IDEAS home Printed from https://ideas.repec.org/a/eee/jmvana/v183y2021ics0047259x21000075.html
   My bibliography  Save this article

A non-recursive formula for various moments of the multivariate normal distribution with sectional truncation

Author

Listed:
  • Ogasawara, Haruhiko

Abstract

A unified formula for various moments of the multivariate normal distribution with sectional truncation is derived using a non-recursive method, where sectional truncation is given by several sections (regions) for selection including single and double truncation as special cases. The moments include raw, central, arbitrarily deviated, non-absolute, absolute and partially absolute moments with non-integer orders for variables taking absolute values. The formula is alternatively shown using weighted Kummer’s confluent hypergeometric function and, in the bivariate case, the weighted Gauss hypergeometric function, where the weighted functions have advantages of fast convergence. Numerical illustrations with simulations show that the methods employed are relatively free from accumulating cancellation errors.

Suggested Citation

  • Ogasawara, Haruhiko, 2021. "A non-recursive formula for various moments of the multivariate normal distribution with sectional truncation," Journal of Multivariate Analysis, Elsevier, vol. 183(C).
    • Handle: RePEc:eee:jmvana:v:183:y:2021:i:c:s0047259x21000075
    DOI: 10.1016/j.jmva.2021.104729
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0047259X21000075
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.jmva.2021.104729?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2018. "A multivariate tail covariance measure for elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 81(C), pages 27-35.
    2. Arismendi, Juan C. & Broda, Simon, 2017. "Multivariate elliptical truncated moments," Journal of Multivariate Analysis, Elsevier, vol. 157(C), pages 29-44.
    3. Barry Arnold & Robert Beaver & Richard Groeneveld & William Meeker, 1993. "The nontruncated marginal of a truncated bivariate normal distribution," Psychometrika, Springer;The Psychometric Society, vol. 58(3), pages 471-488, September.
    4. Philippe Artzner & Freddy Delbaen & Jean‐Marc Eber & David Heath, 1999. "Coherent Measures of Risk," Mathematical Finance, Wiley Blackwell, vol. 9(3), pages 203-228, July.
    5. Gupta, Arjun K. & González-Farías, Graciela & Domínguez-Molina, J. Armando, 2004. "A multivariate skew normal distribution," Journal of Multivariate Analysis, Elsevier, vol. 89(1), pages 181-190, April.
    6. Landsman, Zinoviy & Makov, Udi & Shushi, Tomer, 2016. "Multivariate tail conditional expectation for elliptical distributions," Insurance: Mathematics and Economics, Elsevier, vol. 70(C), pages 216-223.
    7. Z. Birnbaum & E. Paulson & F. Andrews, 1950. "On the effect of selection performed on some coordinates of a multi-dimensional population," Psychometrika, Springer;The Psychometric Society, vol. 15(2), pages 191-204, June.
    8. Liseo, Brunero & Loperfido, Nicola, 2003. "A Bayesian interpretation of the multivariate skew-normal distribution," Statistics & Probability Letters, Elsevier, vol. 61(4), pages 395-401, February.
    9. Arismendi, J.C., 2013. "Multivariate truncated moments," Journal of Multivariate Analysis, Elsevier, vol. 117(C), pages 41-75.
    10. William Horrace, 2015. "Moments of the truncated normal distribution," Journal of Productivity Analysis, Springer, vol. 43(2), pages 133-138, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Ogasawara, Haruhiko, 2023. "The density of the sample correlations under elliptical symmetry with or without the truncated variance-ratio," Journal of Multivariate Analysis, Elsevier, vol. 195(C).
    2. Galarza, Christian E. & Matos, Larissa A. & Castro, Luis M. & Lachos, Victor H., 2022. "Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    3. Baishuai Zuo & Chuancun Yin, 2022. "Multivariate doubly truncated moments for generalized skew-elliptical distributions with application to multivariate tail conditional risk measures," Papers 2203.00839, arXiv.org.
    4. Christopher J. Adcock, 2022. "Properties and Limiting Forms of the Multivariate Extended Skew-Normal and Skew-Student Distributions," Stats, MDPI, vol. 5(1), pages 1-42, March.
    5. Baishuai Zuo & Chuancun Yin & Jing Yao, 2023. "Multivariate range Value-at-Risk and covariance risk measures for elliptical and log-elliptical distributions," Papers 2305.09097, arXiv.org.
    6. Ogasawara, Haruhiko, 2023. "The Wishart distribution with two different degrees of freedom," Statistics & Probability Letters, Elsevier, vol. 200(C).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Baishuai Zuo & Chuancun Yin & Jing Yao, 2023. "Multivariate range Value-at-Risk and covariance risk measures for elliptical and log-elliptical distributions," Papers 2305.09097, arXiv.org.
    2. Baishuai Zuo & Chuancun Yin, 2022. "Multivariate doubly truncated moments for generalized skew-elliptical distributions with application to multivariate tail conditional risk measures," Papers 2203.00839, arXiv.org.
    3. Roozegar, Roohollah & Balakrishnan, Narayanaswamy & Jamalizadeh, Ahad, 2020. "On moments of doubly truncated multivariate normal mean–variance mixture distributions with application to multivariate tail conditional expectation," Journal of Multivariate Analysis, Elsevier, vol. 177(C).
    4. Reinaldo B. Arellano-Valle & Adelchi Azzalini, 2022. "Some properties of the unified skew-normal distribution," Statistical Papers, Springer, vol. 63(2), pages 461-487, April.
    5. Sharon Lee & Geoffrey McLachlan, 2013. "On mixtures of skew normal and skew $$t$$ -distributions," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 7(3), pages 241-266, September.
    6. Cheung, Eric C.K. & Peralta, Oscar & Woo, Jae-Kyung, 2022. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Insurance: Mathematics and Economics, Elsevier, vol. 106(C), pages 364-389.
    7. Eric C. K. Cheung & Oscar Peralta & Jae-Kyung Woo, 2021. "Multivariate matrix-exponential affine mixtures and their applications in risk theory," Papers 2201.11122, arXiv.org.
    8. Sharon Lee & Geoffrey McLachlan, 2013. "Model-based clustering and classification with non-normal mixture distributions," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 22(4), pages 427-454, November.
    9. Galarza, Christian E. & Matos, Larissa A. & Castro, Luis M. & Lachos, Victor H., 2022. "Moments of the doubly truncated selection elliptical distributions with emphasis on the unified multivariate skew-t distribution," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    10. Denisa Banulescu-Radu & Christophe Hurlin & Jérémy Leymarie & Olivier Scaillet, 2021. "Backtesting Marginal Expected Shortfall and Related Systemic Risk Measures," Management Science, INFORMS, vol. 67(9), pages 5730-5754, September.
    11. Arismendi, Juan C. & Broda, Simon, 2017. "Multivariate elliptical truncated moments," Journal of Multivariate Analysis, Elsevier, vol. 157(C), pages 29-44.
    12. Samuel Kotz & Donatella Vicari, 2005. "Survey of developments in the theory of continuous skewed distributions," Metron - International Journal of Statistics, Dipartimento di Statistica, Probabilità e Statistiche Applicate - University of Rome, vol. 0(2), pages 225-261.
    13. Baishuai Zuo & Chuancun Yin, 2020. "Conditional tail risk expectations for location-scale mixture of elliptical distributions," Papers 2007.09350, arXiv.org.
    14. Baishuai Zuo & Chuancun Yin, 2021. "Multivariate tail covariance for generalized skew-elliptical distributions," Papers 2103.05201, arXiv.org.
    15. Marcel, Bräutigam & Marie, Kratz, 2018. "On the Dependence between Quantiles and Dispersion Estimators," ESSEC Working Papers WP1807, ESSEC Research Center, ESSEC Business School.
    16. Ansari, Jonathan & Shushi, Tomer & Vanduffel, Steven, 2024. "Up- and down-correlations in normal variance mixture models," Statistics & Probability Letters, Elsevier, vol. 205(C).
    17. Bolance, Catalina & Guillen, Montserrat & Pelican, Elena & Vernic, Raluca, 2008. "Skewed bivariate models and nonparametric estimation for the CTE risk measure," Insurance: Mathematics and Economics, Elsevier, vol. 43(3), pages 386-393, December.
    18. Steven Culpepper, 2012. "Using the Criterion-Predictor Factor Model to Compute the Probability of Detecting Prediction Bias with Ordinary Least Squares Regression," Psychometrika, Springer;The Psychometric Society, vol. 77(3), pages 561-580, July.
    19. Selma Chaker & Nour Meddahi, 2013. "CoMargin," Staff Working Papers 13-47, Bank of Canada.
    20. Kim, Hyoung-Moon & Ryu, Duchwan & Mallick, Bani K. & Genton, Marc G., 2014. "Mixtures of skewed Kalman filters," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 228-251.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:jmvana:v:183:y:2021:i:c:s0047259x21000075. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.elsevier.com/wps/find/journaldescription.cws_home/622892/description#description .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.