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.
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Acknowledgements
This study was supported by the National Research Foundation of Korea under Grant NRF-2018R1A2A1A05023192 and by the 2018 Research Fund of The Catholic University of Korea, for which the authors wish to express their thanks. The authors would also like to thank the Editor in Chief, Associate Editor, and two anonymous reviewers for their constructive suggestions and helpful comments in the reviewing rounds.
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Appendix: Proof of lemma and theorems
Appendix: Proof of lemma and theorems
Proof of Lemma 1
Noting that
and
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
using (6) and
using (7). Again using (7), we get
and
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
and
In addition, we have
for example. We also have
and
Now, for the incomplete moment [r, s, t] and absolute moment \(\nu _{rst}\), we have
and
from Nabeya (1952), Kamat (1953), Kamat (1958), and Johnson and Kotz (1972). Recollecting (59)–(63), we easily get (16) as
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
Now we have
and
Note also that
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). \(\square \)
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Song, I., Lee, S., Kim, Y.H. et al. Explicit formulae and implication of the expected values of some nonlinear statistics of tri-variate Gaussian variables. J. Korean Stat. Soc. 49, 117–138 (2020). https://doi.org/10.1007/s42952-019-00009-9
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DOI: https://doi.org/10.1007/s42952-019-00009-9
Keywords
- Absolute moment
- Impulse function
- Magnitude
- Nonlinear function
- Signum function
- Tri-variate Gaussian distribution