Generate n random matrices, distributed according to the Wishart distribution with parameters Sigma and df, W_p(Sigma, df).

rSingularWishart(n, df, Sigma, covariance = FALSE, simplify = "array")

## Arguments

n integer: the number of replications. numeric parameter, “degrees of freedom”. positive definite ($$p\times p$$) “scale” matrix, the matrix parameter of the distribution. logical on whether a covariance matrix should be generated logical or character string; should the result be simplified to a vector, matrix or higher dimensional array if possible? For sapply it must be named and not abbreviated. The default value, TRUE, returns a vector or matrix if appropriate, whereas if simplify = "array" the result may be an array of “rank” ($$=$$length(dim(.))) one higher than the result of FUN(X[[i]]).

## Value

A numeric array of dimension p * p * n, where each array is a positive semidefinite matrix, a realization of the Wishart distribution W_p(Sigma, df)

## Details

If X_1, ..., X_m is a sample of m independent multivariate Gaussians with mean vector 0, and covariance matrix Sigma, the distribution of M = X'X is W_p(Sigma, m).

## References

Uhlig, Harald. 1994. “On Singular Wishart and Singular Multivariate Beta Distributions.” The Annals of Statistics 22 (1): 395–405. doi:10.1214/aos/1176325375.

## Examples

rSingularWishart(2, 5, diag(1, 20))#> , , 1
#>
#>               [,1]        [,2]        [,3]         [,4]       [,5]        [,6]
#>  [1,]  4.496122747  2.47286117  1.04635474  0.004355063 -2.5315425 -0.92925706
#>  [2,]  2.472861171  3.44413887  1.26959264  1.237882502  1.0950183  0.76673151
#>  [3,]  1.046354737  1.26959264  1.77858542 -0.219811493  0.7659096  0.11335942
#>  [4,]  0.004355063  1.23788250 -0.21981149  4.103538677  3.2429428  2.69337670
#>  [5,] -2.531542500  1.09501834  0.76590960  3.242942750  8.9728584  2.89299031
#>  [6,] -0.929257060  0.76673151  0.11335942  2.693376703  2.8929903  2.19286845
#>  [7,]  3.144114142  1.87358683  1.55462979  1.684476746 -0.4047529  0.64901278
#>  [8,]  1.831155110  0.79849451  0.23015757 -1.194387216 -3.9316539 -0.97237624
#>  [9,] -0.248284314 -1.04228519  1.08281735 -0.880454205  1.2031090 -0.55106578
#> [10,] -0.219195677 -1.18617945 -0.71967980 -0.380840959 -1.1080962 -0.53625095
#> [11,]  0.258481363 -0.41611808 -0.05076391  4.857468708  3.6759255  2.69991251
#> [12,]  0.532383104 -0.59057988  0.61934323 -2.255056011 -2.3371620 -1.49219136
#> [13,] -1.364369600 -0.19811542 -0.50990550  2.435807345 -0.5748421  2.21529820
#> [14,] -3.345898564 -1.66937433 -0.04172636  0.920477293 -1.4043704  2.00517488
#> [15,]  3.446133529  1.15529340  2.52175529 -0.381603256  0.1106376 -0.86786024
#> [16,]  2.214015766  1.40295050  0.11544473  1.940356565  0.6123041  0.58729133
#> [17,]  2.323012017  1.86031402  1.24038189  0.679019787  3.4904505 -0.23026195
#> [18,]  0.339735809 -0.07978953 -2.17926040  1.839844397 -1.9292950  0.65064488
#> [19,] -1.054940520 -1.06096135  1.05970034 -4.200681968 -0.1773063 -2.32179782
#> [20,] -2.071123756 -0.62178843 -0.09315525 -1.488151324 -1.7080956 -0.01356917
#>              [,7]        [,8]       [,9]       [,10]       [,11]       [,12]
#>  [1,]  3.14411414  1.83115511 -0.2482843 -0.21919568  0.25848136  0.53238310
#>  [2,]  1.87358683  0.79849451 -1.0422852 -1.18617945 -0.41611808 -0.59057988
#>  [3,]  1.55462979  0.23015757  1.0828173 -0.71967980 -0.05076391  0.61934323
#>  [4,]  1.68447675 -1.19438722 -0.8804542 -0.38084096  4.85746871 -2.25505601
#>  [5,] -0.40475285 -3.93165394  1.2031090 -1.10809622  3.67592549 -2.33716199
#>  [6,]  0.64901278 -0.97237624 -0.5510658 -0.53625095  2.69991251 -1.49219136
#>  [7,]  3.99681767  0.72029005  0.6643212 -0.36409009  3.40773920 -0.03014889
#>  [8,]  0.72029005  2.33037081 -1.0932861 -0.05091026 -2.39257933  0.97854613
#>  [9,]  0.66432124 -1.09328610  2.6252769  0.18420341  1.35806231  0.89147509
#> [10,] -0.36409009 -0.05091026  0.1842034  0.63766708  0.45717851  0.17839517
#> [11,]  3.40773920 -2.39257933  1.3580623  0.45717851  9.40839733 -2.12302461
#> [12,] -0.03014889  0.97854613  0.8914751  0.17839517 -2.12302461  1.56507368
#> [13,]  0.53747710  0.62335164 -1.7263066 -0.16801722  2.00207835 -1.04850987
#> [14,] -0.54909423  0.89755359 -0.9881515 -0.20275907  0.26436812  0.04971903
#> [15,]  3.96893829  0.01263142  3.0692920 -0.18071099  2.71536171  1.26938398
#> [16,]  2.21242869 -0.13253457 -0.1922637 -0.01416424  3.03971927 -0.90560522
#> [17,]  1.84430652 -1.52585035  1.7864532 -0.39613054  2.15540818 -0.39610824
#> [18,] -0.29834889  0.74045099 -2.7048943  0.63311926  1.19079398 -1.41847302
#> [19,] -2.14757587 -0.15109842  2.0981594 -0.11417940 -4.86958674  2.19656123
#> [20,] -2.02519402  1.21456995 -1.1465792 -0.35286530 -3.93909742  0.74335629
#>            [,13]       [,14]       [,15]       [,16]      [,17]       [,18]
#>  [1,] -1.3643696 -3.34589856  3.44613353  2.21401577  2.3230120  0.33973581
#>  [2,] -0.1981154 -1.66937433  1.15529340  1.40295050  1.8603140 -0.07978953
#>  [3,] -0.5099055 -0.04172636  2.52175529  0.11544473  1.2403819 -2.17926040
#>  [4,]  2.4358073  0.92047729 -0.38160326  1.94035657  0.6790198  1.83984440
#>  [5,] -0.5748421 -1.40437035  0.11063761  0.61230413  3.4904505 -1.92929496
#>  [6,]  2.2152982  2.00517488 -0.86786024  0.58729133 -0.2302620  0.65064488
#>  [7,]  0.5374771 -0.54909423  3.96893829  2.21242869  1.8443065 -0.29834889
#>  [8,]  0.6233516  0.89755359  0.01263142 -0.13253457 -1.5258504  0.74045099
#>  [9,] -1.7263066 -0.98815150  3.06929203 -0.19226366  1.7864532 -2.70489430
#> [10,] -0.1680172 -0.20275907 -0.18071099 -0.01416424 -0.3961305  0.63311926
#> [11,]  2.0020784  0.26436812  2.71536171  3.03971927  2.1554082  1.19079398
#> [12,] -1.0485099  0.04971903  1.26938398 -0.90560522 -0.3961082 -1.41847302
#> [13,]  5.1068123  6.26449529 -2.69778122 -0.26846824 -3.9654463  2.25359137
#> [14,]  6.2644953 10.06865714 -3.42688940 -2.41198968 -6.4641202  0.69815234
#> [15,] -2.6977812 -3.42688940  6.91829103  1.83446232  4.2390647 -3.17493052
#> [16,] -0.2684682 -2.41198968  1.83446232  2.30274016  2.2057695  1.11181292
#> [17,] -3.9654463 -6.46412016  4.23906468  2.20576953  5.7561596 -1.92088694
#> [18,]  2.2535914  0.69815234 -3.17493052  1.11181292 -1.9208869  4.41260418
#> [19,] -3.6174622 -1.68379623  1.16604201 -2.42331854  0.8308740 -3.95816891
#> [20,]  2.1846744  4.74798088 -3.19917649 -2.55553160 -4.0219732 -0.15256591
#>            [,19]       [,20]
#>  [1,] -1.0549405 -2.07112376
#>  [2,] -1.0609614 -0.62178843
#>  [3,]  1.0597003 -0.09315525
#>  [4,] -4.2006820 -1.48815132
#>  [5,] -0.1773063 -1.70809559
#>  [6,] -2.3217978 -0.01356917
#>  [7,] -2.1475759 -2.02519402
#>  [8,] -0.1510984  1.21456995
#>  [9,]  2.0981594 -1.14657923
#> [10,] -0.1141794 -0.35286530
#> [11,] -4.8695867 -3.93909742
#> [12,]  2.1965612  0.74335629
#> [13,] -3.6174622  2.18467440
#> [14,] -1.6837962  4.74798088
#> [15,]  1.1660420 -3.19917649
#> [16,] -2.4233185 -2.55553160
#> [17,]  0.8308740 -4.02197321
#> [18,] -3.9581689 -0.15256591
#> [19,]  5.9389430  1.29286827
#> [20,]  1.2928683  4.05282600
#>
#> , , 2
#>
#>             [,1]        [,2]         [,3]        [,4]        [,5]       [,6]
#>  [1,]  5.4368110  2.99809180  1.479165320 -3.10149298 -1.56908180  3.6768618
#>  [2,]  2.9980918  5.00425046 -0.270553781 -1.24762334  0.02946354  4.5189416
#>  [3,]  1.4791653 -0.27055378  2.084698987 -0.24885346  0.30496693  0.2142017
#>  [4,] -3.1014930 -1.24762334 -0.248853465  2.70664376  1.10053928 -1.9657107
#>  [5,] -1.5690818  0.02946354  0.304966933  1.10053928  2.21492131 -0.2876180
#>  [6,]  3.6768618  4.51894161  0.214201670 -1.96571071 -0.28761801  4.5546636
#>  [7,]  1.1644761  0.90008029 -0.571724039 -0.87324355 -0.86899316  0.6967893
#>  [8,] -3.5866357 -2.64341813 -2.101125554  1.70340059 -1.05564696 -2.9441997
#>  [9,]  0.4177317  3.98788726 -0.003178642  0.83645614  1.68033441  3.2039849
#> [10,] -2.5875393  0.37685467 -2.378863305  0.66399604  0.49810260  0.0611505
#> [11,] -0.6207906  1.71965888 -1.822539915 -0.08154119 -0.27802710  1.4852704
#> [12,] -1.0021484  1.56138242 -1.099216635  0.90653800  1.28145505  0.3602908
#> [13,] -0.9522321 -2.83634683 -1.558322164 -0.87499173 -2.36380036 -2.1392729
#> [14,]  0.3065371 -1.18442349  2.946137391  0.92728936  1.52452484 -0.7534573
#> [15,]  2.0963446  2.37642831  0.665625507 -0.84651109  0.15714053  2.4087790
#> [16,]  0.6348351 -1.32143683  0.642367247 -0.52825281 -0.74349106 -0.9046342
#> [17,] -1.1405063 -2.00081571 -0.477525572 -0.11658182  0.24742690 -2.0401719
#> [18,]  0.4141434 -3.70082836  0.323483005 -2.98457673 -0.43594353 -1.7386781
#> [19,]  0.1105901 -2.38379602 -0.691796234 -1.93852162 -1.49457524 -1.1544674
#> [20,] -0.7574311 -5.05364861  2.346229340  1.04992124 -0.07920224 -4.8743534
#>               [,7]        [,8]         [,9]      [,10]       [,11]       [,12]
#>  [1,]  1.164476136 -3.58663570  0.417731722 -2.5875393 -0.62079056 -1.00214843
#>  [2,]  0.900080293 -2.64341813  3.987887256  0.3768547  1.71965888  1.56138242
#>  [3,] -0.571724039 -2.10112555 -0.003178642 -2.3788633 -1.82253991 -1.09921664
#>  [4,] -0.873243548  1.70340059  0.836456143  0.6639960 -0.08154119  0.90653800
#>  [5,] -0.868993157 -1.05564696  1.680334415  0.4981026 -0.27802710  1.28145505
#>  [6,]  0.696789293 -2.94419965  3.203984931  0.0611505  1.48527041  0.36029079
#>  [7,]  1.118316129 -0.17117164 -0.443343243 -0.3846707  0.05825846  0.84199106
#>  [8,] -0.171171643  4.96303632 -2.012985658  2.5700598  1.51467057 -0.57366109
#>  [9,] -0.443343243 -2.01298566  5.226404581  1.1841707  1.70385382  1.80089970
#> [10,] -0.384670740  2.57005981  1.184170715  4.2141205  3.15093273  0.41798534
#> [11,]  0.058258465  1.51467057  1.703853816  3.1509327  3.01601295  0.10860004
#> [12,]  0.841991055 -0.57366109  1.800899699  0.4179853  0.10860004  3.67695511
#> [13,]  0.432530589  3.85275154 -4.153821830  1.5478717  0.99667925 -2.30460761
#> [14,] -1.492827425 -2.51539514  0.500984188 -2.8623394 -2.71687329 -1.06428745
#> [15,]  0.156132315 -2.15491069  1.971002184 -0.5754599  0.26848149  0.17374492
#> [16,]  0.159897503  0.01224757 -1.928576622 -1.3437015 -1.19780917 -0.96361141
#> [17,]  0.480662081  0.56342233 -2.371529728 -0.3585206 -1.20130973  0.93438687
#> [18,] -0.503440937  0.10987187 -4.941765059  0.4739439 -0.71085395 -3.44769941
#> [19,]  0.002576151  1.81705441 -3.685975323  1.2463115  0.65327258 -2.81273554
#> [20,]  0.112211849 -0.29874747 -4.936679003 -5.1818855 -5.53891152 -0.01693155
#>            [,13]      [,14]      [,15]       [,16]      [,17]      [,18]
#>  [1,] -0.9522321  0.3065371  2.0963446  0.63483513 -1.1405063  0.4141434
#>  [2,] -2.8363468 -1.1844235  2.3764283 -1.32143683 -2.0008157 -3.7008284
#>  [3,] -1.5583222  2.9461374  0.6656255  0.64236725 -0.4775256  0.3234830
#>  [4,] -0.8749917  0.9272894 -0.8465111 -0.52825281 -0.1165818 -2.9845767
#>  [5,] -2.3638004  1.5245248  0.1571405 -0.74349106  0.2474269 -0.4359435
#>  [6,] -2.1392729 -0.7534573  2.4087790 -0.90463421 -2.0401719 -1.7386781
#>  [7,]  0.4325306 -1.4928274  0.1561323  0.15989750  0.4806621 -0.5034409
#>  [8,]  3.8527515 -2.5153951 -2.1549107  0.01224757  0.5634223  0.1098719
#>  [9,] -4.1538218  0.5009842  1.9710022 -1.92857662 -2.3715297 -4.9417651
#> [10,]  1.5478717 -2.8623394 -0.5754599 -1.34370147 -0.3585206  0.4739439
#> [11,]  0.9966792 -2.7168733  0.2684815 -1.19780917 -1.2013097 -0.7108539
#> [12,] -2.3046076 -1.0642874  0.1737449 -0.96361141  0.9343869 -3.4476994
#> [13,]  5.6637826 -2.9326926 -1.8961991  1.06179905  1.1215934  4.4454140
#> [14,] -2.9326926  4.9986310  0.5262937  0.57355423 -0.4983960 -0.1179677
#> [15,] -1.8961991  0.5262937  1.4616347 -0.42307366 -1.2163899 -1.2955730
#> [16,]  1.0617990  0.5735542 -0.4230737  0.95006025  0.6596412  1.6481427
#> [17,]  1.1215934 -0.4983960 -1.2163899  0.65964119  2.3974651  2.4083900
#> [18,]  4.4454140 -0.1179677 -1.2955730  1.64814269  2.4083900 11.3292409
#> [19,]  4.6539045 -1.7332902 -1.1874947  1.05569804  1.1074698  6.8536060
#> [20,]  0.1823178  3.7977208 -1.9006823  2.54162888  3.2595687  1.4392538
#>              [,19]       [,20]
#>  [1,]  0.110590115 -0.75743108
#>  [2,] -2.383796024 -5.05364861
#>  [3,] -0.691796234  2.34622934
#>  [4,] -1.938521615  1.04992124
#>  [5,] -1.494575236 -0.07920224
#>  [6,] -1.154467448 -4.87435336
#>  [7,]  0.002576151  0.11221185
#>  [8,]  1.817054413 -0.29874747
#>  [9,] -3.685975323 -4.93667900
#> [10,]  1.246311514 -5.18188550
#> [11,]  0.653272582 -5.53891152
#> [12,] -2.812735543 -0.01693155
#> [13,]  4.653904470  0.18231779
#> [14,] -1.733290196  3.79772080
#> [15,] -1.187494651 -1.90068229
#> [16,]  1.055698037  2.54162888
#> [17,]  1.107469766  3.25956870
#> [18,]  6.853606027  1.43925384
#> [19,]  5.265165752 -0.40981781
#> [20,] -0.409817805 11.78585618
#>