,Q}), \\
    \mathbf{\Omega}_{i,\lambda }^{(h)}\left( \boldsymbol{\beta },\boldsymbol{\Sigma }_{hi}\right) & =\overline{m}_{h}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})\}\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})) \\
    & \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}\end{aligned}$$and Theorem \[Th2b\] for overdispersed multinomial sampling.

b)

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

Testing linear hypotheses for the PLR coefficients with complex survey design \[secWald\]
=========================================================================================

Based on the asymptotic distribution of the minimum quasi weighted DPD estimator, $\widehat{\boldsymbol{\beta }}_{\lambda ,Q},$ presented in Theorem \[th2\], we now introduce and study a family of Wald-type test statistics for testing $$H_{0}:\boldsymbol{M}^{T}\boldsymbol{\beta =l}\text{ against }H_{0}:\boldsymbol{M}^{T}\boldsymbol{\beta \neq l}\text{ }  \label{A}$$where $\boldsymbol{M}$ is a $d(k+1)\times r$ full row-rank matrix with $r\leq d(k+1)$ and the right-hand-side $r-$vector consists of constants that in many situations are $\boldsymbol{l=0}_{r}.$ For solving the problem of testing given in (\[A\]) we define a family of Wald-type test statistics based on quasi MDPDE as follows.

Let $\widehat{\boldsymbol{\beta }}_{\lambda ,Q}$ the minimum quasi weighted DPD estimator of $\boldsymbol{\beta }$ in the PLR model (\[2.1.0\]) under a complex survey 