1. Differential Geometry of the Freund 4-Manifold F

In [1] we proved that every neighbourhood of an exponential distribution contains a neighbourhood of gamma distributions in the subspace topology of , using information geometry (see Amari et al. [2, 3]) and the affine immersion of Dodson and Matsuzoe [4]. As part of a study of the information geometry of gamma and bivariate gamma stochastic processes [5, 6, 7], we have calculated the geometry of the family of Freund bivariate mixture exponential density functions. The importance of this family lies in the fact that exponential distributions represent intervals between events for Poisson processes on the real line, and the Freund family provides models for bivariate processes with positive and negative covariance.

The Freund family of distributions becomes a Riemannian 4-manifold with the Fisher information metric, and we derive the induced -geometry, that is, the -Ricci curvature, the -scalar curvature, and so forth. We also show that the Freund manifold has a positive constant 0-scalar curvature, so geometrically it constitutes part of a sphere.

1.1. Freund Bivariate Mixture Exponential Distributions

This model was one of the first bivariate distributions to be obtained from reliability considerations. Freund [8] introduced a bivariate exponential mixture distribution arising from the following situation. Suppose that an instrument has two components, and , with lifetimes and having density functions (when both components are in operation) ; , . Then and are dependent, in that a failure of either component changes the parameter of the life distribution of the other component. Thus, when fails, the parameter for becomes ; when fails, the parameter for becomes . There is no other dependence. The joint density function of and is

where .

Define the functions and :

and

Provided that , the marginal density function of is, according to Freund,

and provided that , the marginal density function of is

We can see that the marginal density functions are, in general, not exponential but rather mixtures of exponential distributions if ; otherwise, they are weighted averages. For this reason, this system of distributions should be termed bivariate mixture exponential distributions rather than simply bivariate exponential distributions. The marginal functions and are exponential distributions only in the special case .

Freund discussed the statistics of the special case when and obtained the following joint density function

with marginal density functions:

The covariance and correlation coefficient of and were derived by Freund, as follows:

Note that in general . The correlation coefficient when , , and when and , . In many applications, , that is, lifetime tends to be shorter when the other component is out of action. In such cases the correlation is positive.

1.2. Fisher Information Metric

Proposition 1.1 Let be the set of Freund bivariate mixture exponential distributions, that is,

Then we have:

1. Identifying as a local coordinate system, can be regarded as a 4-manifold.

2. F is a Riemannian space and the Fisher information matrix , where

is given by

1.3. -geometry

In this section we provide the -connection and various -curvature objects of the Freund 4-manifold F: the -curvature tensor, the -sectional curvatures, the -mean curvatures, the -Ricci tensor, and the -scalar curvature.

1.3.1. -connection

For each , the -connection is the torsion-free affine connection with components:

So we have an affine connection defined by

where is the Fisher information metric.

Note that the 0-connection is the Riemannian connection with respect to the Fisher metric.

Proposition 1.2 The functions , are given by

For example, the component is given by

Proposition 1.3 By solving the equations , , we obtain the components of :

So we obtain:

1.3.2 -curvatures

Proposition 1.4 The components of the -curvature tensor , are given by

Since the analytical expression for the curvature tensor is very large, we report the components :

while the other independent components are zero.

Proposition 1.5 The -sectional curvatures:

are given by

Proposition 1.6 The -mean curvatures:

are given by

Contracting with , we obtain the components of the Ricci tensor.

Proposition 1.7 The components of the -Ricci tensor are given by the symmetric matrix .

The -eigenvalues and the -eigenvectors of the -Ricci tensor are given by

Proposition 1.8 The manifold F has a constant -scalar curvature .

Note that the Freund manifold F has a positive constant scalar curvature when , so geometrically it constitutes part of a sphere (compare, for example, Kobayashi and Nomizu [9]).