$ and $\mathbf{\Omega}_{\lambda =0}^{-1}\left( \boldsymbol{\beta }\right) \mathbf{\Psi}_{\lambda =0}\left( \boldsymbol{\beta }\right) \mathbf{\Omega}_{\lambda
=0}^{-1}\left( \boldsymbol{\beta }\right) $ is the inverse of the Fisher information matrix.

Consistency is usually considered as a minimal requirement for an inference procedure. The following result is useful as a tool for estimating $\mathbf{\Omega}_{\lambda }\left( \boldsymbol{\beta }\right) $ and $\mathbf{\Psi}_{\lambda }\left( \boldsymbol{\beta }\right) $ consistently plugging a consistent estimator into $\boldsymbol{\beta }$, and hence also $\mathbf{\Omega}_{\lambda }^{-1}\left( \boldsymbol{\beta }\right) \mathbf{\Psi}_{\lambda }\left( \boldsymbol{\beta }\right) \mathbf{\Omega}_{\lambda
}^{-1}\left( \boldsymbol{\beta }\right) $ again as a (double) plug-in estimator.

\[cor1\]The following ones are (weak) consistent estimators as $n$ goes to infinity:

a)

:   $\widehat{\boldsymbol{\beta }}_{\lambda ,Q}$ is a consistent estimator of the true regression coefficient $\boldsymbol{\beta }_{0}$.

b)

:   $\mathbf{\Psi}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda
    ,Q})=\frac{1}{n}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}\mathbf{\Psi}_{hi,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})$ is a consistent estimator of $\mathbf{\Psi}_{\lambda }\left( \boldsymbol{\beta }_{0}\right) $.

c)

:   $\mathbf{\Omega}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda
    ,Q},\{\widehat{\boldsymbol{\Sigma }}_{hi}\}_{h=1,...,H;i=1,...,n_{h}})=\frac{1}{n}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}\mathbf{\Omega}_{hi,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q},\widehat{\boldsymbol{\Sigma }}_{hi})$ with$$\begin{aligned}
    \mathbf{\Omega}_{hi,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q},\widehat{\boldsymbol{\Sigma }}_{hi})& =w_{hi}^{2}\boldsymbol{\Delta }^{\ast }(\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}))\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}\widehat{\boldsymbol{\Sigma }}_{hi}  \notag \\
    & \times \mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}\boldsymbol{\Delta }^{\ast T}(\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}))\otimes 
    \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T},  \label{Hhi}\end{aligned}$$is a consistent estimator of $\mathbf{\Omega}_{\lambda }\left( \boldsymbol{\beta }_{0}\right) $, whenever $\Sigma _{hi}=\mathrm{Var}[\widehat{\boldsymbol{Y}}_{hi}]$ is consistently estimated through $\widehat{\boldsymbol{\Sigma }}_{hi}$ for all $(h,i)\in \{1,...,H\}\times \{1,...,m_{hi}\}$.

