MarchenkoPasturDistribution

MarchenkoPasturDistribution[λ,σ]

represents a MarchenkoPastur distribution with asymptotic ratio and scale parameter .

MarchenkoPasturDistribution[λ]

represents a MarchenkoPastur distribution with unit scale parameter.

Details

Examples

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Basic Examples  (3)

Probability density function:

Cumulative distribution function:

Mean and variance:

Scope  (7)

Generate a sample of pseudorandom numbers from a MarchenkoPastur distribution with :

Compare its histogram to the PDF:

Generate a sample of pseudorandom numbers from a MarchenkoPastur distribution with :

Compare its cumulative histogram to the CDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

Compare the cumulative histogram of the sample with the CDF of the estimated distribution:

Skewness and kurtosis depend only on :

Different moments with closed forms as functions of parameters:

Moment:

Closed form for symbolic order:

CentralMoment:

Closed form for symbolic order:

FactorialMoment:

Cumulant:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Find the median area:

Applications  (1)

Use MatrixPropertyDistribution to represent the eigenvalues of a Wishart random matrix with identity covariance:

The spectral density converges to the pdf of MarchenkoPasturDistribution[λ] in the limit of large and with the finite ratio :

Properties & Relations  (3)

MarchenkoPasturDistribution is closed under scaling by a positive factor:

MarchenkoPasturDistribution has an atomic weight at 0 when :

MarchenkoPasturDistribution is the limiting distribution of eigenvalues of Wishart matrices. The atomic weight at occurs when the Wishart matrix is singular. Generate a singular Wishart matrix with identity covariance and compute the scaled eigenvalues:

Fit MarchenkoPasturDistribution to the eigenvalues:

Compare the cumulative histogram of the eigenvalues with the CDF:

Possible Issues  (1)

MarchenkoPastur distribution with is a mixed type distribution, which is neither continuous nor discrete:

The CDF for such MarchenkoPastur distributions is discontinuous at :

The probability density function for MarchenkoPastur distribution with is not defined, and PDF returns unevaluated:

Differentiation of the CDF results in a function that does not integrate to one:

Computations with mixed type distributions are fully supported. Compute special moments:

Estimate parameters of Marchenko-Pastur distribution:

Introduced in 2015
 (10.3)
 |
Updated in 2016
 (10.4)