New Methods for Multivariate Normal Moments
Multivariate normal moments are foundational for statistical methods. The derivation and simplification of these moments are critical for the accuracy of various statistical estimates and analyses. Normal moments are the building blocks of the Hermite polynomials, which in turn are the building bloc...
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MDPI AG
2025-06-01
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| Series: | Stats |
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| Online Access: | https://www.mdpi.com/2571-905X/8/2/46 |
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| author | Christopher Stroude Withers |
| author_facet | Christopher Stroude Withers |
| author_sort | Christopher Stroude Withers |
| collection | DOAJ |
| description | Multivariate normal moments are foundational for statistical methods. The derivation and simplification of these moments are critical for the accuracy of various statistical estimates and analyses. Normal moments are the building blocks of the Hermite polynomials, which in turn are the building blocks of the Edgeworth expansions for the distribution of parameter estimates. Isserlis (1918) gave the bivariate normal moments and two special cases of trivariate moments. Beyond that, convenient expressions for multivariate variate normal moments are still not available. We compare three methods for obtaining them, the most powerful being the differential method. We give simpler formulas for the bivariate moment than that of Isserlis, and explicit expressions for the general moments of dimensions 3 and 4. |
| format | Article |
| id | doaj-art-a1e3d3689eb14cdcafe5ae0b622a5bb5 |
| institution | Kabale University |
| issn | 2571-905X |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Stats |
| spelling | doaj-art-a1e3d3689eb14cdcafe5ae0b622a5bb52025-08-20T03:29:44ZengMDPI AGStats2571-905X2025-06-01824610.3390/stats8020046New Methods for Multivariate Normal MomentsChristopher Stroude Withers0Callaghan Innovation (Formerly Industrial Research Ltd.), 101 Allington Road, Wellington 6012, New ZealandMultivariate normal moments are foundational for statistical methods. The derivation and simplification of these moments are critical for the accuracy of various statistical estimates and analyses. Normal moments are the building blocks of the Hermite polynomials, which in turn are the building blocks of the Edgeworth expansions for the distribution of parameter estimates. Isserlis (1918) gave the bivariate normal moments and two special cases of trivariate moments. Beyond that, convenient expressions for multivariate variate normal moments are still not available. We compare three methods for obtaining them, the most powerful being the differential method. We give simpler formulas for the bivariate moment than that of Isserlis, and explicit expressions for the general moments of dimensions 3 and 4.https://www.mdpi.com/2571-905X/8/2/46multivariate normalmomentsHermite polynomialsIsserlisSoper |
| spellingShingle | Christopher Stroude Withers New Methods for Multivariate Normal Moments Stats multivariate normal moments Hermite polynomials Isserlis Soper |
| title | New Methods for Multivariate Normal Moments |
| title_full | New Methods for Multivariate Normal Moments |
| title_fullStr | New Methods for Multivariate Normal Moments |
| title_full_unstemmed | New Methods for Multivariate Normal Moments |
| title_short | New Methods for Multivariate Normal Moments |
| title_sort | new methods for multivariate normal moments |
| topic | multivariate normal moments Hermite polynomials Isserlis Soper |
| url | https://www.mdpi.com/2571-905X/8/2/46 |
| work_keys_str_mv | AT christopherstroudewithers newmethodsformultivariatenormalmoments |