sample design, more general in comparison with Corollary \[cor1b\], since it is not neccessary to get any consistent estimators for $\boldsymbol{\Sigma }_{hi}$.

\[Th2b\]The estimator$$\begin{aligned}
\widehat{\mathbf{\Omega}}_{\lambda }^{(h)}(\widehat{\boldsymbol{\beta }}_{\lambda
,Q})& =\frac{1}{n_{h}}\sum_{i=1}^{n_{h}}\widehat{\mathbf{\Omega}}_{i,\lambda
}^{(h)}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}), \\
\widehat{\mathbf{\Omega}}_{i,\lambda }^{(h)}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})& =\boldsymbol{U}_{\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q},\boldsymbol{x}_{hi})\boldsymbol{U}_{\lambda }^{T}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q},\boldsymbol{x}_{hi}) \\
& =\left[ w_{hi}^{2}\boldsymbol{\Delta }^{\ast }(\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}))\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}\{\widehat{\boldsymbol{Y}}_{hi}-m_{hi}\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}\right. \\
& \left. \{\widehat{\boldsymbol{Y}}_{hi}-m_{hi}\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}^{T}\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}\boldsymbol{\Delta }^{\ast T}(\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}))\right] \otimes \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T},\end{aligned}$$is consistent of $\mathbf{\Omega}_{\lambda }^{(h)}\left( \boldsymbol{\beta }\right) $, as $n_{h}$ goes to infinity, for every $h=1,...,H$.

\[cor3\]Let $\widehat{\boldsymbol{Y}}_{hi}$ be a random variable with overdisepersed multinomial sampling scheme with an overdispersion parameter $\nu _{h}$, specific for each stratum, and $m_{hi}=\overline{m}_{h}$,$$\begin{aligned}
\boldsymbol{\Sigma }_{hi}\left( \boldsymbol{\beta }\right) & =\nu _{h}\overline{m}_{h}\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}\left( 
\boldsymbol{\beta }\right) ), \\
\nu _{h}& =1+\rho _{h}^{2}(\overline{m}_{h}-1),\end{aligned}$$then, for $\nu _{h}$ and $\rho _{h}^{2}$:

a)

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