\text{for }\lambda >0,$$whose expression for the PLR model with complex sampling is$$d_{\lambda }^{\ast }(\widehat{g},f_{\boldsymbol{\beta }},w)=\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}w_{hi}\boldsymbol{\pi }_{hi}^{\lambda ,T}\left( 
\boldsymbol{\beta }\right) \left( m_{hi}\boldsymbol{\pi }_{hi}\left( 
\boldsymbol{\beta }\right) -\frac{\lambda +1}{\lambda }\widehat{\boldsymbol{y}}_{hi}\right) ,\quad \text{for }\lambda >0.  \label{def}$$

Based on $d_{\lambda }^{\ast }(\widehat{g},f_{\boldsymbol{\beta }},w)$ the minimum quasi weighted DPD estimator is formally defined as follows.

The minimum quasi weighted DPD estimator, say $\widehat{\boldsymbol{\beta }}_{\lambda ,Q}$, of $\boldsymbol{\beta }$ is defined as$$\widehat{\boldsymbol{\beta }}_{\lambda ,Q}=\arg \min_{\boldsymbol{\beta }\in 
\mathbb{R}
^{d(k+1)}}d_{\lambda }^{\ast }(\widehat{g},f_{\boldsymbol{\beta }},w).$$

At the particular choice $\lambda\rightarrow 0$, the DPD measure, defined as the limit of (\[def\]) coincides (in limit) with the weighted quasi-loglikelihood, $\mathcal{\ell (}\boldsymbol{\beta },w)$, given in (\[llike\]); thus the minimum quasi weighted DPD estimator of $\boldsymbol{\beta }$ at $\lambda =0$ coincides with the maximum weighted quasi likelihood estimator. With the same philosophy, the following result generalizes $\boldsymbol{u}(\boldsymbol{\beta },\boldsymbol{x}_{hi})$ given in (\[Un\]) which plays an important role for the derivation of the asymptotic distribution of $\widehat{\boldsymbol{\beta}}_{\lambda, Q}$.

\[th:u\] The minimum quasi weighted DPD estimate of $\boldsymbol{\beta }$, say $\widehat{\boldsymbol{\beta }}_{\lambda ,Q}$, can be obtained by solving 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}}\boldsymbol{u}_{\lambda }(\boldsymbol{\beta 
},\boldsymbol{x}_{hi}),  \label{u0} \\
\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( 
