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Histograms of the Levy distance ratio between the sample eigenvalue distribution and the estimates of the population eigenvalue distributions.

The Marčenko Pastur equation describes the relation between the sample eigenvalue distribution and the population eigenvalue distribution in the (GSA) limit, but in practice both N and p are finite.

The MP equation only describes the relation between the sample eigenvalue distribution and the sample eigenvalue distribution for the cases that both the number of samples and their dimensionality are infinite, so using the (MP) equation to remove the bias in practical problems is not straightforward.

The dashed line indicates the population distribution H p, the four solid lines are the empirical sample distributions of G p. In [8] an equation was given, which describes the relation between the sample eigenvalue distributions and the population eigenvalue distribution.

The Marčenko Pastur equation in fact does not give a relation between the sample eigenvalues and the population eigenvalues, but between the corresponding distributions in the (GSA) limit.

Building on the work of many as is described in [7] using (GSA), a relation between the sample eigenvalues and population eigenvalues was determined for a narrow set of sample distributions in[8], the Marčenko Pastur equation.

Both (LSA) and (GSA) apply limit analysis to find a relation between the population eigenvalue set and the sample eigenvalue sets.

First the sample eigenvalue density corresponding to this candidate population eigenvalue set is estimated.

The result is an estimate of the sample eigenvalue density convolved with the Cauchy kernel g ( l ) ̂. Figure 4 Sample eigenvalue density estimation based on population eigenvalues.

Equation 6, the (MP) equation, fully characterizes the sample eigenvalue distribution G ∞  if the population eigenvalue distribution is known.

Figure 7a shows the estimates of the sample eigenvalue distribution for the first setting.