The following two cases of $\mathbf{\Omega}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q},\{\widehat{\boldsymbol{\Sigma }}_{hi}\}_{h=1,...,H;i=1,...,n_{h}})$ are taken into account for two cases of $\boldsymbol{\Sigma }_{hi}$ for which there exists a consistent estimator:

-   if $\widehat{\boldsymbol{Y}}_{hi}$ has (an ordinary) multinomial sampling scheme then$$\mathbf{\Omega}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})=\frac{1}{n}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}\mathbf{\Omega}_{hi,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})$$where $\mathbf{\Omega}_{hi,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})$ is $\mathbf{\Omega}_{hi,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q},\widehat{\boldsymbol{\Sigma }}_{hi})$ given in (\[Hhi\]) with $$\widehat{\boldsymbol{\Sigma }}_{hi}=\boldsymbol{\Sigma }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})=m_{hi}\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}));$$

-   if $\widehat{\boldsymbol{Y}}_{hi}$ has an overdispersed multinomial sampling scheme then$$\mathbf{\Omega}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q},\{\widetilde{\nu }_{hi}\}_{h=1,...,H;i=1,...,n_{h}})=\frac{1}{n}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}\mathbf{\Omega}_{hi,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q},\widetilde{\nu }_{hi}),$$where $\widetilde{\nu }_{hi}$ is a consistent estimator of the overdispersion parameter $\nu _{hi}$ which is established later in Corollary \[cor2\], and $\mathbf{\Omega}_{n,\lambda }(\widetilde{\nu }_{hi},\widehat{\boldsymbol{\beta }}_{\lambda ,Q})$ is $\mathbf{\Omega}_{hi,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q},\widehat{\boldsymbol{\Sigma }}_{hi})$ given in (\[Hhi\]) with $$\widehat{\boldsymbol{\Sigma }}_{hi}=\boldsymbol{\Sigma }_{hi}(\widetilde{\nu 
    }_{hi},\widehat{\boldsymbol{\beta }}_{\lambda ,Q})=\widetilde{\nu }_{hi}m_{hi}\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}));$$

If $\widehat{\boldsymbol{Y}}_{hi}$ has a multinomial sampling scheme, Theorem \[Th1\] can be similarly formulated taking the assumption that $\widehat{\Pr }(\boldsymbol{X=x}_{hi})=\frac{1}{m}$, where $m=\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}m_{hi}$, for each individual of all the clusters in the sample, and taking for the estimation equation $m$ summands rather than $n$, i.e. plugging ${\sum\limits_{j=1}^{m_{hi}}}\boldsymbol{y}_{hij}=\widehat{\boldsymbol{y}}_{hi}$ into (\[u0\]) and considering the system of equations $\boldsymbol{u}_{\lambda }(\boldsymbol{\beta })=\boldsymbol{0}_{d(k+1)}$, where$$\begin{aligned}
\boldsymbol{u}_{\lambda }(\boldsymbol{\beta })&
=\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}{\sum\limits_{j=1}^{m_{hi}}}\boldsymbol{v}_{j,\lambda }(\boldsymbol{\beta },\boldsymbol{x}_{hi}), \\
\boldsymbol{v}_{j,\lambda }(\boldsymbol{\beta },\boldsymbol{x}_{hi})& =\left[
w_{hi}\boldsymbol{\Delta }^{\ast 