# Generate sample eigenvalues from population eigenvalues

### Description

The Marcenko Pastur (MP) law relates the limiting distribution
of the sample eigenvalues to that of the population eigenvalues. In the
finite-dimensional case, the population spectral distribution (PSD) can be
represented as a sum of point masses, and the empirical spectral
distribution (ESD) can be obtained by solving the discretized MP equation.
The QuEST function(see references), uses the quantile function of the ESD
to compute the sample eigenvalues for any given ratio *c = p/n \in
(0,∞)*.

### Usage

1 | ```
lambda_estimate(tau, n)
``` |

### Arguments

`tau` |
(Required) A non-negative numeric vector of population eigenvalues. |

`n` |
(Required) A positive integer representing the number of datapoints
of a hypothetical data matrix with dimension |

### Value

A numeric vector of the same length as `tau`

, containing the
sample eigenvalue estimates, sorted in ascending order.

### References

Ledoit, O. and Wolf, M. (2015). Spectrum estimation: a unified framework for covariance matrix estimation and PCA in large dimensions. Journal of Multivariate Analysis, 139(2)

Ledoit, O. and Wolf, M. (2016). Numerical Implementation of the QuEST function. arXiv:1601.05870 [stat.CO]

### Examples

1 | ```
lambda_estimate(tau = rep(1,200), n = 300)
``` |