lambda_estimate: Generate sample eigenvalues from population eigenvalues

In nlshrink: Non-Linear Shrinkage Estimation of Population Eigenvalues and Covariance Matrices

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

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 c(n, p = length(tau)).

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

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

Example output

  [1] 0.03814089 0.04389033 0.04871140 0.05323002 0.05761071 0.06192319
  [7] 0.06620726 0.07048840 0.07478297 0.07910003 0.08345078 0.08784499
 [13] 0.09227755 0.09676060 0.10130209 0.10588653 0.11053755 0.11524497
 [19] 0.12000656 0.12484467 0.12972694 0.13469270 0.13970789 0.14479921
 [25] 0.14995658 0.15517482 0.16047652 0.16582695 0.17127224 0.17676043
 [31] 0.18234927 0.18797958 0.19371251 0.19948839 0.20536641 0.21129059
 [37] 0.21731507 0.22338978 0.22956233 0.23578946 0.24211185 0.24849309
 [43] 0.25496717 0.26150408 0.26813178 0.27482585 0.28160908 0.28846185
 [49] 0.29540253 0.30241553 0.30951554 0.31669042 0.32395162 0.33129008
 [55] 0.33871430 0.34621816 0.35380718 0.36147839 0.36923397 0.37707457
 [61] 0.38499846 0.39301061 0.40110458 0.40929051 0.41755635 0.42591841
 [67] 0.43435796 0.44289854 0.45151371 0.46023528 0.46902809 0.47793315
 [73] 0.48690575 0.49599678 0.50515154 0.51443097 0.52377051 0.53324067
 [79] 0.54276790 0.55243096 0.56214922 0.57200711 0.58192019 0.59197456
 [85] 0.60208682 0.61233891 0.62265537 0.63310593 0.64363241 0.65428161
 [91] 0.66502481 0.67587315 0.68683874 0.69789005 0.70907863 0.72034015
 [97] 0.73175113 0.74323158 0.75486319 0.76657289 0.77842202 0.79037305
[103] 0.80243820 0.81463832 0.82692662 0.83937139 0.85189762 0.86458024
[109] 0.87736191 0.89027581 0.90332827 0.91647975 0.92979555 0.94320572
[115] 0.95677171 0.97046646 0.98428064 0.99826114 1.01234467 1.02659191
[121] 1.04097820 1.05548963 1.07017635 1.08498066 1.09994564 1.11507620
[127] 1.13032991 1.14576240 1.16135102 1.17707996 1.19299588 1.20906501
[133] 1.22529120 1.24170913 1.25828699 1.27503165 1.29197328 1.30909069
[139] 1.32637789 1.34386834 1.36155539 1.37941618 1.39748456 1.41576613
[145] 1.43424365 1.45292359 1.47182717 1.49095694 1.51029997 1.52986642
[151] 1.54967254 1.56972219 1.59001689 1.61055105 1.63134353 1.65240182
[157] 1.67373166 1.69533930 1.71722975 1.73940721 1.76188613 1.78467535
[163] 1.80778379 1.83122107 1.85499753 1.87912428 1.90361328 1.92847738
[169] 1.95373038 1.97938708 2.00546338 2.03197637 2.05894435 2.08638698
[175] 2.11432537 2.14278220 2.17178976 2.20137555 2.23156425 2.26238728
[181] 2.29389981 2.32613576 2.35913263 2.39297299 2.42768735 2.46337902
[187] 2.50010806 2.53798384 2.57712557 2.61767076 2.65979257 2.70370723
[193] 2.74968511 2.79810358 2.84944603 2.90443938 2.96420207 3.03066296
[199] 3.10790105 3.21099506

nlshrink documentation built on May 1, 2019, 8:42 p.m.