ABSTRACT
In older textbooks on differential and integral calculus (e.g. Edwards, 1922, p. 160; Fichtenholz, 1967, p. 391), the Liouville multiple integral over the positive orthant R + n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351077040/75895dbc-edd8-4bfc-a9cc-799ec9811071/content/eq1320.tif"/> is derived as an extension of the Dirichlet integral (cf. Section 1.4). It appears to the authors of this book that Marshall and Olkin (1979) were the first to use this integral to define what they called the Liouville-Dirichlet distributions. They show that the expectation operator using these distributions preserves the Schur-convexity property for functions on R + n https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781351077040/75895dbc-edd8-4bfc-a9cc-799ec9811071/content/eq1321.tif"/> . Sivazlian (1981) gives some results on marginal distributions and transformation properties concerning the Liouville distributions. Anderson and Fang (1982, 1987) study a subclass of Liouville distributions arising from the distributions of quadratic forms. Gupta and Richards (1987), armed with the Weyl fractional integral and Deny’s theorem in measure theory on locally compact groups, give a more comprehensive treatment of the multivariate Liouville distribution and also extend some results to the matrix analogues. All the authors mentioned above directly employ the Liouville integral in their approach. In this chapter, however, we shall introduce the Liouville distributions following the approach that emerged in Chapter 2 and was used in Chapter 5: a uniform base with a constraint (here a Dirichlet base) multiplied by a positive generating variate. Some unpublished results of K.W. Ng are included.
