PLR model ([2.1.0]{}) under a complex survey. Then we have$$\sqrt{n}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}-\boldsymbol{\beta }_{0})\overset{\mathcal{L}}{\underset{n\mathcal{\rightarrow }\infty }{\longrightarrow }}\mathcal{N}\left( \boldsymbol{0}_{d(k+1)},\mathbf{\Psi}_{\lambda }^{-1}\left( \boldsymbol{\beta }_{0}\right) \mathbf{\Omega}_{\lambda
}\left( \boldsymbol{\beta }_{0}\right) \mathbf{\Psi}_{\lambda }^{-1}\left( 
\boldsymbol{\beta }_{0}\right) \right) ,$$where $\boldsymbol{\beta }_{0}$ is the true parameter value and $$\begin{aligned}
\mathbf{\Omega}_{\lambda }\left( \boldsymbol{\beta }\right) & =\lim_{n\rightarrow
\infty }\mathbf{\Omega}_{n,\lambda }\left( \boldsymbol{\beta }\right)
=\lim_{n\rightarrow \infty }\frac{1}{n}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}\mathbf{\Omega}_{hi,\lambda }\left( \boldsymbol{\beta },\boldsymbol{\Sigma }_{hi}\right) ,  \label{hs} \\
\mathbf{\Psi}_{\lambda }\left( \boldsymbol{\beta }\right) & =\lim_{n\rightarrow
\infty }\mathbf{\Psi}_{n,\lambda }\left( \boldsymbol{\beta }\right)
=\lim_{n\rightarrow \infty }\frac{1}{n}\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}\mathbf{\Psi}_{hi,\lambda }\left( \boldsymbol{\beta }\right) ,  \label{gs}\end{aligned}$$where$$\begin{aligned}
\mathbf{\Omega}_{hi,\lambda }\left( \boldsymbol{\beta },\boldsymbol{\Sigma }_{hi}\right) & =w_{hi}^{2}\boldsymbol{\Delta }^{\ast }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) )\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) \}\boldsymbol{\Sigma 
}_{hi}\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) \}\boldsymbol{\Delta }^{\ast T}(\boldsymbol{\pi }_{hi}\left( 
\boldsymbol{\beta }\right) )\otimes \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T},  \label{hhi} \\
\boldsymbol{\Sigma }_{hi}& =\mathrm{Var}[\widehat{\boldsymbol{Y}}_{hi}], 
\notag \\
\mathbf{\Psi}_{hi,\lambda }\left( \boldsymbol{\beta }\right) & =\left\{ 
\begin{array}{lc}
w_{hi}m_{hi}\boldsymbol{\Delta }^{\ast }(\boldsymbol{\pi }_{hi}\left( 
\boldsymbol{\beta }\right) )\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) \}\boldsymbol{\Delta }^{\ast T}(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) )\otimes \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T}, & \lambda >0 \\ 
w_{hi}m_{hi}\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}^{\ast }\left( 
\boldsymbol{\beta }\right) )\otimes \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T}, & \lambda =0\end{array}\right. .  \label{ghi}\end{aligned}$$

Notice that the expression of $\mathbf{\Psi}_{\lambda =0}\left( \boldsymbol{\beta }\right) $ is the same as the so called Fisher information matrix for multinomial sampling. In addition, if $\widehat{\boldsymbol{Y}}_{hi}$ has a multinomial sampling scheme $\mathbf{\Omega}_{\lambda =0}\left( \boldsymbol{\beta 
}\right) =\mathbf{\Psi}_{\lambda =0}\left( \boldsymbol{\beta }\right) 