proofs of all the results are given in Appendix \[App\].

Maximum Quasi Weighted Likelihood Estimator \[sec:MLE\]
=======================================================

Let us assume that the whole population is partitioned into $H$ distinct strata and the data consist of $n_{h}$ clusters in stratum $h$ for each $h=1,\ldots ,H$. Further, for each cluster $i=1,\ldots ,n_{h}$ in the stratum $h$, we have observed the values of a categorical response variable ($Y$) for $m_{hi}$ units. Assuming $Y$ has $(d+1)$ categories, we denote these observed responses by a $(d+1)$-dimensional classification vector $$\boldsymbol{y}_{hij}=\left( y_{hij1},....,y_{hij,d+1}\right) ^{T},\text{ }h=1,...,H,\text{ }i=1,...,n_{h},\text{ }j=1,...,m_{hi},  \label{2.1}$$with $y_{hijr}$ $=1$ and $y_{hijl}$ $=0$ for $l\in \{1,...,d+1\}-\{r\}$ if the $j$-th unit selected from the $i$-th cluster of the $h$-th stratum falls in the $r$-th category. We also have data on $(k+1)$ explanatory variables which are common for all the individuals in the $i$-th cluster of the $h$-th stratum (very common with dummy or qualitative explanatory variables) to be denoted as $\boldsymbol{x}_{hi}=\left( x_{hi0},x_{hi1},....,x_{hik}\right) ^{T}$; the first one $x_{hi0}=1$ is associated with the intercept. Let us denote the sampling weight from the $i$-th cluster of the $h$-th stratum by $w_{hi}$. For each $i$, $h$ and $j$, the expectation of the $r$-th element of the random variable $\boldsymbol{Y}_{hij}=(Y_{hij1},...,Y_{hij,d+1})^{T}$, corresponding to the realization 