\boldsymbol{\beta }\right) \}\{\widehat{\boldsymbol{y}}_{hi}-m_{hi}\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) \}\right] \otimes 
\boldsymbol{x}_{hi}.  \label{u}\end{aligned}$$

Let $\boldsymbol{U}_{\lambda }(\boldsymbol{\beta },\boldsymbol{X})$ be a random variable generator of (\[u\]) associated with a generic random explanatory variable $\boldsymbol{X}$, with no stratum and cluster assignment. In what is to follow, $$\boldsymbol{U}_{\lambda }(\boldsymbol{\beta },\boldsymbol{x}_{hi})=\left[
w_{hi}\boldsymbol{\Delta }^{\ast }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) )\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}\left( 
\boldsymbol{\beta }\right) \}\{\widehat{\boldsymbol{Y}}_{hi}-m_{hi}\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) \}\right] \otimes 
\boldsymbol{x}_{hi}$$denotes $\boldsymbol{U}_{\lambda }(\boldsymbol{\beta },\boldsymbol{X})|\boldsymbol{X=x}_{hi}$. An important property is that the estimating equation given in Theorem \[th:u\] is unbiased. This property is very important to consider these estimators similar to the maximum quasi weighted likelihood ones in the construction of the asymptotic properties, which was not the case for other distance based estimators such as the one proposed in Castilla et al (2018).

Asymptotic distribution and estimates of the design effect
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The following results are generalized for the PLR model with complex sampling and random explanatory variables. Without any loss of generality it is assumed that $\widehat{\Pr }(\boldsymbol{X=x}_{hi})=\frac{1}{n}$ is estimated from the sample of strata in order to get assymptotic results for the fixed explanatory variables (as a particular case of random explanatory variables).

\[Th1\] Let $\widehat{\boldsymbol{\beta }}_{\lambda ,Q}$ the minimum quasi weighted DPD estimate of $\boldsymbol{\beta }$ in the 