(2015) for details). The weakest assumption is to consider that $\widehat{\boldsymbol{Y}}_{hi}$ has an unknown distribution, with $\boldsymbol{Y}_{hij}$, $j=1,...,m_{hi}$ being possibly correlated but with no specific pattern. It is worth of mentioning that $\nu _{hi}$ plays here a role of nuisance parameter and it is possible to consider a model with additional nuisance parameters, which are more complex than (\[var2\]) but simpler than the option of completely unknown distribution for $\widehat{\boldsymbol{Y}}_{hi}$ (see Morel and Koehler (1995) for details).

Taking into account weights for each cluster, the quasi weighted loglikelihood is defined as $$\mathcal{\ell (}\boldsymbol{\beta },w)={\sum\limits_{h=1}^{H}}{\sum\limits_{i=1}^{n_{h}}}w_{hi}\log \boldsymbol{\pi }_{hi}^{T}\left( 
\boldsymbol{\beta }\right) \widehat{\boldsymbol{Y}}_{hi}.  \label{llike}$$Then, the maximum quasi weighted likelihood estimator of $\boldsymbol{\beta }
$, say $\widehat{\boldsymbol{\beta }}_{P}$, is obtained by maximizing the quasi weighted loglikelihood, $\mathcal{\ell (}\boldsymbol{\beta },w)$, with respect to $\boldsymbol{\beta }$. The corresponding estimating equation is then given by $${\sum\limits_{h=1}^{H}}{\sum\limits_{i=1}^{n_{h}}}w_{hi}\frac{\partial 
\boldsymbol{\pi }_{hi}^{T}\left( \boldsymbol{\beta }\right) }{\partial 
\boldsymbol{\beta }}\mathrm{diag}^{-1}(\boldsymbol{\pi }_{hi}\left( 
\boldsymbol{\beta }\right) )\left[ \widehat{\boldsymbol{Y}}_{hi}-m_{hi}\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) \right] =\boldsymbol{0}_{d(k+1)},  \label{2.4}$$with $$\begin{aligned}
\frac{\partial \boldsymbol{\pi }_{hi}^{T}\left( \boldsymbol{\beta }\right) }{\partial \boldsymbol{\beta }}& =\mathcal{\boldsymbol{\Delta }}^{\ast }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) )\otimes \boldsymbol{x}_{hi}, \quad \boldsymbol{\Delta }^{\ast }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }
\right) ) =\left( \boldsymbol{I}_{d},\boldsymbol{0}_{d}\right) \boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) ) .\end{aligned}$$ The system of equations (\[2.4\]) can be written as 