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Singular Inverse Wishart Distribution with Application to Portfolio Theory

Bodnar, Taras; Mazur, Stepan LU and Podgorski, Krzysztof LU (2015) In Working Papers in Statistics
Abstract
The inverse of the standard estimate of covariance matrix is frequently used in the portfolio theory to estimate the optimal portfolio weights. For this problem, the distribution of the linear transformation of the inverse is needed. We obtain this distribution in the case when the sample size is smaller than the dimension, the underlying covariance matrix is singular, and the vectors of returns are independent and normally distributed. For the result, the distribution of the inverse of covariance estimate is needed and it is derived and referred to as the singular inverse Wishart distribution. We use these results to provide an explicit stochastic representation of an estimate of the mean-variance portfolio weights as well as to derive... (More)
The inverse of the standard estimate of covariance matrix is frequently used in the portfolio theory to estimate the optimal portfolio weights. For this problem, the distribution of the linear transformation of the inverse is needed. We obtain this distribution in the case when the sample size is smaller than the dimension, the underlying covariance matrix is singular, and the vectors of returns are independent and normally distributed. For the result, the distribution of the inverse of covariance estimate is needed and it is derived and referred to as the singular inverse Wishart distribution. We use these results to provide an explicit stochastic representation of an estimate of the mean-variance portfolio weights as well as to derive its characteristic function and the moments of higher order. (Less)
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author
organization
publishing date
type
Working Paper
publication status
published
subject
keywords
singular Wishart distribution, mean-variance portfolio, sample estimate of precision matrix, Moore-Penrose inverse
in
Working Papers in Statistics
issue
2
pages
18 pages
publisher
Department of Statistics, Lund university
language
Swedish
LU publication?
yes
id
326c88db-0128-434b-ab9d-f1b8ec9a4bed (old id 8052806)
alternative location
http://journals.lub.lu.se/index.php/stat/article/view/15033
date added to LUP
2015-10-07 13:31:18
date last changed
2016-04-16 07:11:55
@misc{326c88db-0128-434b-ab9d-f1b8ec9a4bed,
  abstract     = {The inverse of the standard estimate of covariance matrix is frequently used in the portfolio theory to estimate the optimal portfolio weights. For this problem, the distribution of the linear transformation of the inverse is needed. We obtain this distribution in the case when the sample size is smaller than the dimension, the underlying covariance matrix is singular, and the vectors of returns are independent and normally distributed. For the result, the distribution of the inverse of covariance estimate is needed and it is derived and referred to as the singular inverse Wishart distribution. We use these results to provide an explicit stochastic representation of an estimate of the mean-variance portfolio weights as well as to derive its characteristic function and the moments of higher order.},
  author       = {Bodnar, Taras and Mazur, Stepan and Podgorski, Krzysztof},
  keyword      = {singular Wishart distribution,mean-variance portfolio,sample estimate of precision matrix,Moore-Penrose inverse},
  language     = {swe},
  note         = {Working Paper},
  number       = {2},
  pages        = {18},
  publisher    = {Department of Statistics, Lund university},
  series       = {Working Papers in Statistics},
  title        = {Singular Inverse Wishart Distribution with Application to Portfolio Theory},
  year         = {2015},
}