and we denote $$\begin{aligned}
\widehat{\mathbf{Q}}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda
,Q})& =\frac{1}{n}\mathbf{\Psi}_{n,\lambda }^{-1}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\widehat{\mathbf{\Omega}}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\mathbf{\Psi}_{n,\lambda }^{-1}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}) \\
& =\left( n\mathbf{\Psi}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda
,Q})\right) ^{-1}n\widehat{\mathbf{\Omega}}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\left( n\mathbf{\Psi}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\right) ^{-1},\end{aligned}$$where$$\begin{aligned}
n\widehat{\mathbf{\Omega}}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda
,Q})& =\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}\left[ w_{hi}^{2}\boldsymbol{\Delta }^{\ast }(\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}))\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}\{\widehat{\boldsymbol{Y}}_{hi}-m_{hi}\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}\right.  \\
& \left. \times \{\widehat{\boldsymbol{Y}}_{hi}-m_{hi}\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}^{T}\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\}\boldsymbol{\Delta }^{\ast T}(\boldsymbol{\pi }_{hi}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q}))\right. \otimes \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T}, \\
n\mathbf{\Psi}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})&
=\left\{ 
\begin{array}{lc}
\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}w_{hi}m_{hi}\boldsymbol{\Delta }^{\ast }(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) )\mathrm{diag}^{\lambda -1}\{\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta }\right) \}\boldsymbol{\Delta }^{\ast T}(\boldsymbol{\pi }_{hi}\left( \boldsymbol{\beta 
}\right) )\otimes \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T}, & \lambda >0
\\ 
\sum_{h=1}^{H}\sum_{i=1}^{n_{h}}w_{hi}m_{hi}\boldsymbol{\Delta }(\boldsymbol{\pi }_{hi}^{\ast }\left( \boldsymbol{\beta }\right) )\otimes \boldsymbol{x}_{hi}\boldsymbol{x}_{hi}^{T}, & \lambda =0\end{array}\right. .\end{aligned}$$The family of Wald-type test statistics for testing the null hypothesis given in (\[A\]) is given by $$W_{n}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})=(\boldsymbol{M}^{T}\widehat{\boldsymbol{\beta }}_{\lambda ,Q}-\boldsymbol{l})^{T}\left( 
\boldsymbol{M}^{T}\widehat{\mathbf{Q}}_{n,\lambda }(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})\boldsymbol{M}\right) ^{-1}(\boldsymbol{M}^{T}\widehat{\boldsymbol{\beta }}_{\lambda ,Q}-\boldsymbol{l})  \label{B}$$

For $\lambda =0,$ $\widehat{\boldsymbol{\beta }}_{\lambda =0,Q},$ is the maximum quasi weighted likelihood estimator of $\boldsymbol{\beta }$, with estimating equations given in (\[2.4\]). It is not difficult to see that $\mathbf{Q}_{\lambda =0}(\widehat{\boldsymbol{\beta }}_{\lambda =0,Q})$ is the Fisher information matrix and $W_{n}(\widehat{\boldsymbol{\beta }}_{\lambda =0,Q})$ will be the classical Wald test statistic.

Under the null hypothesis given in (\[A\]) the asymptotic 