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The trace of a Wishart matrix, either central or non-central, has important roles in various multi-variate statistical questions. We review several expressions of its distribution given in the literature, establish some new results and provide a discussion on computing methods on the distribution of the ratio: the largest eigenvalue to trace.

Let

population. Classical results show that the sample mean vector

covariance matrix

and they are distributed respectively, as a normal vector

where

If

Several important results in Multivariate analysis are associated with either the determinant, trace or the eigenvalues of this matrix.

For

where

We can also have the matrix

If we consider at the start the

We wish to avoid too technical results in this article, that could digress us from the real purpose of this survey- type article, which is to gather results on the distribution of the trace of the Wishart matrix, that are still scattered in the literature. But several new research results related to this trace, are also presented. In most cases, we will present both the central and the non-central cases, or the null and non-null distributions of a test criterion. It is also natural that we will encounter zonal polynomials, the values of which are not completely known. Finally, due to the extremely complicated mathematical expressions of certain results we will refer the reader to the original publications when this approach appears to be more convenient.

The non-central Wishart distribution has an important role in theoretical Multivariate analysis, but recently has also found some applications, for example in Image Processing [

The Wishart distribution has been generalized in several directions and the most general extension of the Wishart is made by Díaz-García and Guttiérez-Jáimez [

In Section 2 we will first recall several special functions that will be used later. In Section 3 we consider the central Wishart distribution and its trace. Similar results are established for the non-central Wishart and its trace in Section 4. Section 5 studies the moments of the trace while Section 6 considers the Wishartness of some quadratic forms. Section 7 considers the sphericity problem where the trace of the Wishart matrix has an important role. Finally, Section 8 considers the latent roots and their ratios to the trace and shows the need of further research in this area. It also proposes the simulation approach that has proven to be very effective in some of our previous works.

Advanced statistics make use of several special functions and integral transforms: the Humbert function of the second kind and the Lauricella D-function. They are both defined as infinite series, and extended by analytic continuation and are related to each other. We define:

RANDOM VARIABLE | RANDOM MATRIX |
---|---|

Central chi-square | Central Wishart |

Non-central Chi-square | Non-central Wishart |

Central Gamma | Central Gamma |

Randomized Gamma | Non-central Gamma |

Non-central Gamma |

1) The Lauricella D-function, in

where

and

2) Similarly, we define the Humbert function, in

which converges for all values of

We have the Dirichlet distribution (in n + 1 parameters and n variables),

where

The

where

The relation between

An extension of

also known as Picard’s integral for

Formulas (3) and (4) above allow us to use several interesting mathematical results related to Hypergeometric integrals, which are the focus of much recent work by Gelfand, Krapalov and Zelevinsky [

Since some of the results obtained by our research group are highly mathematical we do not reproduce them here but they can be obtained by writing to the third author.

The trace of a square matrix is defined as the sum of its diagonal elements, and is sometimes used to measure the total variance. So, let

For the central Wishart distribution, we will show in the next two sections that when

Essentially there are two cases:

1) The matrix sigma is diagonal,

Bartlett’s classical decomposition of the Wishart matrix,

Since we have

Another approach: Consists in considering the latent roots on the diagonal matrix

dependent, with

REMARK: In the more general case when

linear combination of independent central Chi-squares, each with

PROPOSITION 1. Let

Then we have

can have their densities expressed as G-functions.

PROOF: Immediate from the above results and from [

QED.

2) The matrix sigma is not diagonal,

Results are quite complicated for this case since it involves zonal polynomials, whose expressions are only known for simple cases ([

For

For

where

We have the usual notations:

are respectively the largest and the smallest latent roots of matrix

with

Mathai and Pillai (1980) give another expression, quite similar:

However, using Mellin Transform methods [

where

This distribution is present in many aspects of statistics. Its density is given in our table of densities but below is an alternate expression.

Let

The associated Bessel density is:

A particular case of (6) is the non-central Chi-square density with

Its density is then [

where

Using the above functions Laha [

The density of the product or quotient of two non-central chi square variables can be established in closed form using either Fourier transform [

PROPOSITION 2. Let

where

While the ratio

As given by (2), its trace

First, a simple case is the linear non-central case where the non-centrality parameter is concentrated at one component, can be treated as the central case [

where

and

Hence we have

The above result on Bessel function distributions now allows us to have the density of sums, product and ratios of the traces of independent linear non-central Wishart distributions. We have the following

THEOREM 1. Let

PROOF. Applying (10), we have

For

For the product, we have

QED.

We can use (8) and (9) to graph the density of the product and quotient of the two traces. Some computer algebra software, Maple and Mathematica, for example, can do the computation in (8) (9) as an infinite series. But the computation, especially for (8), is very slow. Here, we approximate (8) by taking a large number of terms.

with the value

where

In the case of planar non-centrality, i.e.

We have the following argument, based on the Moment Generating Function (MGF) of

For

Here we set

where

Using

Using the MGF of the Non-Central Chi-square in our table of densities we have the expression of the trace

where

The density of

The density of a linear combination of non-central chi-square variables has been the subject of investigation by several authors, since it is associated with quadratic forms in normal variables. Ruben [

The approach using Laguerre expansions seems promising, as shown by some authors, including Castano- Martinez and Lopez-Blasquez [

Simulation for the density of the trace of non-central Wishart matrix. Following 4.4, let the covariance matrix be

With the above means and covariance matrix, we have:

and the matrix

We finally have:

Now, two approaches are used to obtain the density of

1) Direct approach 1: We use

We use Matlab command

2) Approach using non-central Chi-squares: We use

where

We use Matlab routine

We can see that the two graphs are very close to each other.

The influence of

fied trace

PROPOSITION 3: Let_{i}, _{i} by taking

PROOF: Using (11) with

Thus, we have

and their sum

For the product:

QED.

Glueck and Muller [

For some values of the parameters, there can be closed form expression for the density of

When the general case we have an expression similar to (5), but preceded by the non-central factor:

with

where

In terms of zonal polynomials, we have Formula (14) of [

For

The trace of

obtain than densities themselves. For example, the r-th cumulant of

the expansion of

also found in [

Some results are unexpected. For example, for

Several other equalities can be found in the same reference.

Letac and Massam [

where

For

There are 3 cases of Quadratic forms:

・

・

・ Similarly,

PROPOSITION 4. ([

and sufficient condition for

The trace

In this section we limit ourselves to the vector case, i.e. of

An interesting property of the Gamma distribution in shape parameter

Let

In testing the hypothesis

1) The classical likelihood ratio criterion (LRC),

The LRC above is hence the ratio of the geometric mean of the eigenvalues to their arithmetic mean. The null-distribution is the distribution of this criterion under

where

as shown by [

where

2) The product of 2 independent beta products [

with

・

・

Their product is

Since these two tests are in fact independent the product of the two criteria gives the above sphericity test criterion. [

In univariate statistics, using

tions are equal, we have Bartlett’s test for homogeneity, based on

with

When the samples have the same size, Glaser [

of independent betas:

n is sample size and

But when these sizes are different [

Gleser ([

Accepting the hypothesis of sphericity allows us to proceed on to other topics, such as analysis of variance using repeated measures. A generalization of this test to

When

1) Khatri and Srivastava [

where

2) [

REMARKS. The non-null density in testing diagonality, as given in [

where

Again, here, the computation of the values of this expression is very complicated and we refer the reader to the original paper.

This distribution has attracted renewed interest lately due to its uses in Physics, on random matrices. Krishnaiah and Shurmann [

The Simulation approach: These distributions have been mentioned by [

1) Central case:

a) When the matrix sigma is diagonal:

Let

b) When the matrix sigma is not diagonal:

This is even more complex and no result is available on this case. The only resort is by simulation, as in Pham-Gia and Turkkan (2010). Simulation of random matrices, using the appropriate technique, can be very accurate, as shown by several articles by Pham-Gia and Turkkan [

2) Non-central case:

This case is naturally more complicated than the previous one and the simulation approach seems to be the only recourse.

ExampleWe give below the simulation results related to the ratio

1)

2)

The simulation results are given by

We have gathered here several important research results related to the trace of a Wishart matrix, and also indicated some potential research topics. Moreover, we have established several connections among these results and proved a few original results. The two main important applications of the trace are the sphericity test and the distribution of the ratio of a latent root to the trace. The lack of results in the second topic clearly shows that research efforts should be made there, as already pointed out by some researchers. Matrix simulation can clearly supply several useful answers.

Finally, as shown in our table of densities, the trace can be further investigated by considering the Gamma random matrix, of which the Wishart is only a special case.