Advanced Search
MyIDEAS: Login to save this paper or follow this series

The Exact Cumulative Distribution Function of a Ratio of Quadratic Forms in Normal Variables with Application to the AR(1) Model

Contents:

Author Info

  • Giovanni Forchini

Abstract

Often neither the exact density nor the exact cumulative distribution function (CDF) of a statistic of interest are available in the statistics and econometrics literature (for example the maximum likelihood estimator of the autocorrelation coefficient in a simple Gaussian AR(1) model with zero start-up value). In other cases the exact CDF of a statistic of interest is very complicated despite the statistic being “simple” (for example the circular serial correlation coefficient, or a quadratic form of a vector uniformly distributed over the unit n-sphere). The first part of the paper tries to explain why this is the case by studying the analytic properties of the CDF of a statistic under very general assumptions. Differential geometric considerations show that there can be points where the CDF of a given statistic is not analytic, and such points do not depend on the parameters of the model but only on the properties of the statistic itself. The second part of the paper derives the exact CDF of a ratio of quadratic forms in normal variables, and for the first time a closed form solution is found. These results are then specialised to the maximum likelihood estimator of the autoregressive parameter in a Gaussian AR(1) model with zero start-up value, which is shown to have precisely those properties highlighted in the first part of the paper.

Download Info

If you experience problems downloading a file, check if you have the proper application to view it first. In case of further problems read the IDEAS help page. Note that these files are not on the IDEAS site. Please be patient as the files may be large.
File URL: http://www.york.ac.uk/media/economics/documents/discussionpapers/2001/0102.pdf
File Function: Main text
Download Restriction: no

Bibliographic Info

Paper provided by Department of Economics, University of York in its series Discussion Papers with number 01/02.

as in new window
Length:
Date of creation:
Date of revision:
Handle: RePEc:yor:yorken:01/02

Contact details of provider:
Postal: Department of Economics and Related Studies, University of York, York, YO10 5DD, United Kingdom
Phone: (0)1904 323776
Fax: (0)1904 323759
Email:
Web page: http://www.york.ac.uk/economics/
More information through EDIRC

Related research

Keywords:

Other versions of this item:

References

No references listed on IDEAS
You can help add them by filling out this form.

Citations

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

Cited by:
  1. Giovanni Forchini, . "The Distribution of a Ratio of Quadratic Forms in Noncentral Normal Variables," Discussion Papers 01/12, Department of Economics, University of York.
  2. Patrick Marsh, . "A Measure of Distance for the Unit Root Hypothesis," Discussion Papers 05/02, Department of Economics, University of York.
  3. Robinson, Peter M. & Rossi, Francesca, 2012. "Improved tests for spatial correlation," MPRA Paper 41835, University Library of Munich, Germany.
  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(01), pages 211-242, February.
  5. Grant Hillier & Federico Martellosio, 2013. "Properties of the maximum likelihood estimator in spacial autoregressive models," CeMMAP working papers CWP44/13, Centre for Microdata Methods and Practice, Institute for Fiscal Studies.
  6. Vougas, Dimitrios V., 2006. "Remark on the asymptotic distribution of the OLS estimator in a simple Gaussian unit-root autoregression," Statistics & Probability Letters, Elsevier, vol. 76(1), pages 27-34, January.

Lists

This item is not listed on Wikipedia, on a reading list or among the top items on IDEAS.

Statistics

Access and download statistics

Corrections

When requesting a correction, please mention this item's handle: RePEc:yor:yorken:01/02. See general information about how to correct material in RePEc.

For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Paul Hodgson).

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 references are entirely missing, you can add them using this form.

If the full references list an item that is present in RePEc, but the system did not link 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 profile, as there may be some citations waiting for confirmation.

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