{"title":"关于奇异的γ、Wishart和β矩阵变量密度函数","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":"{\"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}","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
摘要
当一个p×p$$ p\times p $$ 实正定矩阵S$$ S $$ 遵循Wishart分布,或者更一般地说,遵循形状参数为α的矩阵变量伽马分布$$ \alpha $$ 和正定尺度参数矩阵B$$ B $$ ,可以表示S$$ S $$ 作为XX’$$ X{X}^{\prime } $$ 对于某个矩阵X$$ X $$ 尺寸为p×q$$ p\times q $$ . 当p>q$$ p>q $$ ,$$ S $$ 有一个奇异分布,其性质可以通过X的密度函数来研究$$ X $$ . 当X$$ X $$ 遵循矩阵变量扩展高斯分布,由此产生的奇异分布的密度函数可以通过使用连续变换及其相关的雅可比矩阵得到。然后将奇异Wishart分布作为一种特殊情况得到。由X的任意划分产生的边际和条件密度函数$$ X $$ 也会被考虑。同样的技术也将应用于实和复奇异型- 1和型- 2 β -分布矩阵的密度函数的推导。碰巧的是,所提出的方法是基于涉及某些微分元素的楔形积的操作,通常证明比迄今为止在文献中使用的复杂程序更有效。
On the singular gamma, Wishart, and beta matrix-variate density functions
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.