Appendix
A. The first-order and second-order derivatives of (4) are
$$\begin{aligned} \frac{\partial l_1({\varvec{\theta }})}{\partial \xi }&=\frac{z}{\eta }-\frac{\alpha }{\eta }\,R(z)-\frac{3\beta }{\eta }z^2\,R(z),\quad \frac{\partial l_1({\varvec{\theta }})}{\partial \eta }=\frac{z^2-1}{\eta }-\frac{\alpha }{\eta }z\,R(z)-\frac{3\beta }{\eta }z^3\,R(z),\\ \frac{\partial l_1({\varvec{\theta }})}{\partial \alpha }&=z\,R(z)\quad \text{ and }\quad \frac{\partial l_1({\varvec{\theta }})}{\partial \beta }=z^3\,R(z). \end{aligned}$$
$$\begin{aligned} \frac{\partial ^2 l_1({\varvec{\theta }})}{\partial \xi ^2}= & {} -\frac{1}{\eta ^2}+\frac{(6\beta -\alpha ^3)}{\eta ^2}\,zR(z)-\frac{7\alpha ^2\beta }{\eta ^2}\,z^3R(z)-\frac{15\alpha \beta ^2}{\eta ^2}\,z^5R(z)\\&-\,\frac{9\beta ^3}{\eta ^2}\,z^7R(z)-\frac{\alpha ^2}{\eta ^2}\,R^2(z)-\frac{6\alpha \beta }{\eta ^2}\,z^2R^2(z)-\frac{9\beta ^2}{\eta ^2}\,z^4R^2(z),\\ \frac{\partial ^2 l_1({\varvec{\theta }})}{\partial \eta \partial \xi }= & {} -\frac{2z}{\eta ^2}+\frac{\alpha }{\eta ^2}\,R(z)+\frac{(9\beta -\alpha ^3)}{\eta ^2}\,z^2R(z)-\frac{7\alpha ^2\beta }{\eta ^2}\,z^4R(z)-\frac{15\alpha \beta ^2}{\eta ^2}\,z^6R(z)\\&-\,\frac{9\beta ^3}{\eta ^2}\,z^8R(z)-\frac{\alpha ^2}{\eta ^2}\,zR^2(z)-\frac{6\alpha \beta }{\eta ^2}\,z^3R^2(z)-\frac{9\beta ^2}{\eta ^2}\,z^5R^2(z),\\ \frac{\partial ^2 l_1({\varvec{\theta }})}{\partial \alpha \partial \xi }= & {} -\frac{1}{\eta }\,R(z)+\frac{\alpha ^2}{\eta }\,z^2R(z)+\frac{4\alpha \beta }{\eta }\,z^4R(z)+\frac{3\beta ^2}{\eta }\,z^6R(z)+\frac{\alpha }{\eta }\,zR^2(z)\\&+\,\frac{3\beta }{\eta }\,z^3R^2(z),\\ \frac{\partial ^2 l_1({\varvec{\theta }})}{\partial \beta \partial \xi }= & {} -\frac{3}{\eta }\,z^2R(z)+\frac{\alpha ^2}{\eta }\,z^4R(z)+\frac{4\alpha \beta }{\eta }\,z^6R(z)+\frac{3\beta ^2}{\eta }\,z^8R(z)+\frac{\alpha }{\eta }\,z^3R^2(z),\\&+\,\frac{3\beta }{\eta }\,z^5R^2(z),\\ \frac{\partial ^2 l_1({\varvec{\theta }})}{\partial \eta ^2}= & {} \frac{1}{\eta ^2}-\frac{3z^2}{\eta ^2}+\frac{2\alpha }{\eta ^2}\,zR(z)+\frac{(12\beta -\alpha ^3)}{\eta ^2}\,z^3R(z)-\frac{7\alpha ^2\beta }{\eta ^2}\,z^5R(z)\\&-\,\frac{15\alpha \beta ^2}{\eta ^2}\,z^7R(z)-\frac{9\beta ^3}{\eta ^2}\,z^9R(z)-\frac{6\alpha \beta }{\eta ^2}\,z^4R^2(z)-\frac{\alpha ^2}{\eta ^2}\,z^2R^2(z)\\&-\,\frac{9\beta ^2}{\eta ^2}\,z^6R^2(z),\\ \frac{\partial ^2 l_1({\varvec{\theta }})}{\partial \alpha \partial \eta }= & {} -\frac{1}{\eta }\,zR(z)+\frac{\alpha ^2}{\eta }\,z^3R(z)+\frac{4\alpha \beta }{\eta }\,z^5R(z)+\frac{3\beta ^2}{\eta }\,z^7R(z)+\frac{\alpha }{\eta }\,z^2R^2(z),\\&+\,\frac{3\beta }{\eta }\,z^4R^2(z),\\ \frac{\partial ^2 l_1({\varvec{\theta }})}{\partial \beta \partial \eta }= & {} -\frac{3}{\eta }\,z^3R(z)+\frac{\alpha ^2}{\eta }\,z^5R(z)+\frac{4\alpha \beta }{\eta }\,z^7R(z)+\frac{3\beta ^2}{\eta }\,z^9R(z)+\frac{\alpha }{\eta }\,z^4R^2(z),\\&+\,\frac{3\beta }{\eta }\,z^6R^2(z),\\ \frac{\partial ^2 l_1({\varvec{\theta }})}{\partial \alpha ^2}= & {} -\alpha z^3\,R(z)-\beta z^5\,R(z)-z^2\,R^2(z),\\ \frac{\partial ^2 l_1({\varvec{\theta }})}{\partial \beta \partial \alpha }= & {} -\alpha z^5\,R(z)-\beta z^7\,R(z)-z^4\,R^2(z), \\ \frac{\partial ^2 l_1({\varvec{\theta }})}{\partial \beta ^2}= & {} -\alpha z^7\,R(z)-\beta z^9\,R(z)-z^6\,R^2(z). \end{aligned}$$
where \(z=\eta ^{-1}(x-\xi )\), \(R(z)=\phi (\alpha z+\beta z^3)/\Phi (\alpha z+\beta z^3)\) and \(l_1({\varvec{\theta }}):=l_1(x;{\varvec{\theta }})\).
B. For the purpose of writing the derivatives with respect to \({\widetilde{\alpha }}\) of the likelihood function as simply as possible, in each reparametrization, we consider the following expressions and corresponding notation: taking \(y:=y({\widetilde{\theta }})=\frac{x-{\widetilde{\xi }}+K_{11}{\widetilde{\alpha }}}{{\widetilde{\eta }}-\frac{1}{2}K_{22}{\widetilde{\alpha }}^2}\), where \({\widetilde{\theta }}=({\widetilde{\xi }},{\widetilde{\eta }},{\widetilde{\alpha }},{\widetilde{\beta }})\), we obtain
$$\begin{aligned} \frac{dy}{d{\widetilde{\alpha }}}= & {} y_{{\widetilde{\alpha }}}=\frac{K_{11}+K_{22}{\widetilde{\alpha }} y}{{\widetilde{\eta }}-\frac{1}{2}K_{22}{\widetilde{\alpha }}^2},\\ \frac{d^2y}{d{\widetilde{\alpha }}^2}= & {} y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}=\frac{K_{22}(y+{\widetilde{\alpha }}y_{{\widetilde{\alpha }}}) }{{\widetilde{\eta }}-\frac{1}{2}K_{22}{\widetilde{\alpha }}^2}+\frac{K_{22}{\widetilde{\alpha }} y_{{\widetilde{\alpha }}}}{{\widetilde{\eta }}-\frac{1}{2}K_{22}{\widetilde{\alpha }}^2},\\ \frac{d^3y}{d{\widetilde{\alpha }}^3}= & {} y_{(3{\widetilde{\alpha }})}=K_{22}\frac{(3y_{{\widetilde{\alpha }}}+2{\widetilde{\alpha }}y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}) }{{\widetilde{\eta }}-\frac{1}{2}K_{22}{\widetilde{\alpha }}^2}+K_{22}^2\frac{(y+2{\widetilde{\alpha }} y_{{\widetilde{\alpha }}}){\widetilde{\alpha }}}{({\widetilde{\eta }}-\frac{1}{2}K_{22}{\widetilde{\alpha }}^2)^2},\\ \frac{d^4y}{d{\widetilde{\alpha }}^4}= & {} y_{(4{\widetilde{\alpha }})}=K_{22}\frac{5y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}+2{\widetilde{\alpha }}y_{(3{\widetilde{\alpha }})} }{{\widetilde{\eta }}-\frac{1}{2}K_{22}{\widetilde{\alpha }}^2}+K_{22}^2\frac{y+8{\widetilde{\alpha }} y_{{\widetilde{\alpha }}}+4{\widetilde{\alpha }}^2y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}}{({\widetilde{\eta }}-\frac{1}{2}K_{22}{\widetilde{\alpha }}^2)^2}\\&+\,K_{22}^3\frac{2{\widetilde{\alpha }}^2(y+2{\widetilde{\alpha }} y_{{\widetilde{\alpha }}})}{({\widetilde{\eta }}-\frac{1}{2}K_{22}{\widetilde{\alpha }}^2)^3} \end{aligned}$$
Now, evaluating in \({\varvec{\theta }}={\varvec{\theta }}^{*}=(\xi ^{*},\eta ^{*},0,0)\) we have:
$$\begin{aligned} y_{{\widetilde{\alpha }}}( {\varvec{\theta }}^{*})= & {} \frac{K_{11}}{{\widetilde{\eta }}}=\sqrt{\frac{2}{\pi }},\quad \quad y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}( {\varvec{\theta }}^{*})=\frac{K_{22}}{{\widetilde{\eta }}}Z^*=-\frac{2}{\pi }Z^*,\\ y_{(3{\widetilde{\alpha }})}( {\varvec{\theta }}^{*})= & {} -3\left( \frac{2}{\pi }\right) ^{3/2},\quad \quad y_{(4{\widetilde{\alpha }})}( {\varvec{\theta }}^{*})=\frac{24}{\pi ^2}Z^*. \end{aligned}$$
For the third reparameterization we consider the following function log-likelihood
$$\begin{aligned} L:=l_1({\widetilde{\theta }})=\log (2)-\log ({\widetilde{\eta }})+\log (\phi (y))+\log (\Phi (w)), \end{aligned}$$
(7)
where \(w={\widetilde{\alpha }}y+{\widetilde{\beta }}y^3\) and consider \(R(w)=\frac{\phi }{\Phi }(w)\) with the idea of calculating \(\frac{d^3}{d{\widetilde{\alpha }}^3}L=L_{(3{\widetilde{\alpha }})}\) for L defined in (7).
We start with calculations of the derivatives with respect to \({\widetilde{\alpha }}\) of w and R and then evaluate in \({\varvec{\theta }}={\varvec{\theta }}^{*}\).
$$\begin{aligned} w_{{\widetilde{\alpha }}}= & {} y+{\widetilde{\alpha }}y_{{\widetilde{\alpha }}}+3{\widetilde{\beta }}y^2y_{{\widetilde{\alpha }}}\\ w_{{\widetilde{\alpha }}{\widetilde{\alpha }}}= & {} 2y_{{\widetilde{\alpha }}}+{\widetilde{\alpha }}y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}+6{\widetilde{\beta }}yy_{{\widetilde{\alpha }}}^2+3{\widetilde{\beta }}y^2y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}\\ w_{(3{\widetilde{\alpha }})}= & {} 3y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}+{\widetilde{\alpha }}y_{(3{\widetilde{\alpha }})}+6{\widetilde{\beta }}y_{{\widetilde{\alpha }}}^3+18{\widetilde{\beta }}yy_{{\widetilde{\alpha }}}y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}+3{\widetilde{\beta }}y^2y_{(3{\widetilde{\alpha }})}\\ R_{{\widetilde{\alpha }}}(w)= & {} -w_{{\widetilde{\alpha }}}[wR(w)+R^2(w)],\\ R_{{\widetilde{\alpha }}{\widetilde{\alpha }}}(w)= & {} -w_{{\widetilde{\alpha }}{\widetilde{\alpha }}}[wR(w)+R^2(w)]-w_{{\widetilde{\alpha }}}[w_{{\widetilde{\alpha }}}R(w)+wR_{{\widetilde{\alpha }}}(w)+2R(w)R_{{\widetilde{\alpha }}}(w)]. \end{aligned}$$
Evaluating in \({\varvec{\theta }}={\varvec{\theta }}^{*}\) we have
\(w_{{\widetilde{\alpha }}}({\varvec{\theta }}^{*})=Z^*\), \(w_{{\widetilde{\alpha }}{\widetilde{\alpha }}}({\varvec{\theta }}^{*})=2\sqrt{\frac{2}{\pi }}\) and \(w_{(3{\widetilde{\alpha }})}({\varvec{\theta }}^{*})=-\frac{6}{\pi }Z^*\).
\(R(w({\varvec{\theta }}^{*}))=\sqrt{\frac{2}{\pi }}\), \(R_{{\widetilde{\alpha }}}(w({\varvec{\theta }}^{*}))=-\frac{2}{\pi }Z^*\) and \(R_{{\widetilde{\alpha }}{\widetilde{\alpha }}}(w({\varvec{\theta }}^{*}))=-2\left( \frac{2}{\pi }\right) ^{3/2} -\sqrt{\frac{2}{\pi }}{Z^*}^2+2\left( \frac{2}{\pi }\right) ^{3/2}{Z^*}^2\).
From the above, we have
\(L_{(3{\widetilde{\alpha }})}=-3y_{{\widetilde{\alpha }}}y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}-yy_{(3{\widetilde{\alpha }})}+R_{{\widetilde{\alpha }}{\widetilde{\alpha }}}(w)w_{{\widetilde{\alpha }}}+2R_{{\widetilde{\alpha }}}(w)w_{{\widetilde{\alpha }}{\widetilde{\alpha }}}+R(w)w_{(3{\widetilde{\alpha }})}\) and when evaluating and simplifying we obtain the expression given in the manuscript, \(L_{(3{\widetilde{\alpha }})}({\varvec{\theta }}^{*})={\widetilde{S}}_3^{(3)}=\frac{\sqrt{2}}{\pi ^{3/2}}[-6Z^*+(4-\pi ){Z^*}^3].\)
In the case of the next reparametrization, the log-likelihood function is given by
$$\begin{aligned} L:=l_1({\widetilde{\theta }})=\log (2)-\log ({\widetilde{\eta }})+\log (\phi (y))+\log (\Phi (s)), \end{aligned}$$
(8)
where \(s={\widetilde{\alpha }}y+({\widetilde{\beta }}-\frac{1}{6}K_{33}{\widetilde{\alpha }}^3)y^3\).
By direct calculations we find
$$\begin{aligned} s_{{\widetilde{\alpha }}}= & {} y+{\widetilde{\alpha }}y_{{\widetilde{\alpha }}}-\frac{1}{2}K_{33}{\widetilde{\alpha }}^2y^3+3{\widetilde{\beta }}y^2y_{{\widetilde{\alpha }}}-\frac{1}{2}K_{33}{\widetilde{\alpha }}^3y^2y_{{\widetilde{\alpha }}},\\ s_{{\widetilde{\alpha }}{\widetilde{\alpha }}}= & {} 2y_{{\widetilde{\alpha }}}+{\widetilde{\alpha }}y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}-K_{33}{\widetilde{\alpha }}y^3-3K_{33}{\widetilde{\alpha }}^2y^2y_{{\widetilde{\alpha }}}-K_{33}{\widetilde{\alpha }}^3yy_{{\widetilde{\alpha }}}^2-\frac{1}{2}K_{33}{\widetilde{\alpha }}^3y^2y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}\\&+\,6{\widetilde{\beta }}yy_{{\widetilde{\alpha }}}^2+3{\widetilde{\beta }}y^2y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}\\ s_{(3{\widetilde{\alpha }})}= & {} 3y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}+{\widetilde{\alpha }}y_{(3{\widetilde{\alpha }})}-K_{33}y^3-9K_{33}{\widetilde{\alpha }}^2yy_{{\widetilde{\alpha }}}^2-\frac{9}{2}K_{33}{\widetilde{\alpha }}^2y^2y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}-K_{33}{\widetilde{\alpha }}^3y_{{\widetilde{\alpha }}}^3\\&-\,3K_{33}{\widetilde{\alpha }}^3yy_{{\widetilde{\alpha }}}y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}-\frac{1}{2}K_{33}{\widetilde{\alpha }}^3y^2y_{(3{\widetilde{\alpha }})}+18{\widetilde{\beta }}yy_{{\widetilde{\alpha }}}y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}+6{\widetilde{\beta }}y_{{\widetilde{\alpha }}}^3\\&+\,3{\widetilde{\beta }}y^2y_{(3{\widetilde{\alpha }})}\\ s_{(4{\widetilde{\alpha }})}= & {} 4y_{(3{\widetilde{\alpha }})}+{\widetilde{\alpha }}y_{(4{\widetilde{\alpha }})}-12K_{33}y^2y_{{\widetilde{\alpha }}}-36K_{33}{\widetilde{\alpha }}yy_{{\widetilde{\alpha }}}^2\\&-\,18K_{33}{\widetilde{\alpha }}y^2y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}-12K_{33}{\widetilde{\alpha }}^2y_{{\widetilde{\alpha }}}^3\\&-\,36K_{33}{\widetilde{\alpha }}^2yy_{{\widetilde{\alpha }}}y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}-6K_{33}{\widetilde{\alpha }}^3y_{{\widetilde{\alpha }}}^2y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}-3K_{33}{\widetilde{\alpha }}^3yy_{{\widetilde{\alpha }}{\widetilde{\alpha }}}^2-4K_{33}{\widetilde{\alpha }}^3yy_{{\widetilde{\alpha }}}y_{(3{\widetilde{\alpha }})}\\&-\,\frac{3}{2}K_{33}{\widetilde{\alpha }}^2y^2y_{(3{\widetilde{\alpha }})}-\frac{1}{2}K_{33}{\widetilde{\alpha }}^3y^2y_{(4{\widetilde{\alpha }})}+36{\widetilde{\beta }}y_{{\widetilde{\alpha }}}^2y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}+18{\widetilde{\beta }}yy_{{\widetilde{\alpha }}{\widetilde{\alpha }}}^2\\&+\,24{\widetilde{\beta }}yy_{{\widetilde{\alpha }}}y_{(3{\widetilde{\alpha }})}+3{\widetilde{\beta }}y^2y_{(4{\widetilde{\alpha }})}\\ R_{{\widetilde{\alpha }}}(s)= & {} -s_{{\widetilde{\alpha }}}[sR(s)+R^2(s)]\\ R_{{\widetilde{\alpha }}{\widetilde{\alpha }}}(s)= & {} -ss_{{\widetilde{\alpha }}{\widetilde{\alpha }}}R(s)-s_{{\widetilde{\alpha }}{\widetilde{\alpha }}}R^2(s)-s_{{\widetilde{\alpha }}}^2R(s)-ss_{{\widetilde{\alpha }}}R_{{\widetilde{\alpha }}}(s)-2s_{{\widetilde{\alpha }}}R(s)R_{{\widetilde{\alpha }}}(s)\\ R_{(3{\widetilde{\alpha }})}(s)= & {} -3s_{{\widetilde{\alpha }}}s_{{\widetilde{\alpha }}{\widetilde{\alpha }}}R(s)-ss_{(3{\widetilde{\alpha }})}R(s)-2ss_{{\widetilde{\alpha }}{\widetilde{\alpha }}}R_{{\widetilde{\alpha }}}(s)-s_{(3{\widetilde{\alpha }})}R^2(s)-4s_{{\widetilde{\alpha }}{\widetilde{\alpha }}}R(s)R_{{\widetilde{\alpha }}}(s)\\&-\,2s_{{\widetilde{\alpha }}}^2R_{{\widetilde{\alpha }}}(s)-ss_{{\widetilde{\alpha }}}R_{{\widetilde{\alpha }}{\widetilde{\alpha }}}(s)-2s_{{\widetilde{\alpha }}}R_{{\widetilde{\alpha }}}^2(s)-2s_{{\widetilde{\alpha }}}R(s)R_{{\widetilde{\alpha }}{\widetilde{\alpha }}}(s) \end{aligned}$$
Now, evaluating in \({\varvec{\theta }}={\varvec{\theta }}^{*}\) we get
$$\begin{aligned} s({\varvec{\theta }}^{*})= & {} 0, \quad s_{{\widetilde{\alpha }}}({\varvec{\theta }}^{*})=Z^*,\quad s_{{\widetilde{\alpha }}{\widetilde{\alpha }}}({\varvec{\theta }}^{*})=2\sqrt{\frac{2}{\pi }},\\ s_{(3{\widetilde{\alpha }})}({\varvec{\theta }}^{*})= & {} -\frac{6}{\pi }Z^*-\frac{4-\pi }{\pi }{Z^*}^3, \quad s_{(4{\widetilde{\alpha }})}({\varvec{\theta }}^{*}){=}{-}12\left( \frac{2}{\pi }\right) ^{3/2}{-}12\sqrt{\frac{2}{\pi }}\frac{4{-}\pi }{\pi }{Z^*}^2\\ R(s({\varvec{\theta }}^{*}))= & {} \sqrt{\frac{2}{\pi }},\quad R_{{\widetilde{\alpha }}}(s({\varvec{\theta }}^{*}))=-\frac{2}{\pi }Z^*\\ R_{{\widetilde{\alpha }}{\widetilde{\alpha }}}(s({\varvec{\theta }}^{*}))= & {} -2\left( \frac{2}{\pi }\right) ^{3/2}-\sqrt{\frac{2}{\pi }}{Z^*}^2+2\left( \frac{2}{\pi }\right) ^{3/2}{Z^*}^2\\ R_{(3{\widetilde{\alpha }})}(s({\varvec{\theta }}^{*}))= & {} \left( -\frac{12}{\pi }-\frac{60}{\pi ^2}\right) Z^*+\left( \frac{6}{\pi }-\frac{16}{\pi ^2}\right) {Z^*}^3 \end{aligned}$$
The fourth derivative with respect to \({\widetilde{\alpha }}\) of the log-likelihood function defined in (8), is as follows
$$\begin{aligned} L_{4{\widetilde{\alpha }}}&=-3y_{{\widetilde{\alpha }}{\widetilde{\alpha }}}^2-4y_{{\widetilde{\alpha }}}y_{(3{\widetilde{\alpha }})}-yy_{(4{\widetilde{\alpha }})}\\&\quad +s_{{\widetilde{\alpha }}}R_{3{\widetilde{\alpha }}}(s)+3s_{{\widetilde{\alpha }}{\widetilde{\alpha }}}R_{{\widetilde{\alpha }}{\widetilde{\alpha }}}(s)+3s_{(3{\widetilde{\alpha }})}R_{{\widetilde{\alpha }}}(s)+s_{(4{\widetilde{\alpha }})}R(s). \end{aligned}$$
Hence, considering the above calculations and evaluating at \({\varvec{\theta }}={\varvec{\theta }}^{*}\), we have that \(L_{4{\widetilde{\alpha }}}({\varvec{\theta }}^{*})={\widetilde{S}}_3^{(4)}=\frac{4}{\pi ^2}[-12+3{Z^*}^2+2{Z^*}^4].\)
C. For the derivation of the Fisher information of (2), the following integrals need to be calculated, which can be done numerically using the (R Development Core Team 2015):
$$\begin{aligned} a_k:= a_k(\alpha ,\beta )=\int _0^{\infty }4z^k\phi (z)\phi (\alpha z+\beta z^3)\,dz,\quad k=0,2,4,6,8. \end{aligned}$$
where \(\phi \) and \(\Phi \) are the pdf and cdf of the standardized normal distribution respectively.
Proposition 4.1
Let \(Z\sim FGSN(\alpha ,\beta )\), then
$$\begin{aligned} c_r:= c_r(\alpha ,\beta )= & {} E\left\{ Z^r\left( \frac{\phi (\alpha Z+\beta Z^3)}{\Phi (\alpha Z+\beta Z^3)}\right) ^2\right\} ,\quad r=0,1,2\ldots 6,\\= & {} \left\{ \begin{array}{ll} d_r, &{} r\quad {\text{ is } \text{ even }}, \\ d_r-\int _0^{\infty }4z^r\frac{\phi ^2(\alpha z+\beta z^3)\phi (z)}{1-\Phi (\alpha z+\beta z^3)}\,dz, &{} r\quad {\text{ is } \text{ odd }}. \end{array} \right. \end{aligned}$$
where \(d_r:= d_r(\alpha ,\beta )=\int _0^{\infty }2z^r\frac{\phi ^2(\alpha z+\beta z^3)\phi (z)}{(1-\Phi (\alpha z+\beta z^3))\Phi (\alpha z+\beta z^3)}\,dz.\)
Proof
By definition of expected value and algebraic manipulation, we have
$$\begin{aligned} E\left\{ Z^r\left[ \frac{\phi (\alpha Z+\beta Z^3)}{\Phi (\alpha Z+\beta Z^3)}\right] ^2\right\}= & {} \int _{-\infty }^{\infty }2z^r\left[ \frac{\phi (\alpha z+\beta z^3)}{\Phi (\alpha z+\beta z^3)}\right] ^2\phi (z)\Phi (\alpha z+\beta z^3)\,dz\\= & {} \int _{-\infty }^{\infty }2z^r\frac{\phi ^2(\alpha z+\beta z^3)\phi (z)}{\Phi (\alpha z+\beta z^3)}\,dz \end{aligned}$$
Using a change of variables, we have the following expression for the expected value \(c_r\)
$$\begin{aligned} c_r=\int _{0}^{\infty }2(-z)^r\frac{\phi ^2(\alpha z+\beta z^3)\phi (z)}{1-\Phi (\alpha z+\beta z^3)}\,dz+\int _{0}^{\infty }2z^r\frac{\phi ^2(\alpha z+\beta z^3)\phi (z)}{\Phi (\alpha z+\beta z^3)}\,dz \end{aligned}$$
If r is even, \(c_r\) is reduced to the expression defined as \(d_r\)
$$\begin{aligned} c_r= & {} \int _{0}^{\infty }2z^r\phi ^2(\alpha z+\beta z^3)\phi (z)\left\{ \frac{1}{1-\Phi (\alpha z+\beta z^3)}+\frac{1}{\Phi (\alpha z+\beta z^3)}\right\} \,dz\\= & {} \int _{0}^{\infty }2z^r\frac{\phi ^2(\alpha z+\beta z^3)\phi (z)}{(1-\Phi (\alpha z+\beta z^3))\Phi (\alpha z+\beta z^3)}\,dz\\= & {} d_r \end{aligned}$$
Otherwise, if r is odd, \(c_r\) is reduced to:
$$\begin{aligned} c_r= & {} \int _{0}^{\infty }2z^r\phi ^2(\alpha z+\beta z^3)\phi (z)\left\{ \frac{-1}{1-\Phi (\alpha z+\beta z^3)}+\frac{1}{\Phi (\alpha z+\beta z^3)}\right\} \,dz\\= & {} \int _{0}^{\infty }2z^r\phi ^2(\alpha z+\beta z^3)\phi (z)\left\{ \frac{1-2\Phi (\alpha z+\beta z^3)}{(1-\Phi (\alpha z+\beta z^3))\Phi (\alpha z+\beta z^3)}\right\} \,dz\\= & {} \int _{0}^{\infty }2z^r\frac{\phi ^2(\alpha z+\beta z^3)\phi (z)}{(1-\Phi (\alpha z+\beta z^3))\Phi (\alpha z+\beta z^3)}\,dz-\int _{0}^{\infty }4z^r\frac{\phi ^2(\alpha z+\beta z^3)\phi (z)}{1-\Phi (\alpha z+\beta z^3)}\,dz\\= & {} d_r-\int _{0}^{\infty }4z^r\frac{\phi ^2(\alpha z+\beta z^3)\phi (z)}{1-\Phi (\alpha z+\beta z^3)}\,dz. \end{aligned}$$
\(\square \)
