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Bimatrix variate beta type IV distribution
In this section we derive the joint pdf of two dependent matrix variate beta type I variates (see (3.3)) given by the Wishart ratios (1.4) and call it the bimatrix variate beta type IV distribution. The marginal and conditional pdfs are derived in Theorem 6.2 and the product moment of the determinants is derived in Theorem 6.3. The latter is used in Theorem 6.4 to derive an exact expression for the pdf of Λ4 = |X1X2| (see (1.10)), the product of two dependent Wilks’ statistics (see Bekker, Roux, Ehlers and Arashi, 2010). The role of the parameters are studied in Section 6.5. This distribution is also known in the literature as the bimatrix variate generalised beta type I distribution and its pdf and some properties were derived independently from this study by Gupta and Nagar (2009b) and Díaz-García and Gutiérrez-Jáimez (2010).
Conclusion
In this thesis a bimatrix group of beta distributions with bounded domain were developed from different dependent Wishart ratios. In this study the goal was to derive the exact pdfs for each of the proposed Wishart ratios ((1.2) to (1.5)) for B having the central or the noncentral Wishart distribution. This was achieved by using symmetrised density functions defined by Greenacre (1973) followed by applying the symmetrised density functions in an inverse way. For each distribution some statistical properties were established; specifically the product moment. The marginal and conditional properties were also studied for the central distributions. In each chapter a direct application in multivariate statistics is presented by linking the results to statistics that are functions of the product of determinants of bimatrix beta variates. By making use of inverse Mellin transforms, exact expressions were derived for the pdfs of the product of the determinants for both the central and noncentral cases. These exact expressions for the pdfs were in terms of zonal polynomials, invariant polynomials, hypergeometric functions with matrix argument, Meijer’s G-function and Fox’s H-function. These functions have recently become more computable due to dynamic programming and the availability of packages and algorithms, therefore the theory is transformed into practice for the user.
1 Introduction
1.1 Wishart ratios
1.2 Application of Wishart ratios
2 Special functions and theory
2.1 Jacobians of matrix transformations
2.2 Liouville distribution and integration
2.3 Zonal polynomials
2.4 Invariant polynomials
2.5 Hypergeometric function
2.6 Hypergeometric function of matrix argument
2.7 Laguerre polynomials
2.8 Mellin transform, Meijer’s G-function and Fox’s H-function
2.9 Symmetrised density function
2.10 Wishart distribution
2.11 Wilks’ statistic
I Central distributions
3 Matrix variate beta type I distribution
3.1 Probability density function
3.2 Moments of the determinant
3.3 Probability density function of the Wilks’ statistic
4 Bimatrix variate beta type I distribution
4.1 Probability density function
4.2 Marginal property and conditional density
4.3 Product moment of the determinants
4.4 Distribution of the product of determinants
4.5 Role of the parameters
5 Bimatrix variate beta type III distribution
5.1 Probability density function
5.2 Marginal property and conditional density
5.3 Product moment of the determinants
5.4 Distribution of the product of determinants
5.5 Role of the parameters
6 Bimatrix variate beta type IV distribution
6.1 Probability density function
6.2 Marginal property and conditional density
6.3 Product moment of the determinants
6.4 Distribution of the product of determinants
6.5 Role of the parameters
7 Bimatrix variate beta type V distribution
7.1 Probability density function
7.2 Marginal property and conditional density
7.3 Product moment of the determinants, (α1 = α2 = α)
7.4 Distribution of the product of determinants, (α1 = α2 = α)
7.5 Role of the parameters
II Noncentral distributions
8 Noncentral matrix variate beta type I distribution
8.1 Probability density function
8.2 Moment of the determinant
8.3 Probability density function of the Wilks’ statistic
9 Noncentral bimatrix variate beta type I distribution
9.1 Probability density function
9.2 Product moment of the determinants
9.3 Distribution of the product of determinants
9.4 Role of the parameters
10 Noncentral bimatrix variate beta type III distribution
10.1 Probability density function
10.2 Product moment of the determinants
10.3 Distribution of the product of determinants
10.4 Bivariate distribution
11 Noncentral bimatrix variate beta type IV distribution
11.1 Probability density function
11.2 Product moment of the determinants
11.3 Distribution of the product of determinants
11.4 Role of the parameters
12 Noncentral bimatrix variate beta type V distribution
12.1 Probability density function
12.2 Product moment of the determinants
12.3 Distribution of the product of determinants
12.4 Bivariate distribution
13 Conclusion
14 List of Figures
15 References
