Correction on\\ Moments of minors of Wishart matrices
aos988.tex
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% Journal "The Annals of Statistics"

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\doi{10.1214/12-AOS988} %kopijuoti is PTS
\volume{40}
\issue{2}
\pubyear{2012}
\firstpage{1283}
\lastpage{1284}

\makeatletter
\newtheorem{theorem}{Theorem}
\newcommand{\E}{\mathrm{E}}
\newcommand{\Var}{\operatorname{Var}}
\newcommand{\trace}{\operatorname{tr}}
\makeatother

\begin{document}
\begin{frontmatter}

\title{Correction on\\
Moments of minors of Wishart matrices}\vspace*{6pt}
\runtitle{Correction}
\pdftitle{Correction on
Moments of minors of Wishart matrices by M. Drton and A. Goia}

\textit{Ann. Statist.} \textbf{36} (2008)
2261--2283


\begin{aug}
\author[A]{\fnms{Mathias} \snm{Drton}\corref{}\ead[label=e1]{drton@galton.uchicago.edu}}
\and
\author[B]{\fnms{Aldo} \snm{Goia}\ead[label=e2]{aldo.goia@eco.unipmn.it}}
\runauthor{M. Drton and A. Goia}
\affiliation{University of Chicago and University of East Piedmont}
\address[A]{Department of Statistics \\
University of Chicago\\
Chicago, Illinois  60637\\
USA\\
\printead{e1}} %adresu isvedimo komanda gale!
\address[B]{Dipartimento di Studi per l'Economia \\
\quad e l'Impresa \\
University of East Piedmont \\
Novara\\
Italy\\
\printead{e2}}
\end{aug}

% HISTORY:
\received{\smonth{3} \syear{2012}}

% ABSTRACT

% KEYWORDS
\begin{keyword}[class=AMS]
\kwd{60E05}
\kwd{62H10}.
\end{keyword}
\begin{keyword}
\kwd{Compound matrix}
\kwd{graphical models}
\kwd{multivariate analysis}
\kwd{random determinant}
\kwd{random matrix}
\kwd{tetrad}.
\end{keyword}

\end{frontmatter}


\def\thetheorem{5.7}

Theorem 5.7 in \cite{drton} gives a formula for the variance of a minor
(i.e., a subdeterminant) of a Wishart random matrix. The formula has to
be corrected as follows.


\begin{theorem}
\label{thm:var-gen}  Let $I,J\in{r\brace m}$ have intersection
$C:=I\cap J$ of  cardinality $c=|C|=|I\cap J|$. Define $\bar
I=I\setminus(I\cap J)$,  $\bar J=J\setminus(I\cap J)$ and $\bar I\bar
J=\bar I \cup\bar J$.  Then the minor
$\det(S_{I\times J})=\det(S_{IJ})$ of the Wishart matrix $S\sim\mathcal{W}%
_{r}(n,\Sigma)$ has  variance
%% D1 = \det(\Sigma_{C\times C})
%% D2 = \det(\bar\Sigma_{\bar I\bar J})
%% traceterm =  \det(\bar\Sigma_{\bar I\bar J\times \bar I\bar J}) *
%%    (
%%      \sum_{k=0}^{m-c-1}
%%      (m-c-k)!\frac{(n+2-c)!}{(n+2-c-k)!}(-1)^{k}
%%       \trace\{ (
%%         \bar{\Sigma}_{\bar{I}\bar{J}}\bar{\Sigma}^{\bar{I}\bar{J}})^{(k)}
%%             \})
%% NOTE:  D1*D2 = \det(\Sigma_{IJ})
\begin{eqnarray*}
&&\Var[\det(  S_{IJ})]
\\
&&\qquad= \det(\Sigma_{IJ})^{2} \frac{n!}{(n-m)!} \biggl[
\frac{(n+2)!}{(n+2-m)!} - \frac{n!}{(n-m)!} \biggr]  \\
&&\quad\qquad{} + \det(\Sigma_{C\times C})^{2} \det(\bar\Sigma_{\bar I\bar
J\times\bar I\bar
J}) \frac{(n+2)!}{(n+2-c)!} \cdot\frac{n!}{(n-m)!} \\
&&\qquad\qquad{}\times \Biggl[  \sum_{k=0}^{m-c-1}
(m-c-k)!\frac{(n+2-c)!}{(n+2-c-k)!}(-1)^{k}
\trace\bigl\{  (  \bar{\Sigma}_{\bar{I}\bar{J}}\bar{\Sigma}^{\bar{I}%
\bar{J}}) ^{(k)} \bigr\}  \Biggr] ,
\end{eqnarray*}
where $ \bar\Sigma=\Sigma_{([r]\setminus C)\times([r]\setminus C)}-
\Sigma_{([r]\setminus C)\times C}\Sigma_{C\times C}^{-1}\Sigma_{C\times
([r]\setminus C)}$.
\end{theorem}

\begin{pf}
Define $\bar S$ in analogy to $\bar \Sigma$.  Since $\det(S_{I J}) =
\det(S_{C\times C})\det(\bar S_{\bar I\times \bar J})$, and $S_{C\times
C}$ and $\bar S_{\bar I\times \bar J}$ are independent (Lemma~5.2),
\begin{eqnarray*}
\Var[  \det(  S_{IJ})  ]   &=&\bigl(  \Var[ \det( S_{CC}) ] +\E[ \det( S_{CC})
] ^{2}\bigr) \Var[ \det( \bar{S}_{\bar
{I}\bar{J}})  ] \\
&&{}+\Var[  \det(  S_{CC})  ]  \E[ \det( \bar{S}_{\bar{I}\bar{J}}) ] ^{2}.
\end{eqnarray*}
The claim now follows from Corollary~4.2 and Propositions~5.1 and
5.5.
\end{pf}






\def\bibname{Reference}

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\begin{thebibliography}{2}
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\bibitem{drton}
\begin{barticle}[mr]
\bauthor{\bsnm{Drton},~\bfnm{Mathias}\binits{M.}},
  \bauthor{\bsnm{Massam},~\bfnm{H{\'e}l{\`e}ne}\binits{H.}} \AND
  \bauthor{\bsnm{Olkin},~\bfnm{Ingram}\binits{I.}}
(\byear{2008}).
\btitle{Moments of minors of {W}ishart matrices}.
\bjournal{Ann. Statist.}
\bvolume{36}
\bpages{2261--2283}.
\bid{doi={10.1214/07-AOS522}, issn={0090-5364}, mr={2458187}}
\bptok{imsref}%
\end{barticle}
\endbibitem

\end{thebibliography}




\printaddresses

\end{document}

1208.3360.extracted_bib
\begin{thebibliography}{2}
% BibTex style file: ims.bst, 2011-05-30
% Default style options (sort=0,type=number).
% Used options (sort=1,type=number).

\bibitem{drton}
\begin{barticle}[mr]
\bauthor{\bsnm{Drton},~\bfnm{Mathias}\binits{M.}},
  \bauthor{\bsnm{Massam},~\bfnm{H{\'e}l{\`e}ne}\binits{H.}} \AND
  \bauthor{\bsnm{Olkin},~\bfnm{Ingram}\binits{I.}}
(\byear{2008}).
\btitle{Moments of minors of {W}ishart matrices}.
\bjournal{Ann. Statist.}
\bvolume{36}
\bpages{2261--2283}.
\bid{doi={10.1214/07-AOS522}, issn={0090-5364}, mr={2458187}}
\bptok{imsref}%
\end{barticle}
\endbibitem

\end{thebibliography}