}}_{\lambda ,Q}), \\
    \mathbf{\Omega}_{hi,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})& =\overline{m}w_{hi}^{2}\boldsymbol{\Delta }^{\ast }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) )\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) \}\boldsymbol{\Delta 
    }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) ) \\
    & \times \mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}\left( 
    \boldsymbol{\beta }\right) \}\boldsymbol{\Delta }^{\ast T}(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) )\otimes \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T},\end{aligned}$$and Theorem \[Th1b\] for overdispersed multinomial sampling.

b)

:   robust and consistent estimators based on the method of moments are given by$$\begin{aligned}
    \widetilde{\nu }_{n,\lambda }^{M}& =\widetilde{\nu }_{n,\lambda }^{M}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})=\frac{1}{nd}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}\sum_{j=1}^{d+1}\frac{(\widehat{Y}_{hij}-\overline{m}\pi
    _{hij}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}))^{2}}{\overline{m}\pi
    _{hij}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})},  \label{vM} \\
    \widetilde{\rho }_{n,\lambda }^{2,M}& =\frac{\widetilde{\nu }_{n,\lambda
    }^{M}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})-1}{\overline{m}-1}.  \notag\end{aligned}$$

The preceding results were established for $n=\sum\nolimits_{h=1}^{H}n_{h}$ tending to infinity which implies in practice that $H$ is fixed and there exists $\eta _{h}=\lim_{n\rightarrow \infty }\frac{n_{h}}{n}\in (0,1)$, $\sum\nolimits_{h=1}^{H}\eta _{h}=1$. This means that it is approriate to consider in the place of Theorem \[Th1\], the following one, assuming that $\widehat{\Pr }(\boldsymbol{X}_{h}\boldsymbol{=x}_{hi})=\frac{1}{n_{h}}$.

\[th2\] Let $\widehat{\boldsymbol{\beta }}_{\lambda ,Q}$ the quasi MDPDE of $\boldsymbol{\beta }$ in the PLR model (\[2.1.0\]) under a complex survey. Then we have$$\sqrt{n}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}-\boldsymbol{\beta }_{0})\overset{\mathcal{L}}{\underset{n\mathcal{\rightarrow }\infty }{\longrightarrow }}\mathcal{N}\left( \boldsymbol{0}_{d(k+1)},\mathbf{\Psi}_{\lambda }^{-1}\left( \boldsymbol{\beta }_{0}\right) \mathbf{\Omega}_{\lambda
}\left( \boldsymbol{\beta }_{0}\right) \mathbf{\Psi}_{\lambda }^{-1}\left( 
\boldsymbol{\beta }_{0}\right) \right) ,$$where $\boldsymbol{\beta }_{0}$ is the true parameter value and$$\begin{aligned}
\mathbf{\Omega}_{\lambda }\left( \boldsymbol{\beta }\right) & =\sum_{h=1}^{H}\eta
_{h}\mathbf{\Omega}_{\lambda }^{(h)}\left( \boldsymbol{\beta }\right)
=\sum_{h=1}^{H}\eta _{h}\lim_{n_{h}\rightarrow \infty }\frac{1}{n_{h}}\sum_{i=1}^{n_{h}}\mathbf{\Omega}_{i,\lambda 