Improved score tests for exponential family nonlinear models

Abstract

This paper focuses on the corrections to the score test statistic under the exponential family nonlinear model. We use Monte Carlo simulations to compare the corrected statistics and their uncorrected versions and to examine the impact of the number of nuisance parameters on their finite-sample behaviors for the normal nonlinear regression model. The numerical results have shown that the corrected score statistic performs better than the uncorrected version. Finally, we perform a statistical analysis with real data by using the approach proposed in the article.

Keywords:

• Bartlett-type correction
• exponential family
• Monte Carlo simulation
• nonlinear model
• score test

Acknowledgments

This paper is based on part of Cavalcanti’s doctoral thesis submitted to the Instituto de Matemática e Estatística, Universidade de São Paulo, under the supervision of the second author. The authors thank the Editors and referees for their constructive comments on an earlier version of this manuscript which resulted in this improved version. Also, we thank Prof. Dr. Michelli Barros (UFCG) for valuable help in simulation program.

Appendix

 

A. Cumulants of log-likelihood derivatives in exponential family nonlinear models

Let $\mathcal{L}=\mathcal{L}(\mathit{\beta },\mathit{\delta },\gamma )$ be the log-likelihood (4(4) $$\mathcal{L}(\mathit{\beta },\mathit{\delta },\gamma )=-\frac{1}{2}\sum _{\ell =1}^n\left[\frac{q_{\ell }}{\gamma }{d}_{\ell }+\mathrm{t}(y_{\ell })+\hspace{0.17em}\text{log}\hspace{0.17em}\left(\frac{\gamma }{q_{\ell }}\right)\right],\hspace{1em}\ell =1,\dots ,n$$(4) ). We hall use the standard notation for joint cumulants of log-likelihood derivatives, i.e., ${\kappa }_{rs}=\mathrm{E}({\partial }^2\mathcal{L}/\partial {\beta }_r\partial {\beta }_s),\hspace{0.5em}{\kappa }_{r,s}=\mathrm{E}(\partial \mathcal{L}/\partial {\beta }_r\partial \mathcal{L}/\partial {\beta }_s),\hspace{0.5em}{\kappa }_{\mathit{rst}}=\mathrm{E}({\partial }^3\mathcal{L}/\partial {\beta }_r\partial {\beta }_s\partial {\beta }_t),\hspace{0.5em}{\kappa }_{r,s,t}=\mathrm{E}(\partial \mathcal{L}/\partial {\beta }_r\partial \mathcal{L}/\partial {\beta }_s\partial \mathcal{L}/\partial {\beta }_t),\hspace{0.5em}{\kappa }_{r,st}=\mathrm{E}(\partial \mathcal{L}/\partial {\beta }_r{\partial }^2\mathcal{L}/\partial {\beta }_s\partial {\beta }_t),\hspace{0.5em}{\kappa }_{rs}^{(t)}=\partial {\kappa }_{rs}/\partial {\beta }_t,$ and so on. We obtain the following cumulants: $${\kappa }_{rs}=-\frac{1}{\gamma }\sum _{\ell =1}^n{x}_{\ell r}^{*}{q}_{\ell }{w}_{\ell }{x}_{\ell s}^{*},\hspace{1em}{\kappa }_{RS}=-\frac{1}{2}\sum _{\ell =1}^n\frac{q_{{\ell }_{RS}}}{q_{\ell }},\hspace{1em}{\kappa }_{\gamma \gamma }=-\frac{n}{2{\gamma }^2},\hspace{1em}{\kappa }_{rS}={\kappa }_{r\gamma }={\kappa }_{R\gamma }=0$$ where $x_{\ell r}^{*}=\partial {\eta }_{\ell }/\partial {\beta }^r$ and $q_{{\ell }_{RS}}={\partial }^2{q}_{\ell }/\partial {\delta }^R\partial {\delta }^S,$ for $r=1,\dots ,p$ and $R,S=1,\dots ,q.$ The third-order cumulants are $$\begin{array}{c}{\kappa }_{\mathit{rst}}=-\frac{1}{\gamma }\sum _{\ell =1}^n\{q_{\ell }(f_{\ell }+2g_{\ell })x_{\ell r}^{*}{x}_{\ell s}^{*}{x}_{\ell t}^{*}+q_{\ell }{w}_{\ell }(x_{\ell rs}^{**}{x}_{\ell t}^{*}+x_{\ell rt}^{**}{x}_{\ell s}^{*}+x_{\ell st}^{**}{x}_{\ell r}^{*})\},\hspace{1em}{\kappa }_{\gamma \gamma \gamma }=\frac{2n}{{\gamma }^3},\\ {\kappa }_{\mathit{RST}}=-\frac{1}{2}\sum _{\ell =1}^n\frac{q_{{\ell }_{\mathit{RST}}}}{q_{\ell }},\hspace{1em}{\kappa }_{\mathit{rsT}}=-\frac{1}{\gamma }\sum _{\ell =1}^n{x}_{\ell r}^{*}{q}_{{\ell }_T}{w}_{\ell }{x}_{\ell s}^{*},\hspace{1em}{\kappa }_{rs\gamma }=\frac{1}{{\gamma }^2}\sum _{\ell =1}^n{x}_{\ell r}^{*}{q}_{\ell }{w}_{\ell }{x}_{\ell s}^{*}\\ {\kappa }_{RS\gamma }=\frac{1}{2\gamma }\sum _{\ell =1}^n\frac{q_{{\ell }_{RS}}}{q_{\ell }},\hspace{1em}{\kappa }_{\gamma ,\gamma \gamma }=-\frac{n}{{\gamma }^3},\hspace{1em}{\kappa }_{\gamma ,\gamma ,\gamma }=\frac{n}{{\gamma }^3},\hspace{1em}{\kappa }_{\gamma ,RS}=-\frac{1}{2\gamma }\sum _{\ell =1}^n\frac{q_{{\ell }_{RS}}}{q_{\ell }}\\ {\kappa }_{R\gamma \gamma }={\kappa }_{r\gamma \gamma }={\kappa }_{\mathit{RSt}}={\kappa }_{rS\gamma }=0,\hspace{1em}{\kappa }_{r,s,t}=\frac{1}{\gamma }\sum _{\ell =1}^n{q}_{\ell }(f_{\ell }-g_{\ell })x_{\ell r}^{*}{x}_{\ell s}^{*}{x}_{\ell t}^{*},\\ {\kappa }_{R,S,T}=\frac{1}{2}\sum _{\ell =1}^n\{\frac{q_{{\ell }_{\mathit{RST}}}}{q_{\ell }}-\frac{q_{{\ell }_{RS}}{q}_{{\ell }_T}}{q_{\ell }^2}-\frac{q_{{\ell }_{RT}}{q}_{{\ell }_S}}{q_{\ell }^2}-\frac{q_{{\ell }_{ST}}{q}_{{\ell }_R}}{q_{\ell }^2}\},\hspace{1em}{\kappa }_{r,s,T}=-\frac{1}{\gamma }\sum _{\ell =1}^n{q}_{{\ell }_T}{x}_{\ell r}^{*}{x}_{\ell s}^{*}{w}_{\ell },\\ {\kappa }_{r,s,\gamma }=\frac{1}{{\gamma }^2}\sum _{\ell =1}^n{q}_{\ell }{x}_{\ell r}^{*}{x}_{\ell s}^{*}{w}_{\ell },\hspace{1em}{\kappa }_{R,S,\gamma }=\frac{1}{\gamma }\sum _{\ell =1}^n\frac{q_{{\ell }_{RS}}}{q_{\ell }},\hspace{1em}{\kappa }_{r,S,\gamma }={\kappa }_{r,\gamma ,\gamma }={\kappa }_{r,S,T}={\kappa }_{R,\gamma ,\gamma }=0,\\ {\kappa }_{r,st}=\frac{1}{\gamma }\sum _{\ell =1}^n{q}_{\ell }(g_{\ell }{x}_{\ell r}^{*}{x}_{\ell s}^{*}{x}_{\ell t}^{*}+w_{\ell }{x}_{\ell r}^{*}{x}_{{\ell }_{st}}^{**}),\hspace{1em}{\kappa }_{R,ST}=\frac{1}{2}\sum _{\ell =1}^n\frac{q_{{\ell }_R}{q}_{{\ell }_{ST}}}{q_{\ell }^2},\\ {\kappa }_{r,sT}=\frac{1}{\gamma }\sum _{\ell =1}^n{x}_{\ell r}^{*}{q}_{{\ell }_T}{w}_{\ell }{x}_{\ell s}^{*},\hspace{1em}{\kappa }_{r,s\gamma }=-\frac{1}{{\gamma }^2}\sum _{\ell =1}^n{q}_{\ell }{x}_{\ell r}^{*}{x}_{\ell s}^{*}{w}_{\ell },\hspace{1em}{\kappa }_{R,S\gamma }=-\frac{1}{2\gamma }\sum _{\ell =1}^n\frac{q_{{\ell }_R}{q}_{{\ell }_S}}{q_{\ell }^2},\\ {\kappa }_{r,ST}={\kappa }_{r,S\gamma }={\kappa }_{r,\gamma \gamma }={\kappa }_{R,st}={\kappa }_{R,St}={\kappa }_{R,s\gamma }={\kappa }_{R,\gamma \gamma }={\kappa }_{\gamma ,rs}={\kappa }_{\gamma ,rS}={\kappa }_{\gamma ,\gamma r}={\kappa }_{\gamma ,\gamma R}=0\end{array}$$ where $x_{\ell rs}^{**}={\partial }^2{\eta }_{\ell }/\partial {\beta }^r\partial {\beta }^s,q_{{\ell }_{\mathit{RST}}}={\partial }^3{q}_{\ell }/\partial {\delta }^R\partial {\delta }^S\partial {\delta }^T,$ for $r,s,t=1,\dots ,p$ and $R,S,T=1\dots ,q$ and the quantities $q_{\ell },f_{\ell }$ and $g_{\ell },$ $\ell =1,\dots ,n,$ are defined by Equations (4)(4) $$\mathcal{L}(\mathit{\beta },\mathit{\delta },\gamma )=-\frac{1}{2}\sum _{\ell =1}^n\left[\frac{q_{\ell }}{\gamma }{d}_{\ell }+\mathrm{t}(y_{\ell })+\hspace{0.17em}\text{log}\hspace{0.17em}\left(\frac{\gamma }{q_{\ell }}\right)\right],\hspace{1em}\ell =1,\dots ,n$$(4) and (9)(9) $$\begin{array}{c}{f}_{\ell }=V_{\ell }^{-1}\left(\frac{d{\mu }_{\ell }}{d{\eta }_{\ell }}\right)\frac{d^2{\mu }_{\ell }}{d{\eta }_{\ell }^2},\hspace{1em}{g}_{\ell }=V_{\ell }^{-1}\left(\frac{d{\mu }_{\ell }}{d{\eta }_{\ell }}\right)\frac{d^2{\mu }_{\ell }}{d{\eta }_{\ell }^2}-V_{\ell }^{-2}{\left(\frac{d{\mu }_{\ell }}{d{\eta }_{\ell }}\right)}^3\frac{d{V}_{\ell }}{d{\mu }_{\ell }},\\ b_{\ell }=V_{\ell }^{-3}{\left(\frac{d{\mu }_{\ell }}{d{\eta }_{\ell }}\right)}^4\left\{{(\frac{d{V}_{\ell }}{d{\mu }_{\ell }})}^2+V_{\ell }\frac{d^2{V}_{\ell }}{d{\mu }_{\ell }^2}\right\}\end{array}$$(9) , respectively. Further, the fourth-order cumulants are given by $$\begin{array}{c}{\kappa }_{\mathit{rstu}}=-\frac{1}{\gamma }\sum _{\ell =1}^n{q}_{\ell }\{{\left[\left(\frac{{\partial }^{2}w}{\partial {\eta }^2}\right)+\left(\frac{{\partial }^2\mu }{\partial {\eta }^2}\right)\left(\frac{{\partial }^2\theta }{\partial {\eta }^2}\right)+2\left(\frac{\partial \mu }{\partial \eta }\right)\left(\frac{{\partial }^3\theta }{\partial {\eta }^3}\right)\right]}_{\ell }{x}_{{\ell }_r}^{*}{x}_{{\ell }_s}^{*}{x}_{{\ell }_t}^{*}{x}_{{\ell }_u}^{*}\\ +w_{\ell }\left(x_{{\ell }_{\mathit{rst}}}^{***}{x}_{{\ell }_u}^{*}+x_{{\ell }_{\mathit{rsu}}}^{***}{x}_{{\ell }_t}^{*}+x_{{\ell }_{\mathit{rut}}}^{***}{x}_{{\ell }_s}^{*}+x_{{\ell }_{\mathit{ust}}}^{***}{x}_{{\ell }_r}^{*}+x_{{\ell }_{rs}}^{**}{x}_{{\ell }_{tu}}^{**}+x_{{\ell }_{rt}}^{**}{x}_{{\ell }_{su}}^{**}+x_{{\ell }_{ru}}^{**}{x}_{{\ell }_{st}}^{**}\right)\\ +\left(x_{{\ell }_{rs}}^{**}{x}_{{\ell }_t}^{*}{x}_{{\ell }_u}^{*}+x_{{\ell }_{ts}}^{**}{x}_{{\ell }_r}^{*}{x}_{{\ell }_u}^{*}+x_{{\ell }_{us}}^{**}{x}_{{\ell }_r}^{*}{x}_{{\ell }_t}^{*}+x_{{\ell }_{ru}}^{**}{x}_{{\ell }_s}^{*}{x}_{{\ell }_t}^{*}+x_{{\ell }_{tu}}^{**}{x}_{{\ell }_r}^{*}{x}_{{\ell }_s}^{*}+x_{{\ell }_{rt}}^{**}{x}_{{\ell }_s}^{*}{x}_{{\ell }_u}^{*}\right)\\ \times {\left[2\left(\frac{\partial \mu }{\partial \eta }\right)\left(\frac{{\partial }^2\theta }{\partial {\eta }^2}\right)+\left(\frac{\partial \theta }{\partial \eta }\right)\left(\frac{{\partial }^2\mu }{\partial {\eta }^2}\right)\right]}_{\ell }\},\\ {\kappa }_{\mathit{RSTU}}=-\frac{1}{2}\sum _{\ell =1}^n\frac{q_{{\ell }_{\mathit{RSTU}}}}{q_{\ell }},\hspace{1em}{\kappa }_{\mathit{\text{RSγγ}}}=-\frac{1}{{\gamma }^2}\sum _{\ell =1}^n\frac{q_{{\ell }_{RS}}}{q_{\ell }}\hspace{1em}{\kappa }_{r,s,t,u}=\frac{1}{\gamma }\sum _{\ell =1}^n{q}_{\ell }{b}_{\ell }{x}_{{\ell }_r}^{*}{x}_{{\ell }_s}^{*}{x}_{{\ell }_t}^{*}{x}_{{\ell }_u}^{*},\\ {\kappa }_{R,S,T,U}=\frac{1}{2}\sum _{\ell =1}^n(\frac{q_{{\ell }_{\mathit{RSTU}}}}{q_{\ell }}-\frac{q_{{\ell }_{\mathit{RST}}}{q}_{{\ell }_U}}{q_{\ell }^2}-\frac{q_{{\ell }_{\mathit{RSU}}}{q}_{{\ell }_T}}{q_{\ell }^2}-\frac{q_{{\ell }_{\mathit{RTU}}}{q}_{{\ell }_S}}{q_{\ell }^2}-\frac{q_{{\ell }_{\mathit{STU}}}{q}_{{\ell }_R}}{q_{\ell }^2}\\ +\frac{2q_{{\ell }_R}{q}_{{\ell }_S}{q}_{{\ell }_{TU}}}{q_{\ell }^3}+\frac{2q_{{\ell }_R}{q}_{{\ell }_T}{q}_{{\ell }_{SU}}}{q_{\ell }^3}+\frac{2q_{{\ell }_R}{q}_{{\ell }_U}{q}_{{\ell }_{ST}}}{q_{\ell }^3}+\frac{2q_{{\ell }_S}{q}_{{\ell }_T}{q}_{{\ell }_{RU}}}{q_{\ell }^3}\\ +\frac{2q_{{\ell }_S}{q}_{{\ell }_U}{q}_{{\ell }_{RT}}}{q_{\ell }^3}+\frac{2q_{{\ell }_T}{q}_{{\ell }_U}{q}_{{\ell }_{RS}}}{q_{\ell }^3}-\frac{q_{{\ell }_{RS}}{q}_{{\ell }_{TU}}}{q_{\ell }^2}-\frac{q_{{\ell }_{RT}}{q}_{{\ell }_{SU}}}{q_{\ell }^2}-\frac{q_{{\ell }_{RU}}{q}_{{\ell }_{ST}}}{q_{\ell }^2}),\end{array}$$ $$\begin{array}{c}{\kappa }_{r,s,T,U}=\frac{2}{\gamma }\sum _{\ell =1}^n{q}_{\ell }{w}_{\ell }{x}_{{\ell }_r}^{*}{x}_{{\ell }_s}^{*}\frac{q_{{\ell }_T}{q}_{{\ell }_U}}{q_{\ell }^2},\hspace{1em}{\kappa }_{r,s,\gamma ,\gamma }=\frac{2}{{\gamma }^3}\sum _{\ell =1}^n{q}_{\ell }{w}_{\ell }{x}_{{\ell }_r}^{*}{x}_{{\ell }_s}^{*},\hspace{1em}{\kappa }_{R,S,\gamma ,\gamma }=\frac{3}{{\gamma }^2}\sum _{\ell =1}^n\frac{q_{{\ell }_{RS}}}{q_{\ell }},\\ {\kappa }_{r,s,tu}=\frac{1}{\gamma }\sum _{\ell =1}^n{q}_{\ell }\left\{(h_{\ell }-b_{\ell })x_{{\ell }_r}^{*}{x}_{{\ell }_s}^{*}{x}_{{\ell }_t}^{*}{x}_{{\ell }_u}^{*}+(f_{\ell }-g_{\ell })x_{{\ell }_r}^{*}{x}_{{\ell }_s}^{*}{x}_{{\ell }_{tu}}^{**}\right\},\hspace{1em}{\kappa }_{R,S,\gamma \gamma }=-\frac{2}{{\gamma }^2}\sum _{\ell =1}^n\frac{q_{{\ell }_{RS}}}{q_{\ell }},\\ {\kappa }_{R,S,TU}=-\sum _{\ell =1}^n\frac{q_{{\ell }_R}{q}_{{\ell }_S}{q}_{{\ell }_{TU}}}{q_{\ell }^3},\hspace{1em}{\kappa }_{r,s,TU}=-\frac{1}{\gamma }\sum _{\ell =1}^n{w}_{\ell }{x}_{{\ell }_r}^{*}{x}_{{\ell }_s}^{*}{q}_{{\ell }_{TU}},\hspace{1em}{\kappa }_{r,s,\gamma \gamma }=-\frac{2}{{\gamma }^3}\sum _{\ell =1}^n{q}_{\ell }{w}_{\ell }{x}_{{\ell }_r}^{*}{x}_{{\ell }_s}^{*}\end{array}$$ where $x_{\ell \mathit{rst}}^{***}={\partial }^3{\eta }_{\ell }/\partial {\beta }^r\partial {\beta }^s{\beta }^t,q_{{\ell }_{\mathit{RSTU}}}={\partial }^4{q}_{\ell }/\partial {\delta }^R\partial {\delta }^S\partial {\delta }^T\partial {\delta }^U,$ for $r,s,t,u=1,\dots ,p$ and $R,S,T,U=1,\dots ,q,$ and the quantities $b_{\ell }$ and $h_{\ell }$ defined by Equations (9)(9) $$\begin{array}{c}{f}_{\ell }=V_{\ell }^{-1}\left(\frac{d{\mu }_{\ell }}{d{\eta }_{\ell }}\right)\frac{d^2{\mu }_{\ell }}{d{\eta }_{\ell }^2},\hspace{1em}{g}_{\ell }=V_{\ell }^{-1}\left(\frac{d{\mu }_{\ell }}{d{\eta }_{\ell }}\right)\frac{d^2{\mu }_{\ell }}{d{\eta }_{\ell }^2}-V_{\ell }^{-2}{\left(\frac{d{\mu }_{\ell }}{d{\eta }_{\ell }}\right)}^3\frac{d{V}_{\ell }}{d{\mu }_{\ell }},\\ b_{\ell }=V_{\ell }^{-3}{\left(\frac{d{\mu }_{\ell }}{d{\eta }_{\ell }}\right)}^4\left\{{(\frac{d{V}_{\ell }}{d{\mu }_{\ell }})}^2+V_{\ell }\frac{d^2{V}_{\ell }}{d{\mu }_{\ell }^2}\right\}\end{array}$$(9) and (10)(10) $$h_{\ell }=V_{\ell }^{-2}\frac{d{V}_{\ell }}{d{\mu }_{\ell }}{\left(\frac{d{\mu }_{\ell }}{d{\eta }_{\ell }}\right)}^2\frac{d^2{\mu }_{\ell }}{d{\eta }_{\ell }^2}+V_{\ell }^{-2}\frac{d^2{V}_{\ell }}{d{\mu }_{\ell }^2}{\left(\frac{d{\mu }_{\ell }}{d{\eta }_{\ell }}\right)}^4$$(10) , respectively.

The equation $m_{\ell }=\hspace{0.17em}\text{exp}\hspace{0.17em}(z_{\ell }^{\top }\delta )$ gives $q_{\ell }=\hspace{0.17em}\text{exp}\hspace{0.17em}\{-{(z_{\ell }-\overline{z})}^{\top }\delta \},q_{{\ell }_R}=-{(z_{\ell }-\overline{z})}_R\hspace{0.17em}\text{exp}\hspace{0.17em}\{-{(z_{\ell }-\overline{z})}^{\top }\delta \}=-{(z_{\ell }-\overline{z})}_R{q}_{\ell },q_{{\ell }_{RS}}={(z_{\ell }-\overline{z})}_{RS}{q}_{\ell },q_{{\ell }_{\mathit{RST}}}=-{(z_{\ell }-\overline{z})}_{\mathit{RST}}{q}_{\ell }$ and $q_{{\ell }_{\mathit{RSTU}}}={(z_{\ell }-\overline{z})}_{\mathit{RSTU}}{q}_{\ell },$ where ${(z_{\ell }-\overline{z})}_R=z_{{\ell }_R}-{\overline{z}}_R,$ ${(z_{\ell }-\overline{z})}_{RS}=(z_{{\ell }_R}-{\overline{z}}_R)(z_{{\ell }_S}-{\overline{z}}_S)$ and so on, gives $\overline{z}={({\overline{z}}_1,\dots ,{\overline{z}}_q)}^{\top }$ and ${\overline{z}}_R=(1/n){\sum }_{\ell =1}^n{z}_{{\ell }_R},$ for $R=1,\dots ,q.$ Thus we obtain the following cumulants $$\begin{array}{c}{\kappa }_{RS}=-\frac{1}{2}\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{RS}\hspace{1em}{\kappa }_{\mathit{RST}}=\frac{1}{2}\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{\mathit{RST}},\\ {\kappa }_{\mathit{rsT}}=\frac{1}{\gamma }\sum _{\ell =1}^n{x}_{\ell r}^{*}{w}_{\ell }{x}_{\ell s}^{*}{(z_{\ell }-\overline{z})}_T{q}_{\ell },\hspace{1em}{\kappa }_{RS\gamma }=\frac{1}{2\gamma }\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{RS},\\ {\kappa }_{R,S,T}=\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{\mathit{RST}},\hspace{1em}{\kappa }_{r,s,T}=\frac{1}{\gamma }\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_R{q}_{\ell }{x}_{\ell r}^{*}{x}_{\ell s}^{*}{w}_{\ell },\\ {\kappa }_{R,S,\gamma }=\frac{1}{\gamma }\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{RS},\hspace{1em}{\kappa }_{r,sT}=-\frac{1}{\gamma }\sum _{\ell =1}^n{x}_{\ell r}^{*}{w}_{\ell }{x}_{\ell s}^{*}{(z_{\ell }-\overline{z})}_T{q}_{\ell },\\ {\kappa }_{R,ST}=-\frac{1}{2}\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{\mathit{RST}},\hspace{1em}{\kappa }_{R,S\gamma }=-\frac{1}{2\gamma }\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{RS},\\ {\kappa }_{\gamma ,RS}=-\frac{1}{2\gamma }\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{RS},\hspace{1em}{\kappa }_{\gamma ,RS}=-\frac{1}{2\gamma }\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{RS},\\ {\kappa }_{\mathit{RSTU}}=-\frac{1}{2}\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{\mathit{RSTU}},\hspace{1em}{\kappa }_{\mathit{\text{RSγγ}}}=-\frac{1}{{\gamma }^2}\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{RS},\\ {\kappa }_{R,S,T,U}=3\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{\mathit{RSTU}},\hspace{1em}{\kappa }_{r,s,T,U}=\frac{2}{\gamma }\sum _{\ell =1}^n{q}_{\ell }{w}_{\ell }{x}_{{\ell }_r}^{*}{x}_{{\ell }_s}^{*}{(z_{\ell }-\overline{z})}_{TU},\\ {\kappa }_{R,S,\gamma ,\gamma }=\frac{3}{{\gamma }^2}\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{RS},\hspace{1em}{\kappa }_{R,S,TU}=-\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{\mathit{RSTU}},\\ {\kappa }_{r,s,TU}=-\frac{1}{\gamma }\sum _{\ell =1}^n{w}_{\ell }{x}_{{\ell }_r}^{*}{x}_{{\ell }_s}^{*}{(z_{\ell }-\overline{z})}_{TU}{q}_{\ell },\hspace{1em}{\kappa }_{R,S,\gamma \gamma }=-\frac{2}{{\gamma }^2}\sum _{\ell =1}^n{(z_{\ell }-\overline{z})}_{RS}\end{array}$$

B. Cumulants for the Bartlett correction factor

B.1. Tests for components of β and δ

Suppose that we want to test the null hypothesis ${\mathcal{H}}_0:{\mathit{\beta }}_1={\mathit{\beta }}_1^{(0)},{\mathit{\delta }}_1={\mathit{\delta }}_1^{(0)}$ versus ${\mathcal{H}}_1:{\mathit{\beta }}_1\ne {\mathit{\beta }}_1^{(0)}\hspace{1em}\text{or}\hspace{1em}{\mathit{\delta }}_1\ne {\mathit{\delta }}_1^{(0)}.$ Then, the coefficients for the Bartlett-type corrections can be expressed terms $$\begin{array}{c}{A}_{\mathbf{11}}=3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{F}{\mathbf{Z}}_{\beta }{}_{2d}({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2){\mathbf{Z}}_{\beta }{}_{2d}\mathbf{F}\mathbf{\Phi }\mathbf{1}+6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{P}}_2({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2){\mathbf{Z}}_{\beta }{}_{2d}\mathbf{F}\mathbf{\Phi }\mathbf{1}\\ +3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{P}}_2({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2){\mathbf{P}}_2\mathbf{W}\mathbf{\Phi }\mathbf{1}+3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_{2d}({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){\mathbf{Z}}_{\delta }{}_{2d}\mathbf{1}\\ +3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_{2d}({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){\mathbf{Z}}_{\beta }{}_{2d}\mathbf{W}\mathbf{\Phi }\mathbf{1}+\frac{3}{4}{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_{2d}({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){\mathbf{Z}}_{\delta }{}_{2d}\mathbf{1},\\ A_{\mathbf{12}}=6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{F}{\mathbf{Z}}_{\beta }{}_{2d}{\mathbf{Z}}_{\beta }{}_2{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d(\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}+6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{P}}_2{\mathbf{Z}}_{\beta }{}_2{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d(\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}\\ +6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_{2d}{\mathbf{Z}}_{\delta }{}_2{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{W}\mathbf{\Phi }\mathbf{1}+6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_{2d}{\mathbf{Z}}_{\delta }{}_2{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}\\ +\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{{}_2}_d{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{W}\mathbf{\Phi }\mathbf{1}+\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{{}_2}_d{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}\\ +3{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_{2d}{\mathbf{Z}}_{\delta }{}_2{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{W}\mathbf{\Phi }\mathbf{1}+3{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_{2d}{\mathbf{Z}}_{\delta }{}_2{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}\\ +\frac{6}{n}{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{{}_2}_d{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{W}\mathbf{\Phi }\mathbf{1}+\frac{6}{n}{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{{}_2}_d{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1},\\ A_{\mathbf{13}}=-6{\mathbf{1}}^{\top }\mathbf{\Phi }(2\mathbf{G}-\mathbf{F})\left[{\mathbf{Z}}_{\beta }{}_2^{(2)}\odot ({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)\right]\mathbf{F}\mathbf{\Phi }\mathbf{1}-6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}[({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)\odot {\mathbf{J}}_2]\mathbf{W}\mathbf{\Phi }\mathbf{1}\\ -6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}\left[({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)\odot {\mathbf{C}}_2^{\top }\right]\mathbf{F}\mathbf{\Phi }\mathbf{1}+\frac{9}{2}{\mathbf{1}}^{\top }\left[{\mathbf{Z}}_{\delta }{}_2^{(2)}\odot ({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)\right]\mathbf{1}\\ -6{\mathbf{1}}^{\top }\mathbf{\Phi }(2\mathbf{G}-\mathbf{F})[({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)\odot {\mathbf{C}}_2]\mathbf{W}\mathbf{\Phi }\mathbf{1}+\frac{\mathbf{18}}{n}{\mathbf{1}}^{\top }[{\mathbf{Z}}_{\delta }{}_2\odot ({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)]\mathbf{1}\\ +6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}\left[({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)\odot {\mathbf{Z}}_{\beta }{}_2^{(2)}\right]\mathbf{W}\mathbf{\Phi }\mathbf{1}+\mathbf{12}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}[({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)\odot {\mathbf{Z}}_{\beta }{}_2{\mathbf{Z}}_{\delta }{}_2]\mathbf{W}\mathbf{\Phi }\mathbf{1},\\ A_{\mathbf{14}}=-\mathbf{12}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_{2d}{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}-6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{H}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d{Z}_{\beta }{}_{2d}\mathbf{1}\\ -\mathbf{12}{\mathbf{1}}^{\top }{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d{\mathbf{Z}}_{\delta }{{}_2}_d\mathbf{1}-6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d{\mathbf{Z}}_{\delta }{{}_2}_d\mathbf{1}\\ -6{\mathbf{1}}^{\top }\mathbf{\Phi }(\mathbf{F}-\mathbf{G}){\mathbf{P}}_2{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{1}-\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1},\\ A_{21}=-3{\mathbf{1}}^{\top }\mathbf{\Phi }(\mathbf{F}-\mathbf{G}){({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d{\mathbf{Z}}_{\beta }{}_2{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d(\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}-\frac{6}{n}{\mathbf{1}}^{\top }{\left[{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\right]}^2\mathbf{1}\\ -3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d{\mathbf{Z}}_{\delta }{}_2{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}-3{\mathbf{1}}^{\top }{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d{\mathbf{Z}}_{\delta }{}_2{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{W}\mathbf{\Phi }\mathbf{1}\\ -\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}-3{\mathbf{1}}^{\top }{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d{\mathbf{Z}}_{\delta }{}_2{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}\\ -3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d{\mathbf{Z}}_{\delta }{}_2{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{1}-\frac{6}{n}{\mathbf{1}}^{\top }{\left[\mathbf{\Phi }\mathbf{W}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\right]}^2\mathbf{1},\\ A_{22}=-6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{F}{\mathbf{Z}}_{\beta }{}_{2d}({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2){({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d(\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}\\ -6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_{2d}({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){({\mathbf{Z}}_{\beta }-Z_{\beta }{}_2)}_d\mathbf{W}\mathbf{\Phi }\mathbf{1}+6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_{2d}({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}\\ -3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\delta }{}_{2d}({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{1}-3{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_{2d}({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}\\ -6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{P}}_2({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2){({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d(\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1},\\ A_{23}=-6{\mathbf{1}}^{\top }\mathbf{\Phi }(\mathbf{F}-\mathbf{G})\left[{\mathbf{Z}}_{\beta }{}_2\odot {({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}^{(2)}\right](\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}-\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }\left[{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}^{(2)}\right]\mathbf{1}\\ -6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}\left[{\mathbf{Z}}_{\delta }{}_2\odot {({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}^{(2)}\right]\mathbf{W}\mathbf{\Phi }\mathbf{1}-6{\mathbf{1}}^{\top }\left[{\mathbf{Z}}_{\delta }{}_2\odot {({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}^{(2)}\right]\mathbf{1}\\ -\mathbf{12}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}[{\mathbf{Z}}_{\beta }{}_2\odot ({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)\odot ({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)]\mathbf{W}\mathbf{\Phi }\mathbf{1}-\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}\left[{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}^{(2)}\right]\mathbf{W}\mathbf{\Phi }\mathbf{1},\\ A_{24}=3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{B}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d^2\mathbf{1}+\mathbf{12}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{1}+9{\mathbf{1}}^{\top }{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d^2\mathbf{1},\end{array}$$ $$\begin{array}{c}{A}_{31}=3{\mathbf{1}}^{\top }\mathbf{\Phi }(\mathbf{F}-\mathbf{G}){({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2){({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d(\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}\\ +3{\mathbf{1}}^{\top }{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}+6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}\\ +3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{W}\mathbf{\Phi }\mathbf{1},\\ A_{32}=2{\mathbf{1}}^{\top }\mathbf{\Phi }(\mathbf{F}-\mathbf{G})\left[{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}^{(3)}\right](\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}+2{\mathbf{1}}^{\top }\left[{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}^{(3)}\right]\mathbf{1}\\ +6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}\left[({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)\odot {({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}^{(2)}\right]\mathbf{W}\mathbf{\Phi }\mathbf{1}\end{array}$$

B.2. Tests for components of δ

Here, suppose that we want to test the null hypothesis ${\mathcal{H}}_0:{\mathit{\delta }}_1={\mathit{\delta }}_1^{(0)},$ versus ${\mathcal{H}}_1:{\mathit{\delta }}_1\ne {\mathit{\delta }}_1^{(0)}.$ In this situation, just consider ${\mathbf{Z}}_{\beta }={\mathbf{Z}}_{\beta }{}_2$ and then the coefficients of the corrected score reduce to $$\begin{array}{c}{A}_{\mathbf{11}}=3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){\mathbf{Z}}_{\delta }{}_{2d}\mathbf{1}+3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){\mathbf{Z}}_{\beta }{}_d\mathbf{W}\mathbf{\Phi }\mathbf{1}\\ +\frac{3}{4}{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_{2d}({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){\mathbf{Z}}_{\delta }{}_{2d}\mathbf{1},\\ A_{\mathbf{12}}=6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d{\mathbf{Z}}_{\delta }{}_2{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}+\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}\\ +3{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_{2d}{\mathbf{Z}}_{\delta }{}_2{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}+\frac{6}{n}{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{{}_2}_d{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1},\\ A_{\mathbf{13}}=\frac{9}{2}{\mathbf{1}}^{\top }\left[{\mathbf{Z}}_{\delta }{}_2^{(2)}\odot ({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)\right]\mathbf{1}+\frac{\mathbf{18}}{n}{\mathbf{1}}^{\top }\left[{\mathbf{Z}}_{\delta }{}_2\odot ({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)\right]\mathbf{1}\\ +6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}\left[({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)\odot {\mathbf{Z}}_{\beta }{}^{(2)}\right]\mathbf{W}\mathbf{\Phi }\mathbf{1},\\ A_{\mathbf{14}}=-\mathbf{12}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}-\mathbf{12}{\mathbf{1}}^{\top }{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d{Z}_{\delta }{{}_2}_d\mathbf{1}-\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1},\\ A_{21}=-\frac{6}{n}{\mathbf{1}}^{\top }{\left[{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\right]}^2\mathbf{1}-3{\mathbf{1}}^{\top }{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d{\mathbf{Z}}_{\delta }{}_2{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1},\\ A_{22}=6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1}-3{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_{2d}({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1},\\ A_{23}=-\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }\left[{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}^{(2)}\right]\mathbf{1}-6{\mathbf{1}}^{\top }\left[{\mathbf{Z}}_{\delta }{}_2\odot {({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}^{(2)}\right]\mathbf{1},\\ A_{24}=9{\mathbf{1}}^{\top }{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d^2\mathbf{1},\\ A_{31}=3{\mathbf{1}}^{\top }{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2){({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}_d\mathbf{1},\\ A_{32}=2{\mathbf{1}}^{\top }\left[{({\mathbf{Z}}_{\delta }-{\mathbf{Z}}_{\delta }{}_2)}^{(3)}\right]\mathbf{1}\end{array}$$ Further, for testing all the components of $\mathit{\delta },$ we have ${\mathbf{Z}}_{\delta }{}_2=\mathbf{0}.$ Hence, $$\begin{array}{c}{A}_{\mathbf{11}}=3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d{\mathbf{Z}}_{\delta }{\mathbf{Z}}_{\beta }{}_d\mathbf{W}\mathbf{\Phi }\mathbf{1},\hspace{1em}{A}_{\mathbf{12}}=\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d{\mathbf{Z}}_{\delta }{}_d\mathbf{1},\\ A_{\mathbf{13}}=6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}\left[{\mathbf{Z}}_{\delta }\odot {\mathbf{Z}}_{\beta }{}^{(2)}\right]\mathbf{W}\mathbf{\Phi }\mathbf{1},\hspace{1em}{A}_{\mathbf{14}}=-\mathbf{12}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d{\mathbf{Z}}_{\delta }{}_d\mathbf{1}-\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_d\mathbf{1},\\ A_{21}=-\frac{6}{n}{\mathbf{1}}^{\top }{\left[{\mathbf{Z}}_{\delta }{}_d\right]}^2\mathbf{1},\hspace{1em}{A}_{22}=6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d{\mathbf{Z}}_{\delta }{\mathbf{Z}}_{\delta }{}_d\mathbf{1},\hspace{1em}{A}_{23}=-\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }\left[{\mathbf{Z}}_{\delta }{}^{(2)}\right]\mathbf{1},\\ A_{24}=9{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_d^2\mathbf{1},\hspace{1em}{A}_{31}=3{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_d{\mathbf{Z}}_{\delta }{\mathbf{Z}}_{\delta }{}_d\mathbf{1},\hspace{1em}{A}_{32}=2{\mathbf{1}}^{\top }\left[{\mathbf{Z}}_{\delta }{}^{(3)}\right]\mathbf{1}\end{array}$$

B.3. Tests for components of β

Finally, suppose that we want to test the null hypothesis ${\mathcal{H}}_0:{\mathit{\beta }}_1={\mathit{\beta }}_1^{(0)}$ versus ${\mathcal{H}}_1:{\mathit{\beta }}_1\ne {\mathit{\beta }}_1^{(0)}.$ If we consider ${\mathbf{Z}}_{\delta }={\mathbf{Z}}_{\delta }{}_2,$ then the coefficients of the corrected score statistic are $$\begin{array}{c}{A}_{\mathbf{11}}=3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{F}{\mathbf{Z}}_{\beta }{}_{2d}({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2){\mathbf{Z}}_{\beta }{}_{2d}\mathbf{F}\mathbf{\Phi }\mathbf{1}+6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{P}}_2({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2){\mathbf{Z}}_{\beta }{}_{2d}\mathbf{F}\mathbf{\Phi }\mathbf{1}\\ +3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{P}}_2({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2){\mathbf{P}}_2\mathbf{W}\mathbf{\Phi }\mathbf{1},\\ A_{\mathbf{12}}=6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{F}{\mathbf{Z}}_{\beta }{}_{2d}{\mathbf{Z}}_{\beta }{}_2{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d(\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}+6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{P}}_2{\mathbf{Z}}_{\beta }{}_2{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d(\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}\\ +6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_{2d}{\mathbf{Z}}_{\delta }{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{W}\mathbf{\Phi }\mathbf{1}+\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{{}_2}_d{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{W}\mathbf{\Phi }\mathbf{1}\\ +3{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_d{\mathbf{Z}}_{\delta }{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{W}\mathbf{\Phi }\mathbf{1}+\frac{6}{n}{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_d{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{W}\mathbf{\Phi }\mathbf{1},\\ A_{\mathbf{13}}=-6{\mathbf{1}}^{\top }\mathbf{\Phi }(2\mathbf{G}-\mathbf{F})[{\mathbf{Z}}_{\beta }{}_2^{(2)}\odot ({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)]\mathbf{F}\mathbf{\Phi }\mathbf{1}-6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}[({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)\odot {\mathbf{J}}_2]\mathbf{W}\mathbf{\Phi }\mathbf{1}\\ -6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}[({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)\odot {\mathbf{C}}_2^{\top }]\mathbf{F}\mathbf{\Phi }\mathbf{1}-6{\mathbf{1}}^{\top }\mathbf{\Phi }(2\mathbf{G}-\mathbf{F})[({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)\odot {\mathbf{C}}_2]\mathbf{W}\mathbf{\Phi }\mathbf{1}\\ +\mathbf{12}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}[({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)\odot {\mathbf{Z}}_{\beta }{}_2{\mathbf{Z}}_{\delta }]\mathbf{W}\mathbf{\Phi }\mathbf{1},\\ A_{\mathbf{14}}=-6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{H}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d{\mathbf{Z}}_{\beta }{}_{2d}\mathbf{1}-6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d{\mathbf{Z}}_{\delta }{}_d\mathbf{1}\\ -6{\mathbf{1}}^{\top }\mathbf{\Phi }(\mathbf{F}-\mathbf{G}){\mathbf{P}}_2{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{1},\\ A_{21}=-3{\mathbf{1}}^{\top }\mathbf{\Phi }(\mathbf{F}-\mathbf{G}){({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d{\mathbf{Z}}_{\beta }{}_2{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d(\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}\\ -3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d{\mathbf{Z}}_{\delta }{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\mathbf{1}-\frac{6}{n}{\mathbf{1}}^{\top }{\left[\mathbf{\Phi }\mathbf{W}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d\right]}^2\mathbf{1},\\ A_{22}=-6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{F}{\mathbf{Z}}_{\beta }{}_{2d}({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2){({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d(\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}\\ -6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{P}}_2({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2){({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d(\mathbf{F}-\mathbf{G})\Phi \mathbf{1},\\ A_{23}=-6{\mathbf{1}}^{\top }\mathbf{\Phi }(\mathbf{F}-\mathbf{G})\left[{\mathbf{Z}}_{\beta }{}_2\odot {({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}^{(2)}\right](\mathbf{F}-\mathbf{G})\Phi \mathbf{1}\\ -6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}[{\mathbf{Z}}_{\delta }\odot {({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}^{(2)}]\mathbf{W}\mathbf{\Phi }\mathbf{1}\\ -\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}\left[{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}^{(2)}\right]\mathbf{W}\mathbf{\Phi }\mathbf{1},\\ A_{24}=3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{B}{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d^2\mathbf{1},\\ A_{31}=3{\mathbf{1}}^{\top }\mathbf{\Phi }(\mathbf{F}-\mathbf{G}){({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2){({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}_d(F-G)\mathbf{\Phi }\mathbf{1},\\ A_{32}=2{\mathbf{1}}^{\top }\mathbf{\Phi }(\mathbf{F}-\mathbf{G})\left[{({\mathbf{Z}}_{\beta }-{\mathbf{Z}}_{\beta }{}_2)}^{(3)}\right](\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}\end{array}$$ Finally, for testing all the components of $\mathit{\beta },$ we have ${\mathbf{Z}}_{\beta }{}_2={\mathbf{P}}_2={\mathbf{C}}_2={\mathbf{J}}_2=0.$ Hence, $$\begin{array}{c}{A}_{\mathbf{11}}=A_{\mathbf{13}}=A_{22}=0,\hspace{1em}{A}_{\mathbf{12}}=3{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_d{\mathbf{Z}}_{\delta }{\mathbf{Z}}_{\beta }{}_d\mathbf{W}\mathbf{\Phi }\mathbf{1}+\frac{6}{n}{\mathbf{1}}^{\top }{\mathbf{Z}}_{\delta }{}_d{\mathbf{Z}}_{\beta }{}_d\mathbf{W}\mathbf{\Phi }\mathbf{1},\\ A_{\mathbf{14}}=-6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d{\mathbf{Z}}_{\delta }{}_d\mathbf{1},\hspace{1em}{A}_{21}=-3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d{\mathbf{Z}}_{\delta }{\mathbf{Z}}_{\beta }{}_d\mathbf{1}-\frac{6}{n}{\mathbf{1}}^{\top }{[\mathbf{\Phi }\mathbf{W}{\mathbf{Z}}_{\beta }{}_d]}^2\mathbf{1},\\ A_{23}=-6{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}\left[{\mathbf{Z}}_{\delta }\odot {\mathbf{Z}}_{\beta }{}^{(2)}\right]\mathbf{W}\mathbf{\Phi }\mathbf{1}-\frac{\mathbf{12}}{n}{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{W}\left[{\mathbf{Z}}_{\beta }{}^{(2)}\right]\mathbf{W}\mathbf{\Phi }\mathbf{1},\hspace{1em}{A}_{24}=3{\mathbf{1}}^{\top }\mathbf{\Phi }\mathbf{B}{\mathbf{Z}}_{\beta }{}_d^2\mathbf{1},\\ A_{31}=3{\mathbf{1}}^{\top }\mathbf{\Phi }(\mathbf{F}-\mathbf{G}){\mathbf{Z}}_{\beta }{}_d{\mathbf{Z}}_{\beta }{\mathbf{Z}}_{\beta }{}_d(\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1},\hspace{1em}{A}_{32}=2{\mathbf{1}}^{\top }\mathbf{\Phi }(\mathbf{F}-\mathbf{G}){\mathbf{Z}}_{\beta }{}^{(3)}(\mathbf{F}-\mathbf{G})\mathbf{\Phi }\mathbf{1}\end{array}$$

Additional information

Funding

This study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001 and Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) [grant number 303133/2014-7].