IDEAS home Printed from https://ideas.repec.org/p/azt/cemmap/16-04.html
   My bibliography  Save this paper

Spatial design matrices and associated quadratic forms: structure and properties

Author

Listed:
  • Grant Hillier
  • Federico Martellosio

Abstract

The paper provides significant simplifications and extensions of results obtained by Gorsich, Genton, and Strang (J. Multivariate Anal. 80 (2002) 138) on the structure of spatial design matrices. These are the matrices implicitly defined by quadratic forms that arise naturally in modelling intrinsically stationary and isotropic spatial processes. We give concise structural formulae for these matrices, and simple generating functions for them. The generating functions provide formulae for the cumulants of the quadratic forms of interest when the process is Gaussian, second-order stationary and isotropic. We use these to study the statistical properties of the associated quadratic forms, in particular those of the classical variogram estimator, under several assumptions about the actual variogram.

Suggested Citation

  • Grant Hillier & Federico Martellosio, 2004. "Spatial design matrices and associated quadratic forms: structure and properties," CeMMAP working papers 16/04, Institute for Fiscal Studies.
    • Handle: RePEc:azt:cemmap:16/04
    DOI: 10.1920/wp.cem.2004.1604
    as

    Download full text from publisher

    File URL: https://www.cemmap.ac.uk/wp-content/uploads/2020/08/CWP1604.pdf
    Download Restriction: no
    File URL: https://libkey.io/10.1920/wp.cem.2004.1604?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
    ---><---

    References listed on IDEAS

    as
    1. Hillier, Grant, 2001. "THE DENSITY OF A QUADRATIC FORM IN A VECTOR UNIFORMLY DISTRIBUTED ON THE n-SPHERE," Econometric Theory, Cambridge University Press, vol. 17(1), pages 1-28, February.
    2. Gorsich, David J. & Genton, Marc G. & Strang, Gilbert, 2002. "Eigenstructures of Spatial Design Matrices," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 138-165, January.
    3. Ali, Mukhtar M, 1987. "Durbin-Watson and Generalized Durbin-Watson Tests for Autocorrelations and Randomness," Journal of Business & Economic Statistics, American Statistical Association, vol. 5(2), pages 195-203, April.
    Full references (including those not matched with items on IDEAS)

    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. Hillier, Grant & Martellosio, Federico, 2006. "Spatial design matrices and associated quadratic forms: structure and properties," Journal of Multivariate Analysis, Elsevier, vol. 97(1), pages 1-18, January.
    2. Dong Ding & Axel Gandy & Georg Hahn, 2020. "A simple method for implementing Monte Carlo tests," Computational Statistics, Springer, vol. 35(3), pages 1373-1392, September.
    3. J. Roderick McCrorie, 2021. "Moments in Pearson's Four-Step Uniform Random Walk Problem and Other Applications of Very Well-Poised Generalized Hypergeometric Series," Sankhya B: The Indian Journal of Statistics, Springer;Indian Statistical Institute, vol. 83(2), pages 244-281, November.
    4. Hillier, Grant & Kan, Raymond & Wang, Xiaolu, 2009. "Computationally Efficient Recursions For Top-Order Invariant Polynomials With Applications," Econometric Theory, Cambridge University Press, vol. 25(1), pages 211-242, February.
    5. Lu, Zeng-Hua, 2006. "The numerical evaluation of the probability density function of a quadratic form in normal variables," Computational Statistics & Data Analysis, Elsevier, vol. 51(3), pages 1986-1996, December.
    6. Robinson, Peter M. & Rossi, Francesca, 2012. "Improved tests for spatial correlation," MPRA Paper 41835, University Library of Munich, Germany.
    7. Gorsich, David J. & Genton, Marc G. & Strang, Gilbert, 2002. "Eigenstructures of Spatial Design Matrices," Journal of Multivariate Analysis, Elsevier, vol. 80(1), pages 138-165, January.
    8. Zeng-Hua Lu & Maxwell King, 2002. "Improving The Numerical Technique For Computing The Accumulated Distribution Of A Quadratic Form In Normal Variables," Econometric Reviews, Taylor & Francis Journals, vol. 21(2), pages 149-165.
    9. Bogdan-Cristian Chiripuci & Marius Constantin & Maria-Floriana Popescu & Albert Scrieciu, 2020. "The Socio-Economic Impact of Migration on the Labor Market in the Romanian Danube Region," Sustainability, MDPI, vol. 12(20), pages 1-26, October.
    10. Aman Ullah & Yong Bao & Yun Wang, 2014. "Exact Distribution of the Mean Reversion Estimator in the Ornstein-Uhlenbeck Process," Working Papers 201413, University of California at Riverside, Department of Economics.
    11. Grant Hillier & Federico Martellosio, 2013. "Properties of the maximum likelihood estimator in spatial autoregressive models," CeMMAP working papers CWP44/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
    12. Md. Samiul Basir & Samuel Noel & Dennis Buckmaster & Muhammad Ashik-E-Rabbani, 2024. "Enhancing Subsurface Soil Moisture Forecasting: A Long Short-Term Memory Network Model Using Weather Data," Agriculture, MDPI, vol. 14(3), pages 1-24, February.
    13. Genton, Marc G. & Gorsich, David J., 2002. "Nonparametric variogram and covariogram estimation with Fourier-Bessel matrices," Computational Statistics & Data Analysis, Elsevier, vol. 41(1), pages 47-57, November.
    14. A. García-Pérez, 2020. "Saddlepoint approximations for the distribution of some robust estimators of the variogram," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 83(1), pages 69-91, January.
    15. Reinaldo Arellano-Valle & Marc Genton, 2010. "An invariance property of quadratic forms in random vectors with a selection distribution, with application to sample variogram and covariogram estimators," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 62(2), pages 363-381, April.
    16. Genton, Mark G. & Ruiz-Gazen, Anne, 2009. "Visualizing Influential Observations in Dependent Data," TSE Working Papers 09-051, Toulouse School of Economics (TSE).
    17. Patrick Marsh, 2019. "Properties of the power envelope for tests against both stationary and explosive alternatives: the effect of trends," Discussion Papers 19/03, University of Nottingham, Granger Centre for Time Series Econometrics.
    18. repec:cep:stiecm:/2013/565 is not listed on IDEAS
    19. Kim, Hyoung-Moon & Mallick, Bani K., 2003. "Moments of random vectors with skew t distribution and their quadratic forms," Statistics & Probability Letters, Elsevier, vol. 63(4), pages 417-423, July.
    20. Grant Hillier & Federico Martellosio, 2013. "Properties of the maximum likelihood estimator in spatial autoregressive models," CeMMAP working papers 44/13, Institute for Fiscal Studies.
    21. Mihai Dinu & Simona Roxana Pătărlăgeanu & Radu Petrariu & Marius Constantin & Ana-Mădălina Potcovaru, 2020. "Empowering Sustainable Consumer Behavior in the EU by Consolidating the Roles of Waste Recycling and Energy Productivity," Sustainability, MDPI, vol. 12(23), pages 1-24, November.

    More about this item

    Statistics

    Access and download statistics

    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:azt:cemmap:16/04. 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: Dermot Watson (email available below). General contact details of provider: https://edirc.repec.org/data/ifsssuk.html .

    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.