2. Submanifolds of the Freund 4-Manifold F

We consider four submanifolds of the 4-manifold F of Freund bivariate exponential distributions (1) , which includes the case of statistically independent random variables. It also includes the special case of an Absolutely Continuous Bivariate Exponential Distribution called ACBED (or ACBVE) by Block and Basu (compare Hutchinson and Lai [10]). We use the coordinate system for the submanifolds , and the coordinate system for ACBED of the Block and Basu case.

2.1. Submanifold and

The distributions are of the form:

where are the density functions of the one-dimensional exponential distributions with the parameters . This is the case for statistical independence of X and Y, so the space is the direct product of two Riemannian spaces:

and

Proposition 2.1 The Fisher information metric is given by

Proposition 2.2 The components of are given by

while the other components are zero.

Proposition 2.3 The -curvature tensor, -Ricci tensor, and -scalar curvature of are zero.

2.2. Submanifold and

The distributions are of the form:

with parameters . The covariance, correlation coefficient, and marginal functions of and are given by

Note that when .

forms an exponential family, with natural parameters and the potential function

Proposition 2.4 The submanifold is an isometric diffeomorph of the submanifold .

Proof: Since is a potential function, the Fisher information metric is given by the Hessian of , that is,

Then we have the Fisher information metric (10) by a straightforward calculation.

2.2.1 Mutually Dual Foliations

Since , is a -affine coordinate system, and the -affine coordinate system is given by

These coordinates have the potential function

So the coordinates and are mutually dual with respect to the Fisher metric, and the tetrad is a dually flat space. Therefore, has dually orthogonal foliations.

For example, take as a coordinate system for ; then

and the Fisher information metric is

2.2.2 Neighbourhoods of Independence in

An important practical application of the Freund submanifold is the representation of a bivariate stochastic process for which the marginals are identical exponentials. The next result is important because it provides a topological neighbourhood of that subspace W in consisting of the bivariate processes that have zero covariance: we obtain a neighbourhood of independence for random, that is, exponentially distributed processes.

Proposition 2.5 Let be the manifold with Fisher metric and exponential connection . Then can be realized in by the graph of a potential function, namely, can be realized by the affine immersion

where and is the transversal vector field along .

In , the submanifold W consisting of the independent case is represented by the curve

This is illustrated in the graphic that shows , an affine embedding of as a surface in , and an -tubular neighbourhood of W, as the curve in the surface. This curve represents all bivariate distributions having identical exponential marginals and zero covariance; its tubular neighbourhood represents departures from independence. We parametrize here with , .

2.3. Submanifold

The distributions are of the form:

with parameters , . The covariance, correlation coefficient, and marginal functions of and are given by

Note that the correlation coefficient is positive.

Proposition 2.6 The Fisher information metric is given by

Proposition 2.7 The -connection components of are given by

while the other independent components are zero.

Proposition 2.8 The -curvature tensor, -Ricci tensor, and -scalar curvature of are zero.

2.4. Submanifold of Block and Basu

The distributions are of the form:

where the parameters are positive, and . This distribution was derived originally by omitting the singular part of Marshall and Olkin's distribution (compare [10]); Block and Basu called it the ACBED to emphasize that it is an absolutely continuous part of a bivariate exponential distribution. Alternatively, it can be derived from the Freund distribution (1), with

By substitution we obtained the covariance, correlation coefficient, and marginal functions of and :

Note that the correlation coefficient is positive, and the marginal functions are a negative mixture of two exponentials.

The Christoffel symbols, curvature tensor, Ricci tensor, sectional curvatures, mean curvatures, and scalar curvature were computed, but are not listed because they are somewhat cumbersome.

When , this family of distributions becomes an exponential family with natural parameters and potential function , so it would be easy to derive the -geometry, for example,

and the components are as follows:

while the -curvatures vanish.