Explicit formulae and implication of the expected values of some nonlinear statistics of tri-variate Gaussian variables

Abstract

We obtain explicit formulae for the expected values $E \{ \prod \nolimits _{i=1}^{3}g_i ( X_i ) \}$ of standard tri-variate Gaussian random vector ${\underline{X}}= \left( X_1, X_2 , X_3 \right)$ over the set $g_i (x) \in \left\{ \delta (x), \mathrm {sgn}(x), |x|, x \right\}$ of nonlinear and linear functions. Based on the results, we also suggest corrections to long-known formulae for two incomplete moments. Applications of the formulae in practical examples are also illustrated concisely. For easy reference for the readers, the explicit formulae derived in this paper and related well-known results are summarized in tables.

Appendix: Proof of lemma and theorems

Proof of Lemma 1

Noting that

$$\begin{aligned} f_3 (0 , 0, z)= & {} f_3 (0 , 0, 0) \exp \left\{ -\frac{ \left( 1- \rho _{12}^2 \right) }{2 \left| K_3 \right| } z^2\right\} \end{aligned}$$

(52)

and

$$\begin{aligned}&f_3 (0 , y, z) = f_3 (0 , 0, 0) \exp \left\{ - \frac{1}{2} (0 \ y \ z ) K_3^{-1}(0 \ y \ z )^T \right\} \nonumber \\&\quad = \ f_3 (0 , 0, 0) \exp \left\{ -\frac{1}{2 \left( 1- \beta _{23,1}^2 \right) } \left( \frac{y^2}{ 1- \rho _{12}^2 } - \frac{2 \beta _{23,1} yz}{\alpha _{23,1}} + \frac{z^2}{1- \rho _{31}^2 } \right) \right\} , \nonumber \\ \end{aligned}$$

(53)

Lemma 1 is immediately proved, where $\alpha _{ij,k} = \sqrt{ ( 1- \rho _{jk}^2 ) ( 1-\rho _{ki}^2 )}$ for $(i,j,k) \in \{(1, 2, 3), (2, 3, 1), (3, 1, 2)\}$. $\square$

Proof of Theorem 1

It is straightforward to have

$$\begin{aligned} E\left\{ \delta \left( X_1 \right) \delta \left( X_2 \right) \left| X_3 \right| \right\}= & {} \frac{1}{ 2\pi \sqrt{ 1-\rho _{12}^2} }E \left\{ \left| \frac{\sqrt{\left| K_3 \right| }}{ \sqrt{1-\rho _{12}^2}} U \right| \right\} \nonumber \\= & {} \ \frac{ \sqrt{\left| K_3 \right| }}{ 2\pi \left( 1-\rho _{12}^2 \right) }E \{ | U | \} = \frac{ \sqrt{\left| K_3 \right| } }{\sqrt{2 \pi ^3} \left( 1-\rho _{12}^2 \right) } \end{aligned}$$

(54)

using (6) and

$$\begin{aligned} E\left\{ \delta \left( X_1 \right) X_2 X_3 \right\}= & {} \frac{1}{\sqrt{2\pi } } E \left\{ \sqrt{1-\rho _{12}^2} V_1 \sqrt{1-\rho _{31}^2} V_2 \right\} \nonumber \\= & {} \ \frac{\sqrt{1-\rho _{12}^2} \sqrt{1-\rho _{31}^2}}{\sqrt{2\pi } } \beta _{23,1} = \frac{ \rho _{23}-\rho _{31}\rho _{12} }{\sqrt{2\pi } } \end{aligned}$$

(55)

using (7). Again using (7), we get

$$\begin{aligned} E\left\{ \delta \left( X_1 \right) \mathrm {sgn}\left( X_2 \right) X_3 \right\}= & {} \frac{1}{\sqrt{2\pi } } E \left\{ \mathrm {sgn}\left( \sqrt{1-\rho _{12}^2} V_1 \right) \sqrt{1-\rho _{31}^2} V_2 \right\} \nonumber \\= & {} \ \frac{\sqrt{1-\rho _{31}^2}}{\sqrt{2\pi } } E \left\{ V_2 \mathrm {sgn}( V_1 ) \right\} = \frac{\sqrt{1-\rho _{31}^2}}{\sqrt{2\pi } } \sqrt{\frac{2}{\pi }}\beta _{23,1} \nonumber \\= & {} \ \frac{\rho _{23}-\rho _{31}\rho _{12} }{\pi \sqrt{1-\rho _{12}^2} }, \end{aligned}$$

(56)

$$\begin{aligned} E\left\{ \delta \left( X_1 \right) \mathrm {sgn}\left( X_2 \right) \mathrm {sgn}\left( X_3 \right) \right\}= & {} \frac{1}{\sqrt{2\pi } } E \left\{ \mathrm {sgn}\left( \sqrt{1-\rho _{12}^2} V_1 \right) \mathrm {sgn}\left( \sqrt{1-\rho _{31}^2} V_2 \right) \right\} \nonumber \\= & {} \ \frac{1}{\sqrt{2\pi } } E \left\{ \mathrm {sgn}\left( V_1 \right) \mathrm {sgn}\left( V_2 \right) \right\} = \sqrt{ \frac{2}{\pi ^3}} \sin ^{-1} \beta _{23,1},\nonumber \\ \end{aligned}$$

(57)

and

$$\begin{aligned} E\left\{ \delta \left( X_1 \right) \left| X_2 \right| \left| X_3 \right| \right\}= & {} \frac{1}{\sqrt{2\pi } } E \left\{ \left| \sqrt{1-\rho _{12}^2} V_1 \sqrt{1-\rho _{31}^2} V_2 \right| \right\} \nonumber \\= & {} \ \frac{\alpha _{23,1}}{\sqrt{2\pi } } \frac{2}{\pi } \left( \beta _{23,1} \sin ^{-1} \beta _{23,1} + \sqrt{1-\beta _{23,1}^2} \right) \nonumber \\= & {} \ \sqrt{ \frac{2}{\pi ^3 } } \left\{ \sqrt{ \left| K_3 \right| } + \left( \rho _{23}-\rho _{31}\rho _{12} \right) \sin ^{-1} \beta _{23,1} \right\} , \end{aligned}$$

(58)

which complete the proof. $\square$

Proof of Theorem 2: Eqs. (14) and (15)

From the theorem in Price (1958), by letting ${\tilde{\Upsilon }} = E \{ X_1 X_2 | X_3 | \}$, we have $\frac{\partial }{\partial \rho _{12} } {\tilde{\Upsilon }} = E \{ | X_3 |\} = \sqrt{\frac{2}{\pi }}$, $\frac{\partial }{\partial \rho _{23} } {\tilde{\Upsilon }} = E \{ X_1 \mathrm {sgn}( X_3 ) \} = \sqrt{\frac{2}{\pi }} \rho _{31}$, and $\frac{\partial }{\partial \rho _{31} } {\tilde{\Upsilon }} = E\{ X_2 \mathrm {sgn}( X_3 ) \} = \sqrt{\frac{2}{\pi }} \rho _{23}$. In addition, when $\rho _{12} = \rho _{23} = \rho _{31} =0$, we have ${\tilde{\Upsilon }} = E \{ X_1 X_2 | X_3 | \} = E \{ X_1 \} E \{ X_2 \} E \{ | X_3 | \} =0$. Based on these observations, we have (14) after some straightforward steps.

Similarly, let ${\tilde{\Upsilon }} = E\{X_1 | X_2 | \mathrm {sgn}( X_3 ) \}$. Then $\frac{\partial }{\partial \rho _{12}} {\tilde{\Upsilon }} = E\{\mathrm {sgn}( X_2 ) \mathrm {sgn}( X_3 ) \} = \frac{2}{\pi } \sin ^{-1} \rho _{23}$ from the theorem in Price (1958). Now, putting ${\tilde{\Upsilon }} = \frac{2}{\pi } \rho _{12} \sin ^{-1} \rho _{23} + \zeta _1 ( \rho _{23}, \rho _{31} )$, we have $\zeta _1 ( \rho _{23}, \rho _{31}) = \frac{2}{\pi } \rho _{31} \sqrt{ 1- \rho _{23}^2} + \zeta _2 ( \rho _{23})$ since $\frac{\partial }{\partial \rho _{31}} {\tilde{\Upsilon }} = \frac{\partial }{\partial \rho _{31}} \zeta _1 ( \rho _{23}, \rho _{31} ) = 2 E\{ | X_2 | \delta ( X_3 ) \}= \frac{2}{\pi } \sqrt{ 1- \rho _{23}^2}$. To determine $\zeta _1 ( \rho _{23}, \rho _{31} )$ let us next note that $\frac{\partial {\tilde{\Upsilon }} }{\partial \rho _{23}} = 2 E\{X_1 \mathrm {sgn}( X_2 ) \delta ( X_3 ) \}$: since $2 E\{X_1 \mathrm {sgn}( X_2 ) \delta ( X_3 ) \} = \frac{2}{\pi }\frac{\rho _{12} - \rho _{23}\rho _{31}}{ \sqrt{1- \rho _{23}^2}}$ from (11) and $\frac{\partial {\tilde{\Upsilon }} }{\partial \rho _{23}} = \frac{2}{\pi }\frac{\rho _{12}}{ \sqrt{1- \rho _{23}^2}} + \frac{2}{\pi }\frac{\rho _{31} ( -2\rho _{23} )}{ 2\sqrt{1- \rho _{23}^2}} + \zeta _2^{\prime } ( \rho _{23} )$, it is clear that $\zeta _2 ( \rho _{23} )$ is a constant. Subsequently, since $\left. E\{X_1 | X_2 | \mathrm {sgn}( X_3 ) \} \right| _{\rho _{12}=\rho _{23}=\rho _{31}=0} = 0$, we have $\zeta _2 ( \rho _{23} ) =0$. Consequently, we have (15). $\square$

Proof of Theorem 2: Eqs. (16) and (17)

To explicitly denote the dependence of the pdf on the correlation coefficients ${\underline{\rho }} = ( \rho _{12}, \rho _{23}, \rho _{31})$, let us now write $f_3 (x_1 , x_2, x_3 )$ as $g_{{\underline{\rho }}} (x_1 , x_2, x_3 )$. Then we clearly have

$$\begin{aligned} g_{{\underline{\rho }}} \left( -x_1 , x_2, x_3 \right)= & {} \left. g_{{\underline{\rho }}} \left( x_1 , x_2, x_3 \right) \right| _{\langle 1\rangle } , \end{aligned}$$

(59)

$$\begin{aligned} g_{{\underline{\rho }}} \left( x_1 , -x_2, x_3 \right)= & {} \left. g_{{\underline{\rho }}} \left( x_1 , x_2, x_3 \right) \right| _{\langle 2\rangle }, \end{aligned}$$

(60)

and

$$\begin{aligned} g_{{\underline{\rho }}} \left( x_1 , x_2,- x_3 \right)= & {} \left. g_{{\underline{\rho }}} \left( x_1 , x_2, x_3 \right) \right| _{\langle 3\rangle }. \end{aligned}$$

(61)

In addition, we have

$$\begin{aligned}&\int _{0}^{\infty } \int _{0}^{\infty } \int _{-\infty }^0 h\left( x_1 , x_2, x_3 \right) g_{{\underline{\rho }}} \left( x_1 , x_2, x_3 \right) dx_1 dx_2 dx_3 \nonumber \\&\quad = \ \int _{0}^{\infty } \int _{0}^{\infty } \int _{\infty }^0 h\left( x_1 , x_2, -t \right) g_{{\underline{\rho }}}\left( x_1 , x_2, -t \right) dx_1 dx_2 (-dt) \nonumber \\&\quad = \ \left. \int _{0}^{\infty } \int _{0}^{\infty } \int _{0}^{\infty } h\left( x_1 , x_2, -x_3 \right) g_{{\underline{\rho }}} \left( x_1 , x_2, x_3 \right) dx_1 dx_2 dx_3 \right| _{\langle 3\rangle } , \ \ \ \ \end{aligned}$$

(62)

for example. We also have

$$\begin{aligned} \left. \beta _{ij,k} \right| _{\langle i\rangle } = \left. \beta _{ij,k} \right| _{\langle j\rangle }= & {} - \left. \beta _{ij,k} \right| _{\langle k\rangle } = - \beta _{ij,k} , \end{aligned}$$

(63)

$$\begin{aligned} \frac{\partial \sin ^{-1} \beta _{ij,k}}{\partial \rho _{ij}}= & {} \frac{1}{ \sqrt{ \left| K_3 \right| }} , \end{aligned}$$

(64)

and

$$\begin{aligned} \frac{\partial \sin ^{-1} \beta _{ij,k}}{\partial \rho _{jk}}= & {} - \frac{\rho _{ki}-\rho _{ij}\rho _{jk}}{ \left( 1-\rho _{jk}^2 \right) \sqrt{ \left| K_3 \right| } } . \end{aligned}$$

(65)

Now, for the incomplete moment [r, s, t] and absolute moment $\nu _{rst}$, we have

$$\begin{aligned}{}[1,0,0]= & {} \frac{1}{\sqrt{8 \pi ^3}} \left\{ \frac{\pi }{2} + \sin ^{-1} \beta _{23,1} + \rho _{12} \left( \frac{\pi }{2} + \sin ^{-1} \beta _{31,2} \right) \right. \nonumber \\&\left. + \rho _{31} \left( \frac{\pi }{2} + \sin ^{-1} \beta _{12,3} \right) \right\} , \end{aligned}$$

(66)

$$\begin{aligned}= & {} \frac{1}{\sqrt{8 \pi ^3}} \left\{ \sqrt{ \left| K_3 \right| } + \sum ^c \left( \rho _{ij} + \rho _{jk} \rho _{ki} \right) \left( \frac{\pi }{2} + \sin ^{-1} \beta _{ij,k} \right) \right\} , \end{aligned}$$

(67)

$$\begin{aligned} \nu _{211}= & {} \frac{2}{\pi } \left\{ \left( \rho _{23}+2\rho _{12}\rho _{31} \right) \sin ^{-1}\rho _{23} + \left( 1+ \rho _{12}^2+\rho _{31}^2 \right) \sqrt{1-\rho _{23}^2}\right\} , \end{aligned}$$

(68)

and

$$\begin{aligned} \nu _{221}= & {} \sqrt{\frac{2}{\pi }} \left( 1+2\rho _{12}^2 +\rho _{23}^2+\rho _{31}^2 + 4\rho _{12}\rho _{23}\rho _{31} - \rho _{23}^2\rho _{31}^2 \right) \end{aligned}$$

(69)

from Nabeya (1952), Kamat (1953), Kamat (1958), and Johnson and Kotz (1972). Recollecting (59)–(63), we easily get (16) as

$$\begin{aligned} E \left\{ \left| X_1 X_2 X_3 \right| \right\}= & {} \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } \int _{-\infty }^{\infty } \left| x_1 x_2 x_3 \right| g_{{\underline{\rho }}} \left( x_1 , x_2, x_3 \right) dx_1 dx_2 dx_3 \nonumber \\= & {} \ [1, 1, 1] + [1, 1, 1]_{<3>} + [1, 1, 1]_{<2>} + [1, 1, 1]_{<1>} \nonumber \\&+ [1, 1, 1]_{<1>} + [1, 1, 1]_{<2>} + [1, 1, 1]_{<3>} + [1, 1, 1] \nonumber \\= & {} \ \sqrt{\frac{8}{\pi ^3}} \left\{ \sqrt{\left| K_3 \right| } + \sum ^c \left( \rho _{ij} + \rho _{jk} \rho _{ki} \right) \sin ^{-1} \beta _{ij,k} \right\} \end{aligned}$$

(70)

using (67).

The result (17) can similarly be shown using (66): here, we provide another proof for (17). Again applying the theorem in Price (1958), we have

$$\begin{aligned}&E \left\{ \mathrm {sgn}\left( X_1 \right) \mathrm {sgn}\left( X_2 \right) \left| X_3 \right| \right\} = \frac{\partial }{\partial \rho _{12}} E \left\{ \left| X_1 X_2 X_3 \right| \right\} . \end{aligned}$$

(71)

Now we have

$$\begin{aligned}&\qquad \qquad \qquad \frac{\partial \sqrt{ \left| K_3 \right| }}{\partial \rho _{12}} \ = \ - \frac{\rho _{12} - \rho _{23}\rho _{31}}{\sqrt{ \left| K_3 \right| } } , \end{aligned}$$

(72)

$$\begin{aligned}&\frac{\partial }{\partial \rho _{12}} \left\{ \left( \rho _{12} + \rho _{23}\rho _{31} \right) \sin ^{-1} \beta _{12,3} \right\} = \sin ^{-1} \beta _{12,3} + \frac{ \rho _{12} + \rho _{23}\rho _{31} }{ \sqrt{ \left| K_3 \right| }} , \end{aligned}$$

(73)

$$\begin{aligned}&\frac{\partial }{\partial \rho _{12}}\left\{ \left( \rho _{23} + \rho _{31}\rho _{12} \right) \sin ^{-1} \beta _{23,1}\right\} = \rho _{31} \sin ^{-1} \beta _{23,1}\nonumber \\&\qquad + \frac{\left( \rho _{23} + \rho _{31}\rho _{12}\right) \left( -\rho _{31} + \rho _{12}\rho _{23}\right) }{ \sqrt{ \left| K_3 \right| } \left( 1-\rho _{12}^2 \right) } , \end{aligned}$$

(74)

and

$$\begin{aligned}&\frac{\partial }{\partial \rho _{12}} \left\{ \left( \rho _{31} + \rho _{12}\rho _{23} \right) \sin ^{-1} \beta _{31,2}\right\} = \rho _{23} \sin ^{-1} \beta _{31,2} \nonumber \\&\qquad + \frac{\left( \rho _{31} + \rho _{12}\rho _{23}\right) \left( -\rho _{23} + \rho _{31}\rho _{12}\right) }{\sqrt{ \left| K_3 \right| } \left( 1-\rho _{12}^2 \right) }. \end{aligned}$$

(75)

Note also that

$$\begin{aligned}&\frac{ \rho _{12} + \rho _{23}\rho _{31} }{ \sqrt{ \left| K_3 \right| }} + \frac{\left( \rho _{23} + \rho _{31}\rho _{12}\right) \left( -\rho _{31} + \rho _{12}\rho _{23}\right) }{ \sqrt{ \left| K_3 \right| } \left( 1-\rho _{12}^2 \right) } \nonumber \\&\quad \quad + \frac{\left( \rho _{31} + \rho _{12}\rho _{23}\right) \left( -\rho _{23} + \rho _{31}\rho _{12}\right) }{\sqrt{ \left| K_3 \right| } \left( 1-\rho _{12}^2 \right) } \nonumber \\&\quad = \ \frac{ \rho _{12} + \rho _{23}\rho _{31} }{ \sqrt{ \left| K_3 \right| }} - \frac{ 2 \rho _{23}\rho _{31} }{ \sqrt{ \left| K_3 \right| }} = \frac{\rho _{12} - \rho _{23}\rho _{31}}{\sqrt{ \left| K_3 \right| }} \end{aligned}$$

(76)

since $\left( \rho _{23} + \rho _{31}\rho _{12}\right) \left( -\rho _{31} + \rho _{12}\rho _{23}\right) + \left( \rho _{31} + \rho _{12}\rho _{23}\right) \left( -\rho _{23} + \rho _{31}\rho _{12}\right) = -2 \rho _{23} \rho _{31} \left( 1- \rho _{12}^2 \right)$. Using (72)–(76), it is straightforward to get (17) from (71). $\square$