R: Spiked Wishart Eigenvalue Centering and Scaling

WishartSpikePar {RMTstat} | R Documentation

## Spiked Wishart Eigenvalue Centering and Scaling

### Description

Centering and scaling for the sample eigenvalue from a spiked Wishart matrix (sample covariance matrix) with `ndf` degrees of freedom, `pdim` dimensions, and population covariance matrix `diag(spike+var,var,var,...,var)`.

### Usage

      WishartSpikePar( spike, ndf=NA, pdim=NA, var=1, beta=1 )

### Arguments

`spike` | the value of the spike.
`ndf` | the number of degrees of freedom for the Wishart matrix.
`pdim` | the number of dimensions (variables) for the Wishart matrix.
`var` | the population (noise) variance.
`beta` | the order parameter (1 or 2).

### Details

The returned values give appropriate centering and scaling for the largest eigenvalue from a spiked Wishart matrix so that the centered and scaled quantity converges in distribution to a normal random variable with mean 0 and variance 1.

For the spiked distribution to exist, `spike` must be greater than `sqrt(pdim/ndf)*var`.

Supported values for `beta` are `1` for real data and and `2` for complex data.

### Value

`centering` | gives the centering.
`scaleing` | gives the scaling.

### Author(s)

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

### References

Baik, J., Ben Arous, G., and Péché, S. (2005). Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. Annals of Probability 33, 1643–1697.

Baik, J. and Silverstein, J. W. (2006). Eigenvalues of large sample covariance matrices of spiked population models. Journal of Multivariate Analysis 97, 1382-1408.

Paul, D. (2007). Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Statistica Sinica 17, 1617–1642.

### See Also

WishartSpike

[Package RMTstat version 0.3 Index]