R: The Tracy-Widom Distributions

TracyWidom {RMTstat} | R Documentation

## The Tracy-Widom Distributions

### Description

Density, distribution function, quantile function, and random generation for the Tracy-Widom distribution with order parameter `beta`.

### Usage

    dtw(x, beta=1, log = FALSE)
    ptw(q, beta=1, lower.tail = TRUE, log.p = FALSE)
    qtw(p, beta=1, lower.tail = TRUE, log.p = FALSE)
    rtw(n, beta=1)

### 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.
`beta` | the order parameter (1, 2, or 4).
`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

If `beta` is not specified, it assumes the default value of `1`.

The Tracy-Widom law is the edge-scaled limiting distribution of the largest eigenvalue of a random matrix from the beta-ensemble. Supported values for `beta` are `1` (Gaussian Orthogonal Ensemble), `2` (Gaussian Unitary Ensemble), and `4` (Gaussian Symplectic Ensemble).

### Value

`dtw` gives the density, `ptw` gives the distribution function, `qtw` gives the quantile function, and `rtw` generates random deviates.

### Author(s)

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

### Source

The distribution and density functions are computed using a lookup table. They have been pre-computed at 769 values uniformly spaced between `-10` and `6` using MATLAB's `bvp4c` solver to a minimum accuracy of about `3.4e-08`. For all other points, the values are gotten from a cubic Hermite polynomial interpolation. The MATLAB software for computing the grid of values is part of RMLab, a package written by Momar Dieng which is available on his homepage at http://math.arizona.edu/~momar/research.htm.

The quantiles are computed via bisection using uniroot.

Random variates are generated using the inverse CDF.

### References

Dieng, M. (2006). Distribution functions for edge eigenvalues in orthogonal and symplectic ensembles: Painlevé representations. arXiv:math/0506586v2 [math.PR].

Tracy, C.A. and Widom, H. (1994). Level-spacing distributions and the Airy kernel. Communications in Mathematical Physics 159, 151–174.

Tracy, C.A. and Widom, H. (1996). On orthogonal and symplectic matrix ensembles. Communications in Mathematical Phsyics 177, 727–754.

[Package RMTstat version 0.3 Index]