Appendix
1.1 Proof of results
This Subappendix provides a sketch of the proofs of the theoretical derivations of the previous sections.
1.1.1 Proof of Theorem 2
By definition, the estimator \(\widetilde{\varvec{\theta }}_{c}^{\beta }\) , of (14), is obtained by the solution of the system of equations
$$\begin{aligned} \frac{1}{n}\sum _{i=1}^{n}\varPsi _{\beta }({\varvec{Y}}_{i}, \varvec{\theta })+{\varvec{M}}(\varvec{\theta })\varvec{\lambda }= & {} {\mathbf {0}}_{r} \nonumber \\ {\varvec{m}}(\varvec{\theta })= & {} {\mathbf {0}}_{r}, \end{aligned}$$
(48)
where \(\varPsi _{\beta }({\varvec{Y}}_{i},\varvec{\theta })\) is defined by (12) and \(\varvec{\lambda }\) is a vector of Lagrange multipliers, with \(\varvec{\lambda }\in {\mathbb {R}}^{r}\).
The aim now is to derive the asymptotic distribution of the RCMDPDE \( \widetilde{\varvec{\theta }}_{c}^{\beta }\). To proceed in such a direction, let \(\varvec{\theta }_{0}\in \varTheta \). Consider the Taylor expansion of \(\frac{1}{n}\sum _{i=1}^{n}\varPsi _{\beta }\left( {\varvec{Y}}_{i},\widetilde{\varvec{\theta }}_{c}^{\beta }\right) \) around the point \(\varvec{\theta }_{0}\). It gives,
$$\begin{aligned} \frac{1}{n}\sum _{i=1}^{n}\varPsi _{\beta }\left( {\varvec{Y}}_{i}, \widetilde{\varvec{\theta }}_{c}^{\beta }\right)&=\frac{1}{n} \sum _{i=1}^{n}\varPsi _{\beta }({\varvec{Y}}_{i},\varvec{\theta } _{0})+\frac{1}{n}\left( \frac{\partial }{\partial \varvec{\theta }^{T}} \sum _{i=1}^{n}\varPsi _{\beta }({\varvec{Y}}_{i},\varvec{\theta } _{0})\right) \left( \widetilde{\varvec{\theta }}_{c}^{\beta }- \varvec{\theta }_{0}\right) \\&\quad +o\left( \left\| \widetilde{{\varvec{\theta }}}_{c}^{\beta }-\varvec{\theta }_{0}\right\| ^{2}\right) , \end{aligned}$$
or
$$\begin{aligned} \sqrt{n}\frac{1}{n}\sum _{i=1}^{n}\varPsi _{\beta }\left( {\varvec{Y}} _{i},\widetilde{\varvec{\theta }}_{c}^{\beta }\right)= & {} \sqrt{n}\frac{1 }{n}\sum _{i=1}^{n}\varPsi _{\beta }({\varvec{Y}}_{i},{\varvec{\theta }}_{0})\\&+\,\sqrt{n}\frac{1}{n}\left( \frac{\partial }{\partial \varvec{\theta }^{T}}\sum _{i=1}^{n}\varPsi _{\beta }({\varvec{Y}} _{i},\varvec{\theta }_{0})\right) \left( \widetilde{\varvec{\theta }} _{c}^{\beta }-\varvec{\theta }_{0}\right) \\&+\,\sqrt{n}o\left( \left\| \widetilde{\varvec{\theta }}_{c}^{\beta }- \varvec{\theta }_{0}\right\| ^{2}\right) . \end{aligned}$$
Now, we are going to study the convergence in probability of the term \(\frac{ 1}{n}\frac{\partial }{\partial \varvec{\theta }}\sum _{i=1}^{n} \varPsi _{\beta }({\varvec{Y}}_{i},\varvec{\theta }_{0})\). Taking into account that
$$\begin{aligned} {\varvec{u}}(\varvec{\theta },{\varvec{Y}}_{i})=\frac{\partial c\ell (\varvec{\theta },{\varvec{Y}}_{i})}{\partial \varvec{\theta }}= \frac{\partial }{\partial \varvec{\theta }}\log {{\mathcal {C}}}{{\mathcal {L}}}({\varvec{\theta }},{\varvec{Y}}_{i}), \end{aligned}$$
straightforward algebraic manipulations entail that,
$$\begin{aligned}&\frac{1}{n}\frac{\partial }{\partial \varvec{\theta }} \sum _{i=1}^{n}\varPsi _{\beta }({\varvec{Y}}_{i},\varvec{\theta } _{0}) \\&\quad =\frac{1}{n}\frac{\partial }{\partial \varvec{\theta }}\left\{ \sum _{i=1}^{n}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }, {\varvec{Y}}_{\varvec{i}})^{\beta }{\varvec{u}}(\varvec{\theta }, {\varvec{Y}}_{\varvec{i}})-n\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta },{\varvec{y}}){{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}})^{1+\beta }d{\varvec{y}}\right\} \\&\quad =\frac{\beta }{n}\sum _{i=1}^{n}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }, \varvec{{\varvec{Y}}_{i}})^{\beta }{\varvec{u}}(\varvec{\theta }, \varvec{{\varvec{Y}}_{i}}){\varvec{u}}(\varvec{\theta }, \varvec{{\varvec{Y}}_{i}})^{T}\\&\qquad +\,\frac{1}{n}\sum _{i=1}^{n} {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },\varvec{{\varvec{Y}}_{i}})^{\beta } \frac{\partial }{\partial \varvec{\theta }^{T}}{\varvec{u}}( \varvec{\theta },\varvec{{\varvec{Y}}_{i}}) \\&\qquad -\,\int _{{\mathbb {R}}^{m}}\frac{\partial }{\partial \varvec{\theta }^{T}} {\varvec{u}}(\varvec{\theta },{\varvec{y}}){{\mathcal {C}}}{{\mathcal {L}}}({\varvec{\theta }},{\varvec{y}})^{1+\beta }d{\varvec{y}} \\&\qquad -\,(1+\beta )\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta }, {\varvec{y}}){\varvec{u}}(\varvec{\theta },{\varvec{y}})^{T} {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}}\varvec{)} ^{1+\beta }d{\varvec{y}}. \end{aligned}$$
If \(n\rightarrow \infty \), the above expression leads,
$$\begin{aligned} \frac{1}{n}\frac{\partial }{\partial \varvec{\theta }} \sum _{i=1}^{n}\varPsi _{\beta }({\varvec{Y}}_{i},\varvec{\theta } _{0})&\underset{n\rightarrow \infty }{\overset{\textit{P}}{ \longrightarrow }}\beta \int _{{\mathbb {R}}^{m}}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}}\varvec{)}^{1+\beta }{\varvec{u}}( \varvec{\theta },\varvec{y}){\varvec{u}}(\varvec{\theta } ,\varvec{y})^{T}d{\varvec{y}}\\&+\,\int _{{\mathbb {R}}^{m}}\frac{\partial }{ \partial \varvec{\theta }^{T}}{\varvec{u}}(\varvec{\theta } ,\varvec{y}){{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}} \varvec{)}^{1+\beta }d{\varvec{y}} \\&-\,\int _{{\mathbb {R}}^{m}}\frac{\partial }{\partial {\varvec{\theta }}^{T}}{\varvec{u}}(\varvec{\theta },\varvec{y}){{\mathcal {C}}}{{\mathcal {L}}}( \varvec{\theta },{\varvec{y}}\varvec{)}^{1+\beta }d {\varvec{y}} \\&-\,(1+\beta )\int _{{\mathbb {R}}^{m}}{\varvec{u}}({\varvec{\theta }},\varvec{y}){\varvec{u}}(\varvec{\theta },\varvec{y} )^{T}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}} \varvec{)}^{1+\beta }d\varvec{y}, \end{aligned}$$
or
$$\begin{aligned} \frac{1}{n}\frac{\partial }{\partial \varvec{\theta }} \sum _{i=1}^{n}\varPsi _{\beta }({\varvec{Y}}_{i},\varvec{\theta } _{0})\underset{n\rightarrow \infty }{\overset{\textit{p}}{ \longrightarrow }}-\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta } ,\varvec{y}){\varvec{u}}(\varvec{\theta },\varvec{y})^{T} {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}}\varvec{)} ^{1+\beta }d\varvec{y=}-{\varvec{H}}_{\beta }(\varvec{\theta }), \end{aligned}$$
in view of (10). Therefore,
$$\begin{aligned} \sqrt{n}\frac{1}{n}\sum _{i=1}^{n}\varPsi _{\beta }\left( {\varvec{Y}} _{i},\widetilde{\varvec{\theta }}_{c}^{\beta }\right) =&\,\sqrt{n}\frac{1}{n }\sum _{i=1}^{n}\varPsi _{\beta }({\varvec{Y}}_{i},\varvec{\theta } _{0})-\sqrt{n}{\varvec{H}}_{\beta }(\varvec{\theta }_{0})\left( \widetilde{\varvec{\theta }}_{c}^{\beta }-\varvec{\theta } _{0}\right) \\&+o\left( \sqrt{n}\left\| \widetilde{\varvec{\theta }} _{c}^{\beta }-\varvec{\theta }_{0}\right\| ^{2}\right) . \end{aligned}$$
But,
$$\begin{aligned} \frac{1}{n}\sum _{i=1}^{n}\varPsi _{\beta }\left( {\varvec{Y}}_{i}, \widetilde{\varvec{\theta }}_{c}^{\beta }\right) +{\varvec{M}}( \varvec{\theta }_{0})\widetilde{\varvec{\lambda }}_{n}=o_{p}(1), \end{aligned}$$
therefore
$$\begin{aligned} \sqrt{n}\frac{1}{n}\sum _{i=1}^{n}\varPsi _{\beta }\left( {\varvec{Y}} _{i},\varvec{\theta }_{0}\right) -\sqrt{n}{\varvec{H}}_{\beta }( \varvec{\theta }_{0})\left( \widetilde{\varvec{\theta }}_{c}^{\beta }-\varvec{\theta }_{0}\right) +\sqrt{n}{\varvec{M}}(\varvec{\theta }_{0})\widetilde{\varvec{\lambda }}_{n}=o_{p}(1), \end{aligned}$$
(49)
in view of (48). On the other hand, a Taylor series expansion of \( {\varvec{m}}(\widetilde{\varvec{\theta }}_{c}^{\beta })\) around the point \(\varvec{\theta }\) gives, subject to the assumption \( {\varvec{m}}(\varvec{\theta })={\mathbf {0}}_{r}\), that
$$\begin{aligned} \sqrt{n}{\varvec{g}}(\widetilde{\varvec{\theta }}_{c}^{\beta })=\sqrt{n }{\varvec{M}}(\varvec{\theta })^{T}\left( \widetilde{{\varvec{\theta }}}_{c}^{\beta }-\varvec{\theta }\right) +o_{p}(1), \end{aligned}$$
i.e.,
$$\begin{aligned} \sqrt{n}{\varvec{M}}(\varvec{\theta })^{T}\left( \widetilde{ \varvec{\theta }}_{c}^{\beta }-\varvec{\theta }\right) +o_{p}(1)= {\mathbf {0}}_{r}. \end{aligned}$$
(50)
Based on (49) and (50), we have
$$\begin{aligned} \left( \begin{array}{c@{\quad }c} {\varvec{H}}_{\beta }(\varvec{\theta }) &{} {\varvec{M}}( \varvec{\theta }) \\ {\varvec{M}}(\varvec{\theta }_{0})^{T} &{} {\varvec{O}}_{r\times r} \end{array} \right) \left( \begin{array}{c} \sqrt{n}\left( \widetilde{\varvec{\theta }}_{c}^{\beta }-{\varvec{\theta }}\right) \\ \sqrt{n}\varvec{\lambda }_{n} \end{array} \right) =\left( \begin{array}{c} \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\varPsi _{\beta }\left( {\varvec{Y}} _{i},\varvec{\theta }\right) \\ {\mathbf {0}}_{r} \end{array} \right) +o_{p}(1). \end{aligned}$$
Then,
$$\begin{aligned} \left( \begin{array}{c} \sqrt{n}\left( \widetilde{\varvec{\theta }}_{c}^{\beta }-\varvec{\theta }\right) \\ \sqrt{n}\varvec{\lambda }_{n} \end{array}\right) =\left( \begin{array}{c@{\quad }c} {\varvec{H}}_{\beta }(\varvec{\theta }_{0}) &{} {\varvec{M}}( \varvec{\theta }) \\ {\varvec{M}}(\varvec{\theta })^{T} &{} {\varvec{O}}_{r\times r} \end{array}\right) ^{-1}\left( \begin{array}{c} \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\varPsi _{\beta }\left( {\varvec{Y}} _{i},\varvec{\theta }\right) \\ {\mathbf {0}}_{r} \end{array} \right) +o_{p}(1) \end{aligned}$$
and
$$\begin{aligned} \left( \begin{array}{c} \sqrt{n}\left( \widetilde{\varvec{\theta }}_{c}^{\beta }-{\varvec{\theta }}_{0}\right) \\ \sqrt{n}\varvec{\lambda }_{n} \end{array} \right) =\left( \begin{array}{c@{\quad }c} {\varvec{P}}_{\beta }(\varvec{\theta }) &{} {\varvec{Q}}_{\beta }( \varvec{\theta }) \\ {\varvec{Q}}_{\beta }(\varvec{\theta })^{T} &{} {\varvec{R}} _{\beta }(\varvec{\theta }) \end{array} \right) \left( \begin{array}{c} \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\varPsi _{\beta }\left( {\varvec{Y}} _{i},\varvec{\theta }\right) \\ {\mathbf {0}}_{r} \end{array} \right) +o_{p}(1), \end{aligned}$$
(51)
where [cf. Sen and Singer 1993, p. 243, Eq. (5.6.24)],
$$\begin{aligned} {\varvec{P}}_{\beta }(\varvec{\theta })= & {} {\varvec{H}}_{\beta }( \varvec{\theta })^{-1}+{\varvec{H}}_{\beta }(\varvec{\theta } )^{-1}{\varvec{M}}(\varvec{\theta }){\varvec{R}}_{\beta }( \varvec{\theta })^{-1}{\varvec{M}}(\varvec{\theta })^{T} {\varvec{H}}_{\beta }(\varvec{\theta })^{-1} \\ {\varvec{Q}}_{\beta }(\varvec{\theta })= & {} -{\varvec{H}}_{\beta }(\varvec{\theta })^{-1}{\varvec{M}}(\varvec{\theta }) {\varvec{R}}_{\beta }(\varvec{\theta })^{-1} \\ {\varvec{R}}_{\beta }(\varvec{\theta })= & {} -{\varvec{M}}( \varvec{\theta })^{T}{\varvec{H}}_{\beta }(\varvec{\theta } )^{-1}{\varvec{M}}(\varvec{\theta }). \end{aligned}$$
These matrices can be also written as follows,
$$\begin{aligned} {\varvec{P}}_{\beta }(\varvec{\theta })= & {} {\varvec{H}}_{\beta }( \varvec{\theta })^{-1}-{\varvec{Q}}_{\beta }(\varvec{\theta }){\varvec{M}}(\varvec{\theta })^{T}{\varvec{H}}_{\beta }( \varvec{\theta })^{-1} \nonumber \\ {\varvec{Q}}_{\beta }(\varvec{\theta })= & {} -{\varvec{H}}_{\beta }(\varvec{\theta })^{-1}{\varvec{M}}(\varvec{\theta }) {\varvec{R}}_{\beta }(\varvec{\theta })^{-1} \nonumber \\ {\varvec{R}}_{\beta }(\varvec{\theta })= & {} -{\varvec{M}}(\varvec{\theta })^{T}{\varvec{H}}_{\beta }(\varvec{\theta })^{-1}{\varvec{M}}(\varvec{\theta }). \end{aligned}$$
(52)
On the other hand,
$$\begin{aligned} \left( \begin{array}{c} \frac{1}{\sqrt{n}}\sum _{i=1}^{n}\varPsi _{\beta }\left( {\varvec{Y}} _{i},\varvec{\varvec{\theta }}\right) \\ {\mathbf {0}}_{r} \end{array} \right) \underset{n\rightarrow \infty }{\overset{{\mathcal {L}}}{ \longrightarrow }}{\mathcal {N}}\left( \left( \begin{array}{c} {\mathbf {0}}_{p} \\ {\mathbf {0}}_{r} \end{array} \right) ,\left( \begin{array}{c@{\quad }c} {\varvec{J}}_{\beta }(\varvec{\varvec{\theta }}) &{} \varvec{ O} \\ {\varvec{O}} &{} {\varvec{O}} \end{array} \right) \right) , \end{aligned}$$
with \({\varvec{J}}_{\beta }(\varvec{\varvec{\theta }})= {\varvec{E}}_{\varvec{\varvec{\theta }}}\left[ \varPsi _{\beta }\left( {\varvec{Y}},\varvec{\varvec{\theta }}\right) \varPsi _{\beta }\left( {\varvec{Y}},\varvec{\varvec{\theta }}\right) ^{T}\right] \), given by (11). The above limiting relationship and (51) lead
$$\begin{aligned} \left( \begin{array}{c} \sqrt{n}\left( \widetilde{\varvec{\theta }}_{c}^{\beta }-{\varvec{\theta }}\right) \\ \sqrt{n}\varvec{\lambda }_{n} \end{array} \right) \underset{n\rightarrow \infty }{\overset{{\mathcal {L}}}{ \longrightarrow }}{\mathcal {N}}\left( \left( \begin{array}{c} {\mathbf {0}}_{p} \\ {\mathbf {0}}_{r} \end{array} \right) ,\varvec{\varSigma }_{\beta }\varvec{\theta } )\right) , \end{aligned}$$
(53)
where
$$\begin{aligned} \varvec{\varSigma }_{\beta }(\varvec{\varvec{\theta }})=\left( \begin{array}{c@{\quad }c} {\varvec{P}}_{\beta }(\varvec{\theta }) &{} {\varvec{Q}}_{\beta }( \varvec{\theta }) \\ {\varvec{Q}}_{\beta }(\varvec{\theta })^{T} &{} {\varvec{R}} _{\beta }(\varvec{\theta }) \end{array} \right) \left( \begin{array}{c@{\quad }c} {\varvec{J}}_{\beta }(\varvec{\varvec{\theta }}) &{} \varvec{ O} \\ {\varvec{O}} &{} {\varvec{O}} \end{array} \right) \left( \begin{array}{c@{\quad }c} {\varvec{P}}_{\beta }(\varvec{\theta }) &{} {\varvec{Q}}_{\beta }( \varvec{\theta }_{0}) \\ {\varvec{Q}}_{\beta }(\varvec{\theta })^{T} &{} {\varvec{R}} _{\beta }(\varvec{\theta }) \end{array} \right) , \end{aligned}$$
or
$$\begin{aligned} \varvec{\varSigma }_{\beta }(\varvec{\theta })=\left( \begin{array}{c@{\quad }c} {\varvec{P}}_{\beta }(\varvec{\theta }){\varvec{J}}_{\beta }( \varvec{\varvec{\theta }}){\varvec{P}}_{\beta }({\varvec{\theta }}) &{} {\varvec{P}}_{\beta }(\varvec{\theta }){\varvec{J}}_{\beta }(\varvec{\varvec{\theta }}){\varvec{Q}}_{\beta }( \varvec{\theta } ) \\ {\varvec{Q}}_{\beta }(\varvec{\theta })^{T}{\varvec{J}}_{\beta }( \varvec{\varvec{\theta }}){\varvec{P}}_{\beta }({\varvec{\theta }}) &{} {\varvec{Q}}_{\beta }(\varvec{\theta })^{T} {\varvec{J}}_{\beta }(\varvec{\theta }){\varvec{Q}} _{\beta }(\varvec{\theta }) \end{array} \right) . \end{aligned}$$
(54)
The above formulas (52), (53) and (54) lead to the theorem which derives the asymptotic distribution of the RCMDPDE \(\widetilde{ \varvec{\theta }}_{c}^{\beta }\).
1.1.2 Proof of Lemma 1
It is immediate to see that
$$\begin{aligned} {\varvec{E}}\left[ \varPsi _{\beta }({\varvec{Y}},\varvec{\theta }) \right] =\int _{{\mathbb {R}}^{m}}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}})^{\beta +1}{\varvec{u}}(\varvec{\theta },{\varvec{y}})d\varvec{ y}-\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta },{\varvec{y}}) {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}})^{\beta +1}d{\varvec{y}}= {\mathbf {0}} \end{aligned}$$
and
$$\begin{aligned} Cov\left( \varPsi _{\beta }({\varvec{Y}},\varvec{\theta })\right)= & {} {\varvec{E}}\left[ \varPsi _{\beta }\left( {\varvec{Y}},\varvec{\theta } \right) \varPsi _{\beta }\left( {\varvec{Y}},\varvec{\theta }\right) ^{T} \right] \\= & {} {\varvec{E}}\Bigg [ \left( {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{Y}})^{\beta }{\varvec{u}}(\varvec{\theta },{\varvec{Y}})-\int _{ {\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta },{\varvec{y}}){{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}})^{\beta +1}d{\varvec{y}}\right) \\&\times \left( {{\mathcal {C}}}{{\mathcal {L}}}({\varvec{\theta }},{\varvec{Y}})^{\beta }{\varvec{u}}(\varvec{\theta } \varvec{,Y)}-\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta }, {\varvec{y}}){{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}})^{\beta +1}d {\varvec{y}}\right) ^{T}\Bigg ]. \end{aligned}$$
This last expression is simplified, after elementary manipulations, to
$$\begin{aligned} Cov\left( \varPsi _{\beta }({\varvec{Y}},\varvec{\theta })\right) =&\, {\varvec{E}}\left[ {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{Y}} )^{2\beta }{\varvec{u}}(\varvec{\theta },{\varvec{Y}}){\varvec{u}}( \varvec{\theta },{\varvec{Y}})^{T}\right] \\&\text { }-\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta }, {\varvec{y}}){{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}})^{1+\beta }d {\varvec{y}}\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta }, {\varvec{y}})^{T}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}} )^{1+\beta }d{\varvec{y}} \\&-\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta }, {\varvec{y}}){{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}})^{1+\beta }d {\varvec{y}}\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta }, {\varvec{y}})^{T}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}} )^{1+\beta }d{\varvec{y}} \\&+\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta }, {\varvec{y}}){{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}})^{1+\beta }d {\varvec{y}}\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta }, {\varvec{y}})^{T}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}} )^{1+\beta }d{\varvec{y}} \\ =&\,\int _{{\mathbb {R}}^{m}}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}} )^{2\beta +1}{\varvec{u}}(\varvec{\theta },{\varvec{y}})\varvec{u }(\varvec{\theta },{\varvec{y}})^{T}d{\varvec{y}} \\&-\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta }, {\varvec{y}}){{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}})^{1+\beta }d {\varvec{y}}\int _{{\mathbb {R}}^{m}}{\varvec{u}}(\varvec{\theta }, {\varvec{y}})^{T}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta },{\varvec{y}} )^{1+\beta }d{\varvec{y}}. \end{aligned}$$
1.1.3 Proof of Theorem 3
By (18) it is immediate to see that
$$\begin{aligned} \sqrt{n}{\varvec{U}}_{n}^{\beta }\left( \varvec{\theta }_{0}\right) ^{T}\underset{n\rightarrow \infty }{\overset{{\mathcal {L}}}{\longrightarrow }} {\mathcal {N}}\left( {\varvec{0}},{\varvec{J}}_{\beta }(\varvec{\theta } _{0})\right) , \end{aligned}$$
and the result follows.
1.1.4 Proof of Theorem 4
Based on (53) we have
$$\begin{aligned} \sqrt{n}\widetilde{\varvec{\lambda }}_{n}\underset{n\rightarrow \infty }{ \overset{{\mathcal {L}}}{\longrightarrow }}{\mathcal {N}}\left( {\varvec{0}}, {\varvec{Q}}_{\beta }(\varvec{\theta })^{T}{\varvec{J}}_{\beta }( \varvec{\theta }){\varvec{Q}}_{\beta }(\varvec{\theta })\right) . \end{aligned}$$
(55)
By (48),
$$\begin{aligned} \frac{1}{n}\sum _{i=1}^{n}\varPsi _{\beta }\left( {\varvec{Y}}_{i}, \widetilde{\varvec{\theta }}_{c}^{\beta }\right) +{\varvec{M}}\left( \widetilde{\varvec{\theta }}_{c}^{\beta }\right) \widetilde{\varvec{\lambda }}_{n}=o_{p}(1), \end{aligned}$$
and by (19),
$$\begin{aligned} {\varvec{U}}_{n}^{\beta }\left( \widetilde{\varvec{\theta }} _{c}^{\beta }\right) =\frac{1}{n}\sum _{i=1}^{n}\varPsi _{\beta }\left( {\varvec{Y}}_{i},\widetilde{\varvec{\theta }}_{c}^{\beta }\right) . \end{aligned}$$
Therefore,
$$\begin{aligned} {\varvec{U}}_{n}^{\beta }\left( \widetilde{\varvec{\theta }} _{c}^{\beta }\right) ^{T}=-\widetilde{\varvec{\lambda }}_{n}^{T} {\varvec{M}}\left( \widetilde{\varvec{\theta }}_{c}^{\beta }\right) ^{T}+o_{p}(1) \end{aligned}$$
and
$$\begin{aligned} {\varvec{U}}_{n}^{\beta }\left( \widetilde{\varvec{\theta }} _{c}^{\beta }\right) ^{T}{\varvec{Q}}_{\beta }\left( \widetilde{ \varvec{\theta }}_{c}^{\beta }\right) =&\,\widetilde{\varvec{\lambda } }_{n}^{T}{\varvec{M}}\left( \widetilde{\varvec{\theta }}_{c}^{\beta }\right) ^{T}{\varvec{H}}\left( \widetilde{\varvec{\theta }} _{c}^{\beta }\right) ^{-1}{\varvec{M}}\left( \widetilde{\varvec{\theta }}_{c}^{\beta }\right) \\&\times \left[ {\varvec{M}}\left( \widetilde{{\varvec{\theta }}}_{c}^{\beta }\right) ^{T}{\varvec{H}}\left( \widetilde{ \varvec{\theta }}_{c}^{\beta }\right) ^{-1}{\varvec{M}}\left( \widetilde{\varvec{\theta }}_{c}^{\beta }\right) \right] ^{-1}+o_{p}(1) \\ =&\,\widetilde{\varvec{\lambda }}_{n}^{T}+o_{p}(1). \end{aligned}$$
hence,
$$\begin{aligned} R_{n}^{\beta }\left( \widetilde{\varvec{\theta }}_{c}^{\beta }\right) =n \widetilde{\varvec{\lambda }}_{n}^{T}\left( {\varvec{Q}}_{\beta }\left( \widetilde{\varvec{\theta }}_{c}^{\beta }\right) {\varvec{J}} _{\beta }\left( \widetilde{\varvec{\theta }}_{c}^{\beta }\right) {\varvec{Q}}_{\beta }\left( \widetilde{\varvec{\theta }}_{c}^{\beta }\right) \right) ^{-1}\widetilde{\varvec{\lambda }}_{n}+o_{p}(1), \end{aligned}$$
and the result follows by (55).
1.2 Computations in Sect. 4.1
We want to compute
$$\begin{aligned} \varvec{\varPsi } _{\beta }\left( {\varvec{Y}},\varvec{\theta }_{0}\right) = {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{\beta }{\varvec{u}}( \varvec{\theta }_{0},{\varvec{y}})-\int _{{\mathbb {R}}^{q}}{\varvec{u}}( \varvec{\theta }_{0},{\varvec{y}}){{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta } _{0},{\varvec{y}})^{1+\beta }d{\varvec{y}}. \end{aligned}$$
Accordingly to (22) in Martín et al. (2018)
$$\begin{aligned} {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})= & {} \prod \limits _{r=2}^{q}f_{Y_{r-1},Y_{r}\text { }}\left( y_{r-1},y_{y}\right) \\= & {} \prod \limits _{r=2}^{q}\frac{\sqrt{1-\rho ^{2}}}{2\pi \sigma ^{2}}\exp \left\{ -\frac{1}{2\sigma ^{2}}Q\left( y_{r-1},y_{r}\right) \right\} \\= & {} \frac{(1-\rho ^2)^{(q-1)/2}}{\sqrt{2\pi }^{(q-1)}\sigma ^{2(q-1)}}\exp \left\{ -\frac{1}{2\sigma ^{2}}Q\left( y_{r-1},y_{r}\right) \right\} \end{aligned}$$
with \(Q\left( y_{r-1},y_{r}\right) =(y_{r-1}-\mu )^{2}-2\rho (y_{r-1}-\mu )(y_{r}-\mu )+(y_{r}-\mu )^{2}.\)
The first term of \(\varvec{\varPsi } _{\beta }\left( {\varvec{Y}},\varvec{\theta }_{0}\right) \) is then easily computed as
$$\begin{aligned} {\varvec{u}}(\varvec{\theta },{\varvec{y}})&=\frac{\partial \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial {\varvec{\theta }}}\\&=\left( \frac{\partial \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \mu },\frac{\partial \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \sigma ^2},\frac{\partial \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \rho } \right) ^T, \end{aligned}$$
with
$$\begin{aligned} \frac{\partial \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \mu }&=\frac{1-\rho }{\sigma ^2}\sum _{r=2}^q\left\{ (y_{r}-\mu )+(y_{r-1}-\mu )\right\} ,\\ \frac{\partial \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \sigma ^2}&=-\frac{(q-1)}{\sigma ^2}\\&\quad +\frac{1}{2\sigma ^4}\sum _{r=2}^q\left\{ (y_{r}-\mu )^2-2\rho (y_{r-1}-\mu )(y_{r}-\mu )+(y_{r-1}-\mu )^2\right\} ,\\ \frac{\partial \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \rho }&=-\frac{(q-1)\rho }{1-\rho ^2}+\frac{1}{\sigma ^2}\sum _{r=2}^q(y_{r-1}-\mu )(y_{r}-\mu ). \end{aligned}$$
Further,
$$\begin{aligned} \int _{{\mathbb {R}}^{m}}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_0,{\varvec{y}}\varvec{)}^{\beta +1}{\varvec{u}}(\varvec{\theta }_0,{\varvec{y}}\varvec{)}d{\varvec{y}}&= \int _{{\mathbb {R}}^{m}}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_0,{\varvec{y}}\varvec{)}^{\beta +1}\frac{\partial \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_0,{\varvec{y}}\varvec{)}}{\partial \varvec{\theta }}d{\varvec{y}}\\&=\int _{{\mathbb {R}}^{m}}{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_0,{\varvec{y}}\varvec{)}^{\beta }\frac{\partial {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_0,{\varvec{y}}\varvec{)}}{\partial \varvec{\theta }_0}d{\varvec{y}} \nonumber \\&=\int _{{\mathbb {R}}^{m}}\frac{1}{\beta +1}\frac{\partial {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_0,{\varvec{y}}\varvec{)}^{\beta +1}}{\partial \varvec{\theta }}d{\varvec{y}}\\&=\frac{1}{\beta +1} \frac{\partial }{\partial \varvec{\theta }}\int _{{\mathbb {R}}^{m}} {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_0,{\varvec{y}}\varvec{)}^{\beta +1}d{\varvec{y}}, \nonumber \end{aligned}$$
so we need to compute
$$\begin{aligned} \frac{1}{\beta +1} \frac{\partial }{\partial \varvec{\theta }}\int _{{\mathbb {R}}^{m}} {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_0,{\varvec{y}}\varvec{)}^{\beta +1}d{\varvec{y}}. \end{aligned}$$
But
$$\begin{aligned} f_{Y_{r-1},Y_{r}\text { }}^{1+\beta }\left( y_{r-1},y_{y}\right)= & {} \left( \frac{1}{1+\beta }\right) ^{1+\beta }\left( \frac{\sqrt{1-\rho ^{2}}}{2\pi \sigma _{*}^{2}}\right) ^{1+\beta }\exp \left\{ -\frac{1}{2\sigma _{*}^{2}}Q\left( y_{r-1},y_{r}\right) \right\} \\= & {} \left( \frac{1}{1+\beta }\right) ^{1+\beta }\left( \frac{1}{2\pi \sigma _{**}^{2}\sqrt{1-\rho ^{2}}}\right) ^{\beta }g_{Y_{r-1},Y_{r}\text { } }\left( y_{r-1},y_{y}\right) \end{aligned}$$
with \(\sigma _{*}^{2}=\frac{\sigma ^{2}}{1+\beta },\) \(\sigma _{**}^{2}=\frac{\sigma _{*}^{2}}{1-\rho ^{2}}\) and \(g_{Y_{r-1},Y_{r}\text { } }\left( y_{r-1},y_{y}\right) \) the density function corresponding to a bidimensional normal density function with vector mean \(\varvec{\mu }=\left( \mu ,\mu \right) ^{T}\) and variance–covariance matrix
$$\begin{aligned} \left( \begin{array}{c@{\quad }c} \sigma _{**}^{2} &{} \rho \sigma _{**}^{2} \\ \rho \sigma _{**}^{2} &{} \sigma _{**}^{2} \end{array} \right) . \end{aligned}$$
It is, therefore, easy to see that
$$\begin{aligned} \int _{{\mathbb {R}}^{m}} {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_0,{\varvec{y}}\varvec{)}^{\beta +1}d{\varvec{y}}&=\left( \frac{1}{1+\beta }\right) ^{(1+\beta )(q-1) }\left( \frac{1}{2\pi \sigma _{**}^{2}\sqrt{1-\rho ^{2}}}\right) ^{\beta (q-1)}\\&=\left( \frac{1}{1+\beta }\right) ^{(q-1)}\left( \frac{\sqrt{1-\rho ^2}}{\sqrt{2\pi }\sigma ^2}\right) ^{\beta (q-1)}, \end{aligned}$$
and the second term of \(\varvec{\varPsi } _{\beta }\left( {\varvec{Y}},\varvec{\theta }_{0}\right) \) is given by
$$\begin{aligned}&\frac{1}{\beta +1} \frac{\partial }{\partial \varvec{\theta }}\int _{{\mathbb {R}}^{m}} {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_0,{\varvec{y}}\varvec{)}^{\beta +1}d{\varvec{y}}\\&\quad =\left( 0,-\frac{\left( q-1\right) \beta }{\left( 1+\beta \right) ^{q}\sigma ^{2}} \left( \frac{\sqrt{1-\rho ^{2}}}{\sqrt{2\pi } \sigma ^{2}}\right) ^{\left( q-1\right) \beta },-\frac{\left( q-1\right) \beta \rho }{\left( 1+\beta \right) ^{q}\left( 1-\rho ^{2}\right) }\left( \frac{\sqrt{1-\rho ^{2}}}{\sqrt{2\pi } \sigma ^{2}} \right) ^{\left( q-1\right) \beta } \right) . \end{aligned}$$
1.2.1 Computations for complex composite null hypothesis
We want to compute the matrix
$$\begin{aligned} \widehat{{\varvec{H}}}_{\beta }(\varvec{\theta }_{0})=\frac{1}{n} \sum \limits _{i=1}^{n}\frac{\partial \varvec{\varPsi }_{\beta }({\varvec{Y}}_{i},\varvec{\theta }_{0})}{\partial \varvec{\theta }^{T}}, \end{aligned}$$
so we need to obtain the value of
$$\begin{aligned} \frac{\partial \varvec{\varPsi }_{\beta }({\varvec{Y}},\varvec{\theta }_{0})}{\partial \varvec{\theta }^{T}}&=\frac{\partial }{\partial \varvec{\theta }^{T}} \left\{ {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{\beta }{\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})-\int _{{\mathbb {R}}^{q}}{\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}}){{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{1+\beta }d{\varvec{y}}\right\} \\&=\frac{\partial {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{\beta }{\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \varvec{\theta }^{T}}-\frac{\partial }{\partial \varvec{\theta }^{T}}\left\{ \int _{{\mathbb {R}}^{q}}{\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}}){{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{1+\beta }d{\varvec{y}}\right\} . \end{aligned}$$
On the one hand
$$\begin{aligned} \frac{\partial {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{\beta }{\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \varvec{\theta }^{T}}&=\frac{\partial {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{\beta }}{\partial \varvec{\theta }^{T}}{\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})+{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{\beta }\frac{\partial {\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \varvec{\theta }^{T}}\\&=\beta {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{\beta }{\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})^T{\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})+{{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{\beta }\frac{\partial {\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \varvec{\theta }^{T}}\\&={{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{\beta }\left\{ \beta {\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})^T{\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})+\frac{\partial {\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \varvec{\theta }^{T}} \right\} . \end{aligned}$$
Let \(\varvec{\varUpsilon }=\dfrac{\partial {\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \varvec{\theta }^{T}}\), the elements of \(\varvec{\varUpsilon }\) are given by
$$\begin{aligned} \varvec{\varUpsilon }_{(1,1)}&=\frac{\partial ^2 \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \mu ^2}=\frac{2(\rho -1)(q-1)}{\sigma ^2}\\ \varvec{\varUpsilon }_{(2,2)}&=\frac{\partial ^2 \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial (\sigma ^2)^2}=\frac{q-1}{\sigma ^2}\\&\quad -\frac{1}{\sigma ^6}\sum _{r=2}^q\left\{ (y_r-\mu )^2-2\rho (y_{r-1}-\mu )(y_r-\mu )+(y_{r-1}-\mu )^2 \right\} \\ \varvec{\varUpsilon }_{(3,3)}&=\frac{\partial ^2 \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \rho ^2}=\frac{(1-q)(1+\rho ^2)}{(1-\rho ^2)^2}\\ \varvec{\varUpsilon }_{(1,2)}&=\varvec{\varUpsilon }_{(2,1)}=\frac{\partial ^2 \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \mu \partial \sigma ^2}=\frac{\rho -1}{\sigma ^4}\sum _{r=2}^q\left\{ (y_r-\mu )+(y_{r-1}-\mu )\right\} \\ \varvec{\varUpsilon }_{(1,3)}&=\varvec{\varUpsilon }_{(3,1)}=\frac{\partial ^2 \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \mu \partial \rho }=\frac{-1}{\sigma ^2}\sum _{r=2}^q\left\{ (y_r-\mu )+(y_{r-1}-\mu )\right\} \\ \varvec{\varUpsilon }_{(2,3)}&=\varvec{\varUpsilon }_{(3,2)}=\frac{\partial ^2 \log {{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})}{\partial \mu \partial \rho }=\frac{-1}{\sigma ^4}\sum _{r=2}^q\left\{ (y_r-\mu )+(y_{r-1}-\mu )\right\} .\\ \end{aligned}$$
On the other hand, let \(\varvec{\varGamma }=\dfrac{\partial }{\partial \varvec{\theta }^{T}}\left\{ \int _{{\mathbb {R}}^{q}}{\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}}){{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{1+\beta }d{\varvec{y}}\right\} \). The elements of \(\varvec{\varGamma }\) are given by
$$\begin{aligned} \varvec{\varGamma }_{(1,1)}&=\varvec{\varGamma }_{(1,2)}=\varvec{\varGamma }_{(2,1)}=\varvec{\varGamma }_{(1,3)}=\varvec{\varGamma }_{(3,1)}=0\\ \varvec{\varGamma }_{(2,2)}&=\dfrac{(q-1)(q+\beta -1)}{(\beta +1)^q\sigma ^4}\left[ \dfrac{\sqrt{1-\rho ^2}}{2\pi \sigma ^2}\right] ^{(q-1)\beta }\\ \varvec{\varGamma }_{(3,3)}&=\dfrac{\beta (q-1)((\beta q -\beta -1)\rho ^2-1)}{(\beta +1)^q(1-\rho ^2)^2}\left[ \dfrac{\sqrt{1-\rho ^2}}{2\pi \sigma ^2}\right] ^{(q-1)\beta }\\ \varvec{\varGamma }_{(2,3)}&=\varvec{\varGamma }_{(3,2)}=\dfrac{\beta ^2(q-1)^2\rho }{(\beta +1)^q(1-\rho ^2)\sigma ^2}\left[ \dfrac{\sqrt{1-\rho ^2}}{2\pi \sigma ^2}\right] ^{(q-1)\beta }\\ \end{aligned}$$
and \(\frac{\partial \varvec{\varPsi }_{\beta }({\varvec{Y}}_{i},\varvec{\theta }_{0})}{\partial \varvec{\theta }^{T}}\) is given by
$$\begin{aligned} \frac{\partial \varvec{\varPsi }_{\beta }({\varvec{Y}}_{i},\varvec{\theta }_{0})}{\partial \varvec{\theta }^{T}}={{\mathcal {C}}}{{\mathcal {L}}}(\varvec{\theta }_{0},{\varvec{y}})^{\beta }\left\{ \beta {\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})^T{\varvec{u}}(\varvec{\theta }_{0},{\varvec{y}})+\varvec{\varUpsilon }\right\} -\varvec{\varGamma }. \end{aligned}$$
1.3 Example R code
The following R codes are provided in order to help the reader to implement the proposed CMDPDE and the corresponding Rao-type tests for any practical applications. These codes were used for the simulation studies presented in Table 4 of Sect. 5 of the paper.
