm x}_i^\top {\bm \beta}, \quad i=1,\ldots,n. $$ For
the computation of the bias-corrected estimators, \(\sigma\) is assumed as
fixed in the jackknife estimator based on the traditional MLE.
The Fisher information matrix for \(\bm \beta\) is given by \({\bm
K}=\sigma^{-2} {\bm X}^\top {\bm W} {\bm X}\), where \({\bm X}=({\bm
x}_1,\ldots,{\bm x}_n)^\top\), \({\bm W}=\mbox{diag}(w_1,\ldots,w_n)\), and
$$ w_i=E\left[\exp\left(\frac{y_i-\log
\lambda_i}{\sigma}\right)\right]=q \times \left\{ 1 - \exp\left[
-L_i^{1/\sigma} \exp(-\mu_i/\sigma) \right] \right\} + (1-q)\times
\left(r/n\right), $$ with \(q = P\left(W_{(r)}\leq \log L_i\right)\) and
\(W_{(r)}\) denoting the \(r\)th order statistic from \(W_1, \ldots,
W_n\), with \(q=1\) and \(q=0\) for types I and II censoring,
respectively. (See Magalhaes et al. 2019 for details).
The bias-corrected maximum likelihood estimator based on the Cox and Snell's
method (say \(\widetilde{\bm \beta}\)) is based on a corrective approach
given by \(\widetilde{\beta}=\widehat{\beta}-B(\widehat{\beta})\), where
$$ B({\bm \beta})= - \frac{1}{2 \sigma^3} {\bm P} {\bm Z}_d \left({\bm
W} + 2 \sigma {\bm W}^{\prime}\right) {\bm 1}, $$ with \({\bm P} = {\bm
K}^{-1} {\bm X}^{\top}\), \({\bm Z} = {\bm X} {\bm K}^{-1} {\bm
X}^{\top}\), \({\bm Z}_d\) is a diagonal matrix with diagonal given by the
diagonal of \({\bm Z}\), \({\bm W}^{\prime} =\) diag\((w_1^{\prime},
\ldots, w_n^{\prime})\), \(w_i^{\prime} = - \sigma^{-1} L_i^{1/\sigma}
\exp\{ -L_i^{1/\sigma} \exp(-\mu_i/\sigma) - \mu_i/\sigma \}\) and \({\bm
1}\) is a \(n\)-dimensional vector of ones.
The bias-corrected maximum likelihood estimator based on the Firth's method
(say \(\check{\bm \beta}\)) is based on a preventive approach, which is
the solution for the equation \({\bm U}_{{\bm \beta}}^{\star} = {\bm 0}\),
where $$ {\bm U}_{{\bm \beta}}^{\star} = {\bm U}_{{\bm \beta}} - {\bm
K}_{{\bm \beta} {\bm \beta}} B({\bm \beta}). $$
The covariance correction is based on the general result of Magalhaes et al.
2021 given by $$ \mbox{{\bf Cov}}_{\bm 2}^{\bm \tau}({\bm
\beta}^{\star}) = {\bm K}^{-1} + {\bm K}^{-1} \left\{ {\bm \Delta} + {\bm
\Delta}^{\top} \right\} {\bm K}^{-1} + \mathcal{O}(n^{-3}) $$ where
\({\Delta} = -0.5 {\Delta}^{(1)} + 0.25 {\Delta}^{(2)} + 0.5 \tau_2
{\Delta}^{(3)}\), with $$ \Delta^{(1)} = \frac{1}{\sigma^4} {\bm
X}^{\top} {\bm W}^{\star} {\bm Z}_{d} {\bm X}, $$ $$ \Delta^{(2)} = -
\frac{1}{\sigma^6} {\bm X}^{\top} \left[ {\bm W} {\bm Z}^{(2)} {\bm W} - 2
\sigma {\bm W} {\bm Z}^{(2)} {\bm W}^{\prime} - 6 \sigma^2 {\bm W}^{\prime}
{\bm Z}^{(2)} {\bm W}^{\prime} \right] {\bm X}, $$ and $$ \Delta^{(3)} =
\frac{1}{\sigma^5} {\bm X}^{\top} {\bm W}^{\prime} {\bm W}^{\star\star} {\bm
X}, $$ where \({\bm W}^{\star} =\) diag\((w_1^{\star}, \ldots,
w_n^{\star})\), \(w_i^{\star} = w_i (w_i -2) - 2 \sigma w_i^{\prime} +
\sigma \tau_1 (w_i^{\prime} + 2 \sigma w_i^{\prime\prime})\), \({\bm
Z}^{(2)} = {\bm Z} \odot {\bm Z}\), with \(\odot\) representing a direct
product of matrices (Hadamard product), \({\bm W}^{\star\star}\) is a
diagonal matrix, with \({\bm Z} ( {\bm W} + 2 \sigma {\bm W}^{\prime})
{\bm Z}_{d} {\bm 1}\) as its diagonal, \({\bm W}^{\prime\prime} =\)
diag\((w_1^{\prime\prime}, \ldots, w_n^{\prime\prime})\),
\(w_i^{\prime\prime} = - \sigma^{-1} w_i^{\prime} \left[ L_i^{1/\sigma}
\exp(-\mu_i/\sigma) - 1 \right]\), \({\bm \tau} = (\tau_1, \tau_2) = (1,
1)\) indicating the second-order covariance matrix of the MLE \({\bm
\beta}^{\star} = \widehat{{\bm \beta}}\) denoted by \(\mbox{{\bf
Cov}}_{\bm 2}(\widehat{{\bm \beta}})\) and \({\bm \tau} = (0, -1)\)
indicating the second-order covariance matrix of the BCE \({\bm
\beta}^{\star} = \widetilde{{\bm \beta}}\) denoted by \(\mbox{{\bf
Cov}}_{\bm 2}(\widetilde{{\bm \beta}})\).