regression with complex sample design, more general in comparison with Corollary \[cor1\], since it is not necessary to get any consistent estimators for $\boldsymbol{\Sigma }_{hi}$.

\[Th1b\]The estimator $\widehat{\mathbf{\Omega}}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda
,Q}) =\frac{1}{n}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}\widehat{\mathbf{\Omega}}_{hi,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})$, with $$\begin{aligned}
\widehat{\mathbf{\Omega}}_{hi,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda
,Q})& =\boldsymbol{U}_{\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q},\boldsymbol{x}_{hi})\boldsymbol{U}_{\lambda }^{T}(\widehat{\boldsymbol{\beta 
}}_{\lambda ,Q},\boldsymbol{x}_{hi}) \\
& =\left[ w_{hi}^{2}\boldsymbol{\Delta }^{\ast }(\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}))\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}\{\widehat{\boldsymbol{Y}}_{hi}-m_{hi}\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}\right. \\
& \left. \times \{\widehat{\boldsymbol{Y}}_{hi}-m_{hi}\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}^{T}\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}\boldsymbol{\Delta }^{\ast T}(\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}))\right] \otimes \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T},\end{aligned}$$is consistent for $\mathbf{\Omega}_{\lambda }\left( \boldsymbol{\beta }\right)$ as $n$ goes to infinity.

\[cor2\]Let $\widehat{\boldsymbol{Y}}_{hi}$ be a random variable with overdisepersed multinomial sampling scheme with a common overdispersion parameter $\nu $ and $m_{hi}=\overline{m}$,$$\begin{aligned}
\boldsymbol{\Sigma }_{hi}& =\boldsymbol{\Sigma }_{hi}\left( \nu ,\boldsymbol{\beta }\right) =\nu \overline{m}\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) ), \\
\nu & =1+\rho ^{2}(\overline{m}-1),\end{aligned}$$then, for $\nu $ and $\rho ^{2}$:

a)

:   robust and consistent estimators based on the estimating equation are given respectively by$$\begin{aligned}
    \widetilde{\nu }_{n,\lambda }^{E}=\widetilde{\nu }_{n,\lambda }^{E}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})& =\frac{1}{d(k+1)}\mathrm{trace}\left( 
    \mathbf{\Omega}_{n,\lambda }^{-1}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\widehat{\mathbf{\Omega}}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda
    ,Q})\right) ,  \label{vE} \\
    \widetilde{\rho }_{n,\lambda }^{2,E}& =\frac{\widetilde{\nu }_{n,\lambda
    }^{E}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})-1}{\overline{m}-1},  \notag\end{aligned}$$where the matrices of interest are as follows, the one associated with multinomial sampling$$\begin{aligned}
    \mathbf{\Omega}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})& =\frac{1}{n}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}\mathbf{\Omega}_{hi,\lambda }(\widehat{\boldsymbol{\beta 