The Tracy-Widom Distributions
Density, distribution function, quantile function, and random
generation for the Tracy-Widom distribution with order parameter
beta.
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)
x,q |
vector of quantiles. |
p |
vector of probabilities. |
n |
number of observations. If |
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]. |
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).
dtw gives the density,
ptw gives the distribution function,
qtw gives the quantile function, and
rtw generates random deviates.
Iain M. Johnstone, Zongming Ma, Patrick O. Perry and Morteza Shahram
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.
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.
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