This package provides an implementation of a kernel-embedding of probability test for elliptical distribution. This is a guide to perform the asymptotic test for elliptical distribution under general alternatives, and the location and shape parameters are assumed to be unknown.
To conduct the test for elliptical distribution, we can directly use the EllKEPT function as follows.
n=200
d=3
## test under a null distribution
X=matrix(rnorm(d*n),nrow=n,ncol=d)
EllKEPT(X,kerU="Gaussian",kerTheta="Gaussian")
#> $stat
#> [1] 0.3487693
#>
#> $pval
#> [1] 0.5984007
#>
#> $lambda
#> [1] 1.088688e-01 6.123149e-02 4.263981e-02 3.497520e-02 2.661992e-02
#> [6] 2.266602e-02 1.961508e-02 1.349981e-02 1.131919e-02 1.052138e-02
#> [11] 8.913155e-03 6.769227e-03 6.568288e-03 5.478614e-03 5.063738e-03
#> [16] 4.519615e-03 4.061917e-03 4.005093e-03 3.212080e-03 3.042056e-03
#> [21] 2.517167e-03 2.164344e-03 2.035871e-03 2.015628e-03 1.759199e-03
#> [26] 1.520858e-03 1.469225e-03 1.339335e-03 1.156372e-03 1.082612e-03
#> [31] 9.926096e-04 9.064540e-04 8.326650e-04 7.749465e-04 7.359378e-04
#> [36] 6.401546e-04 5.656935e-04 5.431664e-04 5.094743e-04 4.504611e-04
#> [41] 4.202145e-04 3.968640e-04 3.488020e-04 3.195805e-04 2.943004e-04
#> [46] 2.844341e-04 2.438304e-04 2.216356e-04 2.113551e-04 1.852163e-04
#> [51] 1.717447e-04 1.660646e-04 1.559455e-04 1.371938e-04 1.253824e-04
#> [56] 1.189404e-04 1.035222e-04 9.334569e-05 9.148783e-05 8.902994e-05
#> [61] 7.895675e-05 7.264327e-05 6.777266e-05 6.163927e-05 5.777148e-05
#> [66] 5.402526e-05 5.380650e-05 4.477338e-05 4.261032e-05 3.970521e-05
#> [71] 3.843923e-05 3.532787e-05 3.262585e-05 2.987325e-05 2.743320e-05
#> [76] 2.639401e-05 2.280254e-05 2.186950e-05 2.064107e-05 2.018225e-05
#> [81] 1.839468e-05 1.745793e-05 1.692289e-05 1.558524e-05 1.478379e-05
#> [86] 1.412910e-05 1.223621e-05 1.141715e-05 1.050745e-05 1.030079e-05
#> [91] 9.476555e-06 8.720226e-06 8.405803e-06 7.789124e-06 7.543411e-06
#> [96] 7.198310e-06 6.149959e-06 5.956285e-06 5.782964e-06 5.545147e-06
#> [101] 5.192935e-06 4.960008e-06 4.621595e-06 4.323634e-06 3.859046e-06
#> [106] 3.689183e-06 3.595934e-06 3.510027e-06 3.327882e-06 3.096183e-06
#> [111] 2.939376e-06 2.855646e-06 2.737331e-06 2.610447e-06 2.427097e-06
#> [116] 2.411481e-06 2.351203e-06 2.076052e-06 2.033755e-06 1.985264e-06
#> [121] 1.912650e-06 1.836713e-06 1.766530e-06 1.737324e-06 1.643632e-06
#> [126] 1.578279e-06 1.575622e-06 1.553056e-06 1.515813e-06 1.440911e-06
#> [131] 1.416407e-06 1.378683e-06 1.344243e-06 1.312518e-06 1.294382e-06
#> [136] 1.274276e-06 1.261906e-06 1.224443e-06 1.216678e-06 1.200128e-06
#> [141] 1.178026e-06 1.155957e-06 1.144968e-06 1.140809e-06 1.122105e-06
#> [146] 1.107937e-06 1.095979e-06 1.093302e-06 1.084455e-06 1.079790e-06
#> [151] 1.071912e-06 1.060838e-06 1.056582e-06 1.054273e-06 1.051504e-06
#> [156] 1.046355e-06 1.044597e-06 1.042802e-06 1.037055e-06 1.035268e-06
#> [161] 1.033249e-06 1.031267e-06 1.029788e-06 1.025921e-06 1.024565e-06
#> [166] 1.023405e-06 1.021004e-06 1.018696e-06 1.018203e-06 1.016784e-06
#> [171] 1.015133e-06 1.014466e-06 1.013594e-06 1.012640e-06 1.012469e-06
#> [176] 1.011935e-06 1.011431e-06 1.011131e-06 1.010517e-06 1.010315e-06
#> [181] 1.010041e-06 1.009879e-06 1.009507e-06 1.009194e-06 1.008978e-06
#> [186] 1.008775e-06 1.008272e-06 1.007943e-06 1.007896e-06 1.007801e-06
#> [191] 1.007535e-06 1.007471e-06 1.007281e-06 1.007192e-06 1.007070e-06
#> [196] 1.007043e-06 1.006944e-06 1.006867e-06 1.006847e-06 1.006729e-06
#>
#> $gamma.U
#> [1] 1.561954
#>
#> $gamma.Theta
#> [1] 0.1692031
EllKEPT(X,kerU="PIQ",kerTheta="PIQ")
#> $stat
#> [1] 0.2795403
#>
#> $pval
#> [1] 0.6034877
#>
#> $lambda
#> [1] 9.886401e-02 4.085687e-02 2.912490e-02 2.379838e-02 2.108489e-02
#> [6] 1.522636e-02 1.288238e-02 1.045090e-02 8.653669e-03 7.690049e-03
#> [11] 6.515209e-03 5.789796e-03 5.210600e-03 4.602971e-03 4.355143e-03
#> [16] 3.903983e-03 3.739027e-03 3.287692e-03 3.202668e-03 2.974350e-03
#> [21] 2.500546e-03 2.288314e-03 2.190611e-03 2.127930e-03 1.969709e-03
#> [26] 1.621817e-03 1.602997e-03 1.429424e-03 1.359800e-03 1.319054e-03
#> [31] 1.266299e-03 1.094376e-03 9.882778e-04 9.301321e-04 9.099254e-04
#> [36] 8.533765e-04 8.083665e-04 7.733439e-04 7.096828e-04 6.680699e-04
#> [41] 6.123831e-04 5.928289e-04 5.765254e-04 5.506757e-04 4.912048e-04
#> [46] 4.524487e-04 4.180577e-04 3.932606e-04 3.673384e-04 3.561637e-04
#> [51] 3.189208e-04 2.975380e-04 2.871208e-04 2.758929e-04 2.613753e-04
#> [56] 2.540522e-04 2.362434e-04 2.267810e-04 2.153569e-04 1.922326e-04
#> [61] 1.876696e-04 1.840515e-04 1.698309e-04 1.568773e-04 1.492295e-04
#> [66] 1.462709e-04 1.406930e-04 1.347799e-04 1.253499e-04 1.209493e-04
#> [71] 1.117954e-04 1.110107e-04 1.038818e-04 1.023668e-04 9.445404e-05
#> [76] 9.353125e-05 8.471844e-05 8.116947e-05 7.839715e-05 7.301110e-05
#> [81] 7.092752e-05 7.025100e-05 6.270226e-05 6.171757e-05 5.968988e-05
#> [86] 5.784649e-05 5.444689e-05 5.294095e-05 5.029791e-05 4.835182e-05
#> [91] 4.462116e-05 4.414201e-05 4.201301e-05 3.921197e-05 3.835393e-05
#> [96] 3.682846e-05 3.486559e-05 3.247636e-05 3.109352e-05 3.050717e-05
#> [101] 2.759141e-05 2.687458e-05 2.511671e-05 2.395394e-05 2.319363e-05
#> [106] 2.242651e-05 2.122783e-05 2.080408e-05 1.997694e-05 1.902686e-05
#> [111] 1.796635e-05 1.759119e-05 1.667032e-05 1.564585e-05 1.518417e-05
#> [116] 1.486583e-05 1.322908e-05 1.280086e-05 1.120080e-05 1.076998e-05
#> [121] 1.066682e-05 1.050112e-05 9.878401e-06 9.761335e-06 9.579974e-06
#> [126] 9.094526e-06 8.679589e-06 8.132899e-06 7.283641e-06 7.177197e-06
#> [131] 6.886504e-06 6.561717e-06 6.486724e-06 6.190228e-06 5.881312e-06
#> [136] 5.597904e-06 5.360347e-06 5.007162e-06 4.938094e-06 4.833128e-06
#> [141] 4.704931e-06 4.391104e-06 4.268655e-06 4.185210e-06 4.123713e-06
#> [146] 3.938453e-06 3.834213e-06 3.615778e-06 3.448224e-06 3.402130e-06
#> [151] 3.138869e-06 3.102637e-06 2.958173e-06 2.728435e-06 2.637233e-06
#> [156] 2.539285e-06 2.436481e-06 2.316290e-06 2.280797e-06 2.171530e-06
#> [161] 2.100791e-06 2.072936e-06 1.969768e-06 1.900393e-06 1.861021e-06
#> [166] 1.824316e-06 1.753765e-06 1.678408e-06 1.602191e-06 1.585078e-06
#> [171] 1.564448e-06 1.504612e-06 1.450016e-06 1.437729e-06 1.403053e-06
#> [176] 1.369179e-06 1.342729e-06 1.299095e-06 1.263812e-06 1.258885e-06
#> [181] 1.239090e-06 1.215309e-06 1.194332e-06 1.169252e-06 1.143084e-06
#> [186] 1.134218e-06 1.121428e-06 1.113694e-06 1.102661e-06 1.087039e-06
#> [191] 1.079501e-06 1.076745e-06 1.068124e-06 1.060306e-06 1.051941e-06
#> [196] 1.047201e-06 1.034481e-06 1.023226e-06 1.020504e-06 1.017399e-06
#>
#> $gamma.U
#> [1] 1.561954
#>
#> $gamma.Theta
#> [1] 0.1692031
## test under an alternative distribution
X=matrix(rchisq(d*n,2),nrow=n,ncol=d)
EllKEPT(X,kerU="Gaussian",kerTheta="Gaussian")
#> Warning in imhof(test.stat.hs, lambda.hs): Note that Qq + abserr is positive.
#> $stat
#> [1] 7.605133
#>
#> $pval
#> [1] 0
#>
#> $lambda
#> [1] 1.911066e-01 4.231062e-02 3.488341e-02 3.287074e-02 2.950553e-02
#> [6] 2.117072e-02 1.576746e-02 1.487788e-02 1.300985e-02 1.021868e-02
#> [11] 8.781429e-03 7.662460e-03 6.518091e-03 5.377544e-03 5.118637e-03
#> [16] 4.393029e-03 3.728366e-03 3.403064e-03 3.135352e-03 2.700196e-03
#> [21] 2.474174e-03 2.396416e-03 2.084952e-03 1.823740e-03 1.669781e-03
#> [26] 1.349915e-03 1.178689e-03 1.074601e-03 9.949751e-04 8.361644e-04
#> [31] 7.788120e-04 6.796781e-04 6.367095e-04 5.769401e-04 5.134022e-04
#> [36] 4.634172e-04 4.331818e-04 4.034492e-04 3.837531e-04 3.215703e-04
#> [41] 3.028370e-04 2.470490e-04 2.378808e-04 2.040319e-04 1.882272e-04
#> [46] 1.560978e-04 1.535627e-04 1.393318e-04 1.292284e-04 1.140983e-04
#> [51] 1.054943e-04 9.740862e-05 8.329689e-05 7.809598e-05 7.451076e-05
#> [56] 6.408038e-05 5.894214e-05 5.617327e-05 5.073704e-05 4.383284e-05
#> [61] 4.069791e-05 3.797307e-05 3.458707e-05 3.234350e-05 2.749838e-05
#> [66] 2.437054e-05 2.149039e-05 2.114796e-05 1.852772e-05 1.691358e-05
#> [71] 1.647841e-05 1.547633e-05 1.392950e-05 1.115353e-05 1.102409e-05
#> [76] 1.043082e-05 9.073738e-06 7.861794e-06 7.525542e-06 6.738438e-06
#> [81] 6.544467e-06 5.952823e-06 5.763012e-06 5.329471e-06 5.141137e-06
#> [86] 4.764803e-06 4.191625e-06 3.785072e-06 3.658319e-06 3.423871e-06
#> [91] 2.918767e-06 2.745054e-06 2.715449e-06 2.534419e-06 2.421669e-06
#> [96] 2.251240e-06 2.166914e-06 2.046343e-06 1.959811e-06 1.942621e-06
#> [101] 1.860725e-06 1.776013e-06 1.642781e-06 1.571391e-06 1.536795e-06
#> [106] 1.476308e-06 1.425321e-06 1.384402e-06 1.356600e-06 1.329116e-06
#> [111] 1.273394e-06 1.248522e-06 1.244742e-06 1.236953e-06 1.206342e-06
#> [116] 1.199983e-06 1.168341e-06 1.142566e-06 1.131420e-06 1.128329e-06
#> [121] 1.120833e-06 1.098451e-06 1.087068e-06 1.076651e-06 1.074256e-06
#> [126] 1.066791e-06 1.064991e-06 1.060006e-06 1.053783e-06 1.048498e-06
#> [131] 1.043000e-06 1.040780e-06 1.036370e-06 1.033772e-06 1.032532e-06
#> [136] 1.028290e-06 1.027605e-06 1.026321e-06 1.021683e-06 1.020493e-06
#> [141] 1.018716e-06 1.017182e-06 1.016317e-06 1.015605e-06 1.015380e-06
#> [146] 1.014140e-06 1.013034e-06 1.012453e-06 1.011871e-06 1.011314e-06
#> [151] 1.010758e-06 1.010458e-06 1.010392e-06 1.010252e-06 1.009848e-06
#> [156] 1.009646e-06 1.009427e-06 1.008952e-06 1.008936e-06 1.008731e-06
#> [161] 1.008626e-06 1.008519e-06 1.008390e-06 1.008218e-06 1.008045e-06
#> [166] 1.007938e-06 1.007794e-06 1.007748e-06 1.007646e-06 1.007634e-06
#> [171] 1.007591e-06 1.007535e-06 1.007498e-06 1.007465e-06 1.007440e-06
#> [176] 1.007397e-06 1.007358e-06 1.007326e-06 1.007261e-06 1.007243e-06
#> [181] 1.007227e-06 1.007194e-06 1.007185e-06 1.007176e-06 1.007160e-06
#> [186] 1.007121e-06 1.007102e-06 1.007075e-06 1.007043e-06 1.007001e-06
#> [191] 1.006970e-06 1.006949e-06 1.006933e-06 1.006911e-06 1.006907e-06
#> [196] 1.006886e-06 1.006844e-06 1.006766e-06 1.006626e-06 1.006554e-06
#>
#> $gamma.U
#> [1] 1.457865
#>
#> $gamma.Theta
#> [1] 0.1754687
EllKEPT(X,kerU="PIQ",kerTheta="PIQ")
#> $stat
#> [1] 5.984837
#>
#> $pval
#> [1] 4.224907e-08
#>
#> $lambda
#> [1] 1.587907e-01 3.002861e-02 2.594621e-02 2.169164e-02 1.940839e-02
#> [6] 1.575223e-02 1.303059e-02 1.229866e-02 1.133497e-02 8.681845e-03
#> [11] 7.451063e-03 5.923897e-03 5.234428e-03 5.077523e-03 4.098476e-03
#> [16] 3.903879e-03 3.388638e-03 3.077116e-03 2.521937e-03 2.392744e-03
#> [21] 2.278023e-03 2.175130e-03 1.845336e-03 1.790299e-03 1.686385e-03
#> [26] 1.451135e-03 1.312968e-03 1.199473e-03 1.057081e-03 1.000013e-03
#> [31] 9.413724e-04 9.015397e-04 7.824578e-04 7.647157e-04 6.899065e-04
#> [36] 6.179036e-04 6.047137e-04 5.465193e-04 4.831844e-04 4.536213e-04
#> [41] 4.383026e-04 4.208397e-04 3.684767e-04 3.520804e-04 3.368720e-04
#> [46] 3.111745e-04 3.078367e-04 2.912928e-04 2.662139e-04 2.447025e-04
#> [51] 2.161478e-04 2.037022e-04 1.882749e-04 1.712073e-04 1.678063e-04
#> [56] 1.575508e-04 1.468703e-04 1.424168e-04 1.291280e-04 1.190302e-04
#> [61] 1.122331e-04 1.074184e-04 1.015538e-04 9.370359e-05 8.504853e-05
#> [66] 7.713214e-05 7.406802e-05 7.087408e-05 6.193313e-05 6.066651e-05
#> [71] 5.804824e-05 5.531079e-05 5.149516e-05 4.857322e-05 4.680782e-05
#> [76] 4.278202e-05 3.982773e-05 3.729122e-05 3.669691e-05 3.472345e-05
#> [81] 3.070045e-05 3.002230e-05 2.789807e-05 2.518639e-05 2.437968e-05
#> [86] 2.321251e-05 2.122527e-05 1.887758e-05 1.857439e-05 1.755139e-05
#> [91] 1.642718e-05 1.603768e-05 1.467441e-05 1.366791e-05 1.275709e-05
#> [96] 1.192825e-05 1.127157e-05 1.068299e-05 1.015948e-05 9.765243e-06
#> [101] 9.048999e-06 8.529574e-06 7.951559e-06 7.474327e-06 6.945429e-06
#> [106] 6.772076e-06 6.263790e-06 5.946702e-06 5.745394e-06 5.441699e-06
#> [111] 4.972942e-06 4.943000e-06 4.670856e-06 4.501454e-06 3.990773e-06
#> [116] 3.901739e-06 3.707903e-06 3.585480e-06 3.299474e-06 3.206585e-06
#> [121] 3.084981e-06 2.901427e-06 2.796370e-06 2.639921e-06 2.509486e-06
#> [126] 2.395684e-06 2.351215e-06 2.278005e-06 2.157049e-06 1.976099e-06
#> [131] 1.961062e-06 1.922161e-06 1.854340e-06 1.808569e-06 1.775827e-06
#> [136] 1.747617e-06 1.665960e-06 1.610246e-06 1.581510e-06 1.576860e-06
#> [141] 1.547574e-06 1.507593e-06 1.479472e-06 1.417084e-06 1.386096e-06
#> [146] 1.349950e-06 1.297703e-06 1.290595e-06 1.282220e-06 1.266469e-06
#> [151] 1.223988e-06 1.207161e-06 1.184854e-06 1.167630e-06 1.152250e-06
#> [156] 1.135909e-06 1.128130e-06 1.126047e-06 1.119095e-06 1.115378e-06
#> [161] 1.096915e-06 1.096535e-06 1.088309e-06 1.083169e-06 1.064273e-06
#> [166] 1.063413e-06 1.057819e-06 1.055687e-06 1.050360e-06 1.048438e-06
#> [171] 1.045597e-06 1.039569e-06 1.036604e-06 1.029660e-06 1.027030e-06
#> [176] 1.025661e-06 1.024211e-06 1.020641e-06 1.018614e-06 1.017267e-06
#> [181] 1.014445e-06 1.012005e-06 1.011695e-06 1.010410e-06 1.009413e-06
#> [186] 1.009166e-06 1.008796e-06 1.008105e-06 1.007884e-06 1.007533e-06
#> [191] 1.007381e-06 1.007232e-06 1.007188e-06 1.007141e-06 1.007069e-06
#> [196] 1.006971e-06 1.006912e-06 1.006833e-06 1.006683e-06 1.006593e-06
#>
#> $gamma.U
#> [1] 1.457865
#>
#> $gamma.Theta
#> [1] 0.1754687