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Functionals of a Wishart matrix and a normal vector and its application to linear discriminant analysis
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| Title: | Functionals of a Wishart matrix and a normal vector and its application to linear discriminant analysis |
| Other Titles: | ウィッシャート行列と正規ベクトルの関数とその線形判別分析への応用 |
| Authors: | 米永, 航志朗 Browse this author |
| Issue Date: | 24-Mar-2022 |
| Publisher: | Hokkaido University |
| Abstract: | In this dissertation, we investigate some functionals of a Wishart matrix and a normal
vector and discuss the application to linear discriminant analysis in a Bayesian framework.
In section 2, we consider the distribution of the product of a Wishart matrix and a
normal vector which are independently distributed. We derive the stochastic representation
of the product which is used to derive the density function and higher order moments of
the product. Based on the higher order moments of the product, we further present an
Edgeworth type expansion for the product. In addition, it turns out that the obtained
stochastic representation, density function and moments of the product remain valid for
the product of a singular Wishart matrix and a normal vector.
In section 3, we consider the distribution of the product of a Wishart matrix and
a conditional normal vector given a Wishart matrix. This type of the product plays
an important role in Bayesian analysis of the optimal portfolio. We derive the novel
stochastic representation for the product and observe from the stochastic representation
that the distribution of the product is closed under conditioning, marginalization, and affine
transformations. Moreover, the formulae for the first four moments, density function and
an Edgeworth type expansion are explicitly presented.
In section 4, we consider discriminant analysis in the case of two multivariate normal
populations with different means and common covariance matrices. We derive the
posterior predictive density function and the first four moments of the population linear
discriminant function under some prior distributions. Based on the derived posterior predictive
density function, we consider the Bayesian estimation for the misclassification rate
associated with a population linear discriminant function, referred to as the optimal error
rate. We obtain an explicit expression of the Bayes estimator of the optimal error rate.
Although the Bayes estimator of the optimal error rate is expressed by the infinite sums and
special functions in general, it is simply expressed under some conditions. In addition, an
Edgeworth type expansion for the Bayes estimator is suggested based on the approximate
posterior predictive distribution of the population linear discriminant function. |
| Conffering University: | 北海道大学 |
| Degree Report Number: | 甲第14922号 |
| Degree Level: | 博士 |
| Degree Discipline: | 経済学 |
| Examination Committee Members: | (主査) 教授 柿沢 佳秀, 教授 高木 真吾, 教授 鈴川 晶夫 |
| Degree Affiliation: | 経済学院（現代経済経営専攻） |
| Type: | theses (doctoral) |
| URI: | http://hdl.handle.net/2115/85619 |
| Appears in Collections: | 課程博士 (Doctorate by way of Advanced Course) > 経済学院(Graduate School of Economics and Business)
学位論文 (Theses) > 博士 （経済学）
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