Centering and scaling for the maximum eigenvalue from a white Wishart
matrix (sample covariance matrix) with with ndf
degrees of freedom,
pdim
dimensions, population variance var
, and order
parameter beta
.
WishartMaxPar(ndf, pdim, var=1, beta=1)
the number of degrees of freedom for the Wishart matrix.
the number of dimensions (variables) for the Wishart matrix.
the population variance.
the order parameter (1 or 2).
gives the centering.
gives the scaling.
If beta
is not specified, it assumes the default value of 1
.
Likewise, var
assumes a default of 1
.
The returned values give appropriate centering and scaling for the largest eigenvalue from a white Wishart matrix so that the centered and scaled quantity converges in distribution to a Tracy-Widom random variable. We use the second-order accurate versions of the centering and scaling given in the references below.
El Karoui, N. (2006). A rate of convergence result for the largest eigenvalue of complex white Wishart matrices. Annals of Probability 34, 2077--2117.
Ma, Z. (2008). Accuracy of the Tracy-Widom limit for the largest eigenvalue in white Wishart matrices. arXiv:0810.1329v1 [math.ST].