distribution of the Wald-type test statistics $W_{n}(\widehat{\boldsymbol{\beta }}_{\lambda ,Q})$, defined in (\[B\]), is chi-square with $r$ degrees of freedom.

The proof is immediate using the asymptotic distribution of the minimum quasi weighted DPD estimator, presented in Theorem \[th2\], and taking into account the consistence of the matrix $\mathbf{Q}_{\lambda }\left( \widehat{\boldsymbol{\beta }}_{\lambda ,Q}\right) $ presented in Corollary \[cor1b\]. Based on the previous theorem the null hypothesis given in (\[A\]) will be rejected if $$W_{n}\left( \widehat{\boldsymbol{\beta }}_{\lambda ,Q}\right) >\chi
_{r,\alpha }^{2}.  \label{C}$$

Next we present a result in order to give an approximation of the power function for the test statisitcs given in (\[C\]). Let $\boldsymbol{\beta }^{\ast }$ $\in \Theta $ be such that $\boldsymbol{M}^{T}\boldsymbol{\beta }^{\ast }\neq \boldsymbol{c,}$ i.e., $\boldsymbol{\beta }^{\ast }$ does not belong to the null parameter space. Let us denote $$l_{\boldsymbol{\beta }_{1}}(\boldsymbol{\beta }_{2})=\left( \boldsymbol{M}^{T}\boldsymbol{\beta }_{1}-\boldsymbol{l}\right) ^{T}\left( \boldsymbol{M}^{T}\widehat{\mathbf{Q}}_{n,\lambda }\left( \boldsymbol{\beta }_{2}\right) 
\boldsymbol{M}\right) ^{-1}\left( \boldsymbol{M}^{T}\boldsymbol{\beta }_{1}-\boldsymbol{l}\right) .$$Then we have the following result.

\[th:c\] Let $\boldsymbol{\beta }^{\ast }\in \Theta $ with $\boldsymbol{M}^{T}\boldsymbol{\beta }^{\ast }\neq \boldsymbol{l}$ the true value of the parameter so that $\widehat{\boldsymbol{\beta }}_{\lambda ,Q}\underset{n\rightarrow \infty }{\overset{P}{\longrightarrow }}\boldsymbol{\beta }^{\ast }$. The power function of the test statistic given in (\[C\]), at $\boldsymbol{\beta }^{\ast }$, is given by $$Po_{W_{n}\left( \widehat{\boldsymbol{\beta }}_{\lambda ,Q}\right) 