R: The Marcenko-Pastur Distribution

MarcenkoPastur {RMTstat} | R Documentation

## The Marcenko-Pastur Distribution

### Description

Density, distribution function, quantile function and random generation for the Marčenko-Pastur distribution, the limiting distribution of the empirical spectral measure for a large white Wishart matrix.

### Usage

    dmp( x, ndf=NA, pdim=NA, var=1, svr=ndf/pdim, log = FALSE )
    pmp( q, ndf=NA, pdim=NA, var=1, svr=ndf/pdim, lower.tail = TRUE, log.p = FALSE )
    qmp( p, ndf=NA, pdim=NA, var=1, svr=ndf/pdim, lower.tail = TRUE, log.p = FALSE )
    rmp( n, ndf=NA, pdim=NA, var=1, svr=ndf/pdim )

### Arguments

`x,q` | vector of quantiles.
`p` | vector of probabilities.
`n` | number of observations. If `length(n) > 1`, the length is taken to be the number required.
`ndf` | the number of degrees of freedom for the Wishart matrix.
`pdim` | the number of dimensions (variables) for the Wishart matrix.
`var` | the population variance.
`svr` | samples to variables ratio; the number of degrees of freedom per dimension.
`log, log.p` | logical; if TRUE, probabilities p are given as log(p).
`lower.tail` | logical; if TRUE (default), probabilities are P[X <= x], otherwise, P[X > x].

### Details

The `concentration` can either be given explicitly, or else computed from the given `ndf` and `pdim`. If `var` is not specified, it assumes the default of `1`.

The Marčenko-Pastur law is the limit of the random probability measure which puts equal mass on all `pdim` eigenvalues of a normalized `pdim`-dimensional white Wishart matrix with `ndf` degrees of freedom and scale parameter `diag(var, var, ..., var)`. It is assumed that `ndf` goes to infinity, and `ndf/pdim` goes to nonzero constant called the "samples-to-variables ratio" (`svr`).

### Value

`dmp` gives the density, `pmp` gives the distribution function, `qmp` gives the quantile function, and `rmp` generates random deviates.

### Author(s)

Iain M. Johnstone, Zongming Ma, Patrick O. Perry and Morteza Shahram

### Source

Other than the density, these functions are relatively slow and imprecise.

The distribution function is computed with integrate. The quantiles are computed via bisection using uniroot. Random variates are generated using the inverse CDF.

### References

Marčenko, V.A. and Pastur, L.A. (1967). Distribution of eigenvalues for some sets of random matrices. Sbornik: Mathematics 1, 457–483.

[Package RMTstat version 0.3 Index]