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] 0.2769771
#>
#> $pval
#> [1] 0.8005293
#>
#> $lambda
#> [1] 1.081735e-01 4.614250e-02 4.397778e-02 3.344607e-02 2.916064e-02
#> [6] 2.104665e-02 1.968491e-02 1.662169e-02 1.304993e-02 1.113304e-02
#> [11] 9.932567e-03 9.093349e-03 6.657533e-03 6.060521e-03 5.744674e-03
#> [16] 5.111374e-03 4.726046e-03 3.953793e-03 3.479026e-03 3.009560e-03
#> [21] 2.970471e-03 2.725364e-03 2.553849e-03 2.242454e-03 1.994572e-03
#> [26] 1.765663e-03 1.698397e-03 1.627769e-03 1.477914e-03 1.269786e-03
#> [31] 1.193930e-03 1.009077e-03 8.416519e-04 7.743184e-04 7.125721e-04
#> [36] 7.014020e-04 6.539012e-04 5.950031e-04 5.686342e-04 5.329094e-04
#> [41] 5.125558e-04 4.695223e-04 4.446937e-04 3.701449e-04 3.431470e-04
#> [46] 3.148787e-04 2.972525e-04 2.708120e-04 2.605834e-04 2.479797e-04
#> [51] 2.281246e-04 2.075319e-04 1.868858e-04 1.511282e-04 1.506122e-04
#> [56] 1.448896e-04 1.358788e-04 1.155455e-04 1.128304e-04 1.071526e-04
#> [61] 9.680568e-05 8.874241e-05 7.626916e-05 7.267959e-05 7.048048e-05
#> [66] 6.039239e-05 5.621815e-05 5.582395e-05 5.255024e-05 5.099450e-05
#> [71] 4.161980e-05 4.061531e-05 3.925478e-05 3.618185e-05 3.397732e-05
#> [76] 3.096492e-05 2.879035e-05 2.810831e-05 2.459755e-05 2.347234e-05
#> [81] 2.249029e-05 2.162283e-05 1.951125e-05 1.759439e-05 1.633400e-05
#> [86] 1.605866e-05 1.465520e-05 1.385973e-05 1.244009e-05 1.148128e-05
#> [91] 1.088174e-05 9.950035e-06 9.688285e-06 8.849622e-06 8.235872e-06
#> [96] 7.613514e-06 6.744350e-06 6.151627e-06 5.962908e-06 5.667942e-06
#> [101] 5.202822e-06 5.038667e-06 4.706066e-06 4.409856e-06 4.317501e-06
#> [106] 3.930106e-06 3.763693e-06 3.594340e-06 3.401357e-06 3.300005e-06
#> [111] 3.171507e-06 3.054443e-06 2.834968e-06 2.789717e-06 2.695248e-06
#> [116] 2.500644e-06 2.434171e-06 2.313492e-06 2.087257e-06 2.061165e-06
#> [121] 2.020072e-06 1.951581e-06 1.905406e-06 1.769122e-06 1.692079e-06
#> [126] 1.643647e-06 1.600226e-06 1.561140e-06 1.507398e-06 1.484624e-06
#> [131] 1.390272e-06 1.374216e-06 1.348051e-06 1.313379e-06 1.300610e-06
#> [136] 1.281128e-06 1.260226e-06 1.217128e-06 1.209138e-06 1.203072e-06
#> [141] 1.179635e-06 1.158567e-06 1.156472e-06 1.133637e-06 1.130701e-06
#> [146] 1.125300e-06 1.116087e-06 1.108413e-06 1.096408e-06 1.090273e-06
#> [151] 1.083008e-06 1.077296e-06 1.071747e-06 1.065459e-06 1.058204e-06
#> [156] 1.053544e-06 1.051803e-06 1.050574e-06 1.045169e-06 1.038003e-06
#> [161] 1.033778e-06 1.031279e-06 1.028613e-06 1.027183e-06 1.026614e-06
#> [166] 1.025454e-06 1.023971e-06 1.021321e-06 1.018712e-06 1.018152e-06
#> [171] 1.015969e-06 1.015827e-06 1.015236e-06 1.013401e-06 1.013255e-06
#> [176] 1.012375e-06 1.012162e-06 1.011406e-06 1.010827e-06 1.009982e-06
#> [181] 1.009742e-06 1.009658e-06 1.009464e-06 1.008902e-06 1.008640e-06
#> [186] 1.008584e-06 1.008357e-06 1.007979e-06 1.007889e-06 1.007860e-06
#> [191] 1.007796e-06 1.007618e-06 1.007536e-06 1.007456e-06 1.007332e-06
#> [196] 1.007276e-06 1.007221e-06 1.006991e-06 1.006966e-06 1.006732e-06
#>
#> $gamma.U
#> [1] 1.72166
#>
#> $gamma.Theta
#> [1] 0.1841406
EllKEPT(X,kerU="PIQ",kerTheta="PIQ")
#> $stat
#> [1] 0.2006694
#>
#> $pval
#> [1] 0.8809612
#>
#> $lambda
#> [1] 9.117211e-02 3.236201e-02 2.935093e-02 2.404879e-02 2.315553e-02
#> [6] 1.635279e-02 1.297738e-02 1.164860e-02 9.909406e-03 8.502158e-03
#> [11] 7.663264e-03 6.539185e-03 5.771858e-03 5.364589e-03 4.797875e-03
#> [16] 4.301296e-03 3.878238e-03 3.422083e-03 3.028029e-03 2.750117e-03
#> [21] 2.657611e-03 2.529549e-03 2.379314e-03 2.286554e-03 2.209114e-03
#> [26] 1.872685e-03 1.777012e-03 1.676203e-03 1.611795e-03 1.532515e-03
#> [31] 1.350778e-03 1.191451e-03 1.098651e-03 1.058913e-03 9.734122e-04
#> [36] 9.130092e-04 8.482878e-04 8.404540e-04 7.815143e-04 7.722023e-04
#> [41] 6.620711e-04 6.372454e-04 6.302412e-04 5.665097e-04 5.440907e-04
#> [46] 5.232880e-04 4.979290e-04 4.520641e-04 4.280064e-04 3.978030e-04
#> [51] 3.857207e-04 3.722795e-04 3.556303e-04 3.470276e-04 3.271471e-04
#> [56] 3.098270e-04 2.914465e-04 2.782805e-04 2.511474e-04 2.371861e-04
#> [61] 2.205208e-04 2.090337e-04 2.044733e-04 1.934398e-04 1.821770e-04
#> [66] 1.773781e-04 1.664709e-04 1.599964e-04 1.497366e-04 1.398637e-04
#> [71] 1.384685e-04 1.238954e-04 1.161469e-04 1.127275e-04 1.064389e-04
#> [76] 1.025509e-04 9.867166e-05 9.050101e-05 8.801249e-05 8.558702e-05
#> [81] 7.775634e-05 7.500199e-05 7.076367e-05 7.047439e-05 6.824224e-05
#> [86] 6.447277e-05 6.045121e-05 5.918784e-05 5.448295e-05 5.327267e-05
#> [91] 5.002906e-05 4.750418e-05 4.425027e-05 4.343969e-05 4.267399e-05
#> [96] 4.010822e-05 3.667343e-05 3.539239e-05 3.489728e-05 3.194502e-05
#> [101] 3.084770e-05 2.932240e-05 2.747687e-05 2.684993e-05 2.597245e-05
#> [106] 2.512452e-05 2.459579e-05 2.398956e-05 2.224454e-05 2.070421e-05
#> [111] 2.000518e-05 1.816688e-05 1.761786e-05 1.642118e-05 1.613195e-05
#> [116] 1.545136e-05 1.512776e-05 1.341621e-05 1.325875e-05 1.314548e-05
#> [121] 1.263826e-05 1.237298e-05 1.139013e-05 1.080697e-05 1.052005e-05
#> [126] 9.815876e-06 9.438800e-06 9.164871e-06 8.940866e-06 8.399443e-06
#> [131] 7.962403e-06 7.525094e-06 7.303568e-06 6.757273e-06 6.399056e-06
#> [136] 6.108321e-06 5.832777e-06 5.781316e-06 5.583019e-06 5.356705e-06
#> [141] 5.164459e-06 4.732894e-06 4.337012e-06 4.283600e-06 4.219961e-06
#> [146] 3.949155e-06 3.874082e-06 3.556845e-06 3.510667e-06 3.376191e-06
#> [151] 3.244781e-06 3.121502e-06 2.941246e-06 2.872129e-06 2.810286e-06
#> [156] 2.732688e-06 2.669611e-06 2.500996e-06 2.355219e-06 2.285777e-06
#> [161] 2.249681e-06 2.138617e-06 2.128047e-06 2.084449e-06 1.992650e-06
#> [166] 1.962338e-06 1.873853e-06 1.826448e-06 1.643185e-06 1.633237e-06
#> [171] 1.594088e-06 1.552291e-06 1.535194e-06 1.521735e-06 1.497089e-06
#> [176] 1.425886e-06 1.408003e-06 1.384778e-06 1.313508e-06 1.312900e-06
#> [181] 1.258762e-06 1.231499e-06 1.223257e-06 1.211910e-06 1.178444e-06
#> [186] 1.172381e-06 1.159664e-06 1.150662e-06 1.130767e-06 1.126836e-06
#> [191] 1.110599e-06 1.099052e-06 1.073229e-06 1.066380e-06 1.039961e-06
#> [196] 1.026157e-06 1.019074e-06 1.015461e-06 1.012285e-06 1.011150e-06
#>
#> $gamma.U
#> [1] 1.72166
#>
#> $gamma.Theta
#> [1] 0.1841406
## test under an alternative distribution
X=matrix(rchisq(d*n,2),nrow=n,ncol=d)
EllKEPT(X,kerU="Gaussian",kerTheta="Gaussian")
#> $stat
#> [1] 5.34332
#>
#> $pval
#> [1] 2.757491e-07
#>
#> $lambda
#> [1] 1.896499e-01 4.889475e-02 4.310687e-02 3.145738e-02 2.519452e-02
#> [6] 2.126273e-02 1.727934e-02 1.504853e-02 1.194347e-02 1.097940e-02
#> [11] 9.323372e-03 8.027504e-03 7.109498e-03 6.196992e-03 5.414078e-03
#> [16] 4.997190e-03 4.623671e-03 4.335566e-03 3.399505e-03 3.189029e-03
#> [21] 2.673946e-03 2.480473e-03 1.690455e-03 1.608365e-03 1.465518e-03
#> [26] 1.327224e-03 1.174044e-03 1.112265e-03 1.095839e-03 9.372980e-04
#> [31] 8.071436e-04 7.541639e-04 6.684655e-04 5.323188e-04 5.045668e-04
#> [36] 4.830120e-04 4.547988e-04 3.679397e-04 3.344316e-04 3.162345e-04
#> [41] 2.873702e-04 2.784530e-04 2.498405e-04 1.920534e-04 1.879514e-04
#> [46] 1.684608e-04 1.435898e-04 1.360536e-04 1.252108e-04 1.178558e-04
#> [51] 1.145458e-04 1.112127e-04 1.005442e-04 8.805365e-05 8.081972e-05
#> [56] 7.706502e-05 6.356758e-05 6.099897e-05 5.651984e-05 5.541269e-05
#> [61] 4.599310e-05 4.277016e-05 3.759926e-05 3.282016e-05 3.112799e-05
#> [66] 2.719572e-05 2.628543e-05 2.436990e-05 2.137639e-05 1.935865e-05
#> [71] 1.854806e-05 1.769626e-05 1.556743e-05 1.406297e-05 1.216188e-05
#> [76] 1.103108e-05 1.032562e-05 9.324234e-06 8.622994e-06 8.246011e-06
#> [81] 7.791333e-06 7.076266e-06 6.291386e-06 6.058944e-06 5.961942e-06
#> [86] 5.276798e-06 4.908456e-06 4.701364e-06 4.379558e-06 4.007141e-06
#> [91] 3.766485e-06 3.578684e-06 3.400260e-06 3.255857e-06 3.185382e-06
#> [96] 2.890608e-06 2.546739e-06 2.536525e-06 2.413876e-06 2.263134e-06
#> [101] 2.160251e-06 2.097172e-06 1.984004e-06 1.875296e-06 1.806047e-06
#> [106] 1.683288e-06 1.548702e-06 1.519962e-06 1.480864e-06 1.445768e-06
#> [111] 1.419192e-06 1.403792e-06 1.360609e-06 1.312016e-06 1.282814e-06
#> [116] 1.240259e-06 1.211733e-06 1.205546e-06 1.189159e-06 1.164797e-06
#> [121] 1.159632e-06 1.150747e-06 1.135719e-06 1.131052e-06 1.111257e-06
#> [126] 1.103662e-06 1.091930e-06 1.085897e-06 1.082623e-06 1.068760e-06
#> [131] 1.064754e-06 1.057493e-06 1.049159e-06 1.045085e-06 1.043054e-06
#> [136] 1.040177e-06 1.035101e-06 1.033228e-06 1.032024e-06 1.027584e-06
#> [141] 1.025800e-06 1.020840e-06 1.019992e-06 1.018931e-06 1.017976e-06
#> [146] 1.016997e-06 1.016160e-06 1.015822e-06 1.014205e-06 1.013692e-06
#> [151] 1.013045e-06 1.012270e-06 1.011689e-06 1.010908e-06 1.010506e-06
#> [156] 1.010243e-06 1.010011e-06 1.009616e-06 1.009483e-06 1.009204e-06
#> [161] 1.009121e-06 1.009002e-06 1.008682e-06 1.008547e-06 1.008344e-06
#> [166] 1.008232e-06 1.007964e-06 1.007839e-06 1.007833e-06 1.007766e-06
#> [171] 1.007693e-06 1.007590e-06 1.007563e-06 1.007527e-06 1.007466e-06
#> [176] 1.007417e-06 1.007367e-06 1.007343e-06 1.007324e-06 1.007276e-06
#> [181] 1.007256e-06 1.007230e-06 1.007207e-06 1.007179e-06 1.007161e-06
#> [186] 1.007145e-06 1.007095e-06 1.007082e-06 1.007073e-06 1.007041e-06
#> [191] 1.007026e-06 1.007019e-06 1.007000e-06 1.006956e-06 1.006917e-06
#> [196] 1.006912e-06 1.006868e-06 1.006792e-06 1.006689e-06 1.006608e-06
#>
#> $gamma.U
#> [1] 1.439036
#>
#> $gamma.Theta
#> [1] 0.1657235
EllKEPT(X,kerU="PIQ",kerTheta="PIQ")
#> $stat
#> [1] 4.375077
#>
#> $pval
#> [1] 4.509722e-07
#>
#> $lambda
#> [1] 1.576769e-01 3.799643e-02 3.519573e-02 2.229214e-02 2.036715e-02
#> [6] 1.746360e-02 1.491803e-02 1.311886e-02 1.066510e-02 8.194544e-03
#> [11] 7.496425e-03 6.469179e-03 5.593811e-03 4.884869e-03 4.641783e-03
#> [16] 4.294755e-03 3.692827e-03 3.280286e-03 3.098315e-03 2.942990e-03
#> [21] 2.546715e-03 2.154162e-03 1.907554e-03 1.809852e-03 1.552165e-03
#> [26] 1.472265e-03 1.292313e-03 1.247507e-03 1.140254e-03 1.022731e-03
#> [31] 9.689627e-04 8.683126e-04 8.516396e-04 7.415708e-04 7.229339e-04
#> [36] 6.778548e-04 6.179591e-04 5.809193e-04 5.105989e-04 4.844427e-04
#> [41] 4.508414e-04 4.180578e-04 3.819708e-04 3.553486e-04 3.212439e-04
#> [46] 2.989022e-04 2.870848e-04 2.645050e-04 2.569047e-04 2.529198e-04
#> [51] 2.257889e-04 2.039492e-04 1.989393e-04 1.893909e-04 1.847812e-04
#> [56] 1.597243e-04 1.555241e-04 1.407897e-04 1.365144e-04 1.243553e-04
#> [61] 1.182632e-04 1.090245e-04 1.073299e-04 1.002559e-04 9.535005e-05
#> [66] 8.808133e-05 7.708240e-05 7.547438e-05 7.029370e-05 6.422547e-05
#> [71] 6.030149e-05 5.621572e-05 5.451914e-05 5.209615e-05 4.774915e-05
#> [76] 4.518134e-05 4.251905e-05 4.052199e-05 3.936089e-05 3.803277e-05
#> [81] 3.588012e-05 3.173617e-05 3.044853e-05 2.877773e-05 2.679271e-05
#> [86] 2.532240e-05 2.465772e-05 2.349773e-05 2.180673e-05 1.954098e-05
#> [91] 1.937130e-05 1.826422e-05 1.671363e-05 1.605785e-05 1.503433e-05
#> [96] 1.436470e-05 1.363146e-05 1.293574e-05 1.281231e-05 1.163987e-05
#> [101] 1.125375e-05 1.042714e-05 9.777355e-06 9.304785e-06 8.995857e-06
#> [106] 8.864893e-06 7.490162e-06 7.256584e-06 6.967830e-06 6.658756e-06
#> [111] 6.405472e-06 6.027504e-06 5.656662e-06 5.360372e-06 5.169020e-06
#> [116] 4.991905e-06 4.690191e-06 4.409379e-06 4.166074e-06 4.031814e-06
#> [121] 3.702837e-06 3.637373e-06 3.529331e-06 3.325310e-06 3.220041e-06
#> [126] 3.161539e-06 2.985050e-06 2.774528e-06 2.653381e-06 2.594318e-06
#> [131] 2.470377e-06 2.431815e-06 2.291311e-06 2.226558e-06 2.082184e-06
#> [136] 2.000214e-06 1.962560e-06 1.918376e-06 1.785802e-06 1.746076e-06
#> [141] 1.680303e-06 1.613554e-06 1.559629e-06 1.552506e-06 1.481179e-06
#> [146] 1.447434e-06 1.408955e-06 1.375509e-06 1.362874e-06 1.334338e-06
#> [151] 1.307762e-06 1.275050e-06 1.254193e-06 1.231465e-06 1.213570e-06
#> [156] 1.185517e-06 1.165672e-06 1.148396e-06 1.123535e-06 1.110675e-06
#> [161] 1.106421e-06 1.102317e-06 1.090384e-06 1.079421e-06 1.076269e-06
#> [166] 1.067667e-06 1.058300e-06 1.055011e-06 1.052996e-06 1.048264e-06
#> [171] 1.042413e-06 1.040601e-06 1.037049e-06 1.034393e-06 1.032981e-06
#> [176] 1.030359e-06 1.026096e-06 1.023031e-06 1.022537e-06 1.020814e-06
#> [181] 1.018606e-06 1.017608e-06 1.015044e-06 1.014936e-06 1.013502e-06
#> [186] 1.011142e-06 1.010546e-06 1.009600e-06 1.008871e-06 1.008679e-06
#> [191] 1.007948e-06 1.007868e-06 1.007498e-06 1.007336e-06 1.007119e-06
#> [196] 1.007018e-06 1.006970e-06 1.006838e-06 1.006770e-06 1.006722e-06
#>
#> $gamma.U
#> [1] 1.439036
#>
#> $gamma.Theta
#> [1] 0.1657235