{"title":"On the singular gamma, Wishart, and beta matrix-variate density functions","authors":"Arak M. Mathai, Serge B. Provost","doi":"10.1002/cjs.11710","DOIUrl":null,"url":null,"abstract":"<p>When a <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>×</mo>\n <mi>p</mi>\n </mrow>\n <annotation>$$ p\\times p $$</annotation>\n </semantics></math> real positive definite matrix <math>\n <semantics>\n <mrow>\n <mi>S</mi>\n </mrow>\n <annotation>$$ S $$</annotation>\n </semantics></math> follows a Wishart or, more generally, a matrix-variate gamma distribution with shape parameter <math>\n <semantics>\n <mrow>\n <mi>α</mi>\n </mrow>\n <annotation>$$ \\alpha $$</annotation>\n </semantics></math> and positive definite scale parameter matrix <math>\n <semantics>\n <mrow>\n <mi>B</mi>\n </mrow>\n <annotation>$$ B $$</annotation>\n </semantics></math>, one can represent <math>\n <semantics>\n <mrow>\n <mi>S</mi>\n </mrow>\n <annotation>$$ S $$</annotation>\n </semantics></math> as <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n <msup>\n <mrow>\n <mi>X</mi>\n </mrow>\n <mrow>\n <mo>′</mo>\n </mrow>\n </msup>\n </mrow>\n <annotation>$$ X{X}^{\\prime } $$</annotation>\n </semantics></math> for some matrix <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n </mrow>\n <annotation>$$ X $$</annotation>\n </semantics></math> of dimension <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>×</mo>\n <mi>q</mi>\n </mrow>\n <annotation>$$ p\\times q $$</annotation>\n </semantics></math>. When <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n <mo>></mo>\n <mi>q</mi>\n </mrow>\n <annotation>$$ p>q $$</annotation>\n </semantics></math>, <math>\n <semantics>\n <mrow>\n <mi>S</mi>\n </mrow>\n <annotation>$$ S $$</annotation>\n </semantics></math> has a singular distribution whose properties can be studied via the density function of <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n </mrow>\n <annotation>$$ X $$</annotation>\n </semantics></math>. It will be shown that when <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n </mrow>\n <annotation>$$ X $$</annotation>\n </semantics></math> follows a matrix-variate extended Gaussian distribution, the density function of the resulting singular gamma distribution can be obtained by making use of successive transformations and their associated Jacobians. The singular Wishart distribution will then be obtained as a particular case. The marginal and conditional density functions resulting from an arbitrary partitioning of <math>\n <semantics>\n <mrow>\n <mi>X</mi>\n </mrow>\n <annotation>$$ X $$</annotation>\n </semantics></math> will be considered as well. The same technique will also be applied to the derivation of the density functions of real and complex singular type-1 and type-2 beta-distributed matrices. It so happens that the proposed approach, which is based on manipulations involving the wedge products of certain differential elements, generally proves more efficient than the intricate procedures that have hitherto been employed in the literature.</p>","PeriodicalId":55281,"journal":{"name":"Canadian Journal of Statistics-Revue Canadienne De Statistique","volume":"50 4","pages":"1143-1165"},"PeriodicalIF":0.8000,"publicationDate":"2022-07-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/cjs.11710","citationCount":"5","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Canadian Journal of Statistics-Revue Canadienne De Statistique","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/cjs.11710","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 5
Abstract
When a real positive definite matrix follows a Wishart or, more generally, a matrix-variate gamma distribution with shape parameter and positive definite scale parameter matrix , one can represent as for some matrix of dimension . When , has a singular distribution whose properties can be studied via the density function of . It will be shown that when follows a matrix-variate extended Gaussian distribution, the density function of the resulting singular gamma distribution can be obtained by making use of successive transformations and their associated Jacobians. The singular Wishart distribution will then be obtained as a particular case. The marginal and conditional density functions resulting from an arbitrary partitioning of will be considered as well. The same technique will also be applied to the derivation of the density functions of real and complex singular type-1 and type-2 beta-distributed matrices. It so happens that the proposed approach, which is based on manipulations involving the wedge products of certain differential elements, generally proves more efficient than the intricate procedures that have hitherto been employed in the literature.
期刊介绍:
The Canadian Journal of Statistics is the official journal of the Statistical Society of Canada. It has a reputation internationally as an excellent journal. The editorial board is comprised of statistical scientists with applied, computational, methodological, theoretical and probabilistic interests. Their role is to ensure that the journal continues to provide an international forum for the discipline of Statistics.
The journal seeks papers making broad points of interest to many readers, whereas papers making important points of more specific interest are better placed in more specialized journals. The levels of innovation and impact are key in the evaluation of submitted manuscripts.