represents a Marchenko–Pastur distribution with asymptotic ratio and scale parameter .
represents a Marchenko–Pastur distribution with unit scale parameter.
- MarchenkoPasturDistribution is the limiting spectral density of random matrices from WishartMatrixDistribution.
- The derivative of cumulative distribution function at in a Marchenko–Pastur distribution is proportional to with for between and .
- Marchenko–Pastur distribution has a point mass at with probability when .
- MarchenkoPasturDistribution allows and to be any positive real numbers.
- MarchenkoPasturDistribution allows σ to be a quantity of any unit dimension, and λ to be a dimensionless quantity. »
- MarchenkoPasturDistribution can be used with such functions as Mean, CDF, and RandomVariate.
Examplesopen allclose all
Basic Examples (3)
Compare its histogram to the PDF:
Compare its cumulative histogram to the CDF:
Compare the cumulative histogram of the sample with the CDF of the estimated distribution:
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)
The probability density function for Marchenko–Pastur distribution with is not defined, and PDF returns unevaluated:
Wolfram Research (2015), MarchenkoPasturDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/MarchenkoPasturDistribution.html (updated 2016).
Wolfram Language. 2015. "MarchenkoPasturDistribution." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2016. https://reference.wolfram.com/language/ref/MarchenkoPasturDistribution.html.
Wolfram Language. (2015). MarchenkoPasturDistribution. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/MarchenkoPasturDistribution.html