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] 0.6064348
#>
#> $pval
#> [1] 0.1679897
#>
#> $lambda
#> [1] 1.191800e-01 5.365429e-02 4.387615e-02 3.874089e-02 2.511884e-02
#> [6] 2.083356e-02 1.797081e-02 1.322864e-02 1.031085e-02 9.583424e-03
#> [11] 8.526343e-03 7.348726e-03 6.633554e-03 6.073643e-03 5.857262e-03
#> [16] 4.891460e-03 4.113301e-03 3.353816e-03 3.276652e-03 3.175842e-03
#> [21] 2.604659e-03 2.505808e-03 2.266606e-03 2.043757e-03 1.770673e-03
#> [26] 1.453957e-03 1.289798e-03 1.226505e-03 1.170777e-03 1.114105e-03
#> [31] 9.921664e-04 9.810190e-04 8.463683e-04 7.994458e-04 7.858354e-04
#> [36] 6.553441e-04 5.987093e-04 5.846986e-04 5.160280e-04 4.395103e-04
#> [41] 4.191372e-04 3.680924e-04 3.588200e-04 3.320749e-04 2.990615e-04
#> [46] 2.946034e-04 2.267197e-04 2.114274e-04 2.038046e-04 1.791729e-04
#> [51] 1.606277e-04 1.502999e-04 1.286852e-04 1.169412e-04 1.089193e-04
#> [56] 1.063122e-04 9.732647e-05 9.502945e-05 8.407039e-05 7.531124e-05
#> [61] 7.079015e-05 6.895687e-05 6.108862e-05 5.773873e-05 5.546570e-05
#> [66] 5.115076e-05 4.882849e-05 4.839975e-05 4.441978e-05 3.825563e-05
#> [71] 3.657641e-05 3.488406e-05 3.059186e-05 2.890261e-05 2.621000e-05
#> [76] 2.326737e-05 2.120195e-05 2.023824e-05 1.909289e-05 1.745177e-05
#> [81] 1.612450e-05 1.516613e-05 1.410950e-05 1.289181e-05 1.223781e-05
#> [86] 1.190803e-05 1.137532e-05 1.057835e-05 1.010281e-05 8.972045e-06
#> [91] 7.880981e-06 7.488342e-06 7.247441e-06 6.892950e-06 6.569137e-06
#> [96] 5.787238e-06 5.637044e-06 5.098804e-06 4.861397e-06 4.630135e-06
#> [101] 4.463386e-06 4.127206e-06 3.958653e-06 3.849673e-06 3.647850e-06
#> [106] 3.390155e-06 3.079816e-06 2.989835e-06 2.880983e-06 2.820409e-06
#> [111] 2.671559e-06 2.505156e-06 2.307256e-06 2.289789e-06 2.118256e-06
#> [116] 2.079566e-06 2.072415e-06 1.971776e-06 1.914610e-06 1.837906e-06
#> [121] 1.773105e-06 1.763868e-06 1.644026e-06 1.636333e-06 1.602056e-06
#> [126] 1.555054e-06 1.527651e-06 1.444002e-06 1.400947e-06 1.387081e-06
#> [131] 1.352946e-06 1.317037e-06 1.295658e-06 1.276788e-06 1.263919e-06
#> [136] 1.230346e-06 1.223322e-06 1.195910e-06 1.191610e-06 1.169170e-06
#> [141] 1.155406e-06 1.148151e-06 1.126903e-06 1.114216e-06 1.102969e-06
#> [146] 1.099723e-06 1.093034e-06 1.090867e-06 1.086738e-06 1.077697e-06
#> [151] 1.073711e-06 1.063818e-06 1.059476e-06 1.053932e-06 1.050608e-06
#> [156] 1.047952e-06 1.042827e-06 1.040786e-06 1.038416e-06 1.033317e-06
#> [161] 1.030868e-06 1.028447e-06 1.026560e-06 1.024611e-06 1.023958e-06
#> [166] 1.022896e-06 1.021620e-06 1.019918e-06 1.018121e-06 1.017405e-06
#> [171] 1.016621e-06 1.016169e-06 1.014668e-06 1.013493e-06 1.012781e-06
#> [176] 1.011982e-06 1.011561e-06 1.011365e-06 1.010614e-06 1.010234e-06
#> [181] 1.009914e-06 1.009523e-06 1.009469e-06 1.009049e-06 1.008697e-06
#> [186] 1.008457e-06 1.008301e-06 1.008021e-06 1.007806e-06 1.007648e-06
#> [191] 1.007518e-06 1.007397e-06 1.007277e-06 1.007202e-06 1.007156e-06
#> [196] 1.007090e-06 1.006925e-06 1.006851e-06 1.006718e-06 1.006649e-06
#>
#> $gamma.U
#> [1] 1.639823
#>
#> $gamma.Theta
#> [1] 0.1711223
EllKEPT(X,kerU="PIQ",kerTheta="PIQ")
#> $stat
#> [1] 0.4393197
#>
#> $pval
#> [1] 0.2202305
#>
#> $lambda
#> [1] 1.033840e-01 3.696700e-02 2.984575e-02 2.713360e-02 1.962064e-02
#> [6] 1.344035e-02 1.305813e-02 8.969146e-03 7.591122e-03 6.795743e-03
#> [11] 6.634707e-03 5.479666e-03 5.117740e-03 4.980068e-03 4.415577e-03
#> [16] 4.188952e-03 4.158565e-03 3.495718e-03 3.362340e-03 2.907790e-03
#> [21] 2.506690e-03 2.302322e-03 2.136897e-03 1.981826e-03 1.819510e-03
#> [26] 1.650901e-03 1.552677e-03 1.324727e-03 1.303122e-03 1.192962e-03
#> [31] 1.163788e-03 1.056857e-03 1.036930e-03 9.288310e-04 9.097307e-04
#> [36] 8.183803e-04 7.786176e-04 7.305256e-04 7.135282e-04 6.697789e-04
#> [41] 6.354904e-04 5.879902e-04 5.635099e-04 4.977805e-04 4.808982e-04
#> [46] 4.606171e-04 3.981830e-04 3.872501e-04 3.770568e-04 3.654321e-04
#> [51] 3.335684e-04 3.201190e-04 3.073288e-04 2.928482e-04 2.746345e-04
#> [56] 2.452473e-04 2.387138e-04 2.317648e-04 2.135993e-04 1.997357e-04
#> [61] 1.951521e-04 1.819381e-04 1.702468e-04 1.628102e-04 1.581201e-04
#> [66] 1.385348e-04 1.344372e-04 1.312610e-04 1.253673e-04 1.151346e-04
#> [71] 1.137692e-04 1.046996e-04 9.940895e-05 9.308218e-05 9.029491e-05
#> [76] 8.602521e-05 8.473184e-05 8.095727e-05 7.430997e-05 7.002361e-05
#> [81] 6.895098e-05 5.997905e-05 5.797630e-05 5.525536e-05 5.369477e-05
#> [86] 5.079501e-05 4.796040e-05 4.599567e-05 4.354619e-05 4.217066e-05
#> [91] 4.029730e-05 3.955824e-05 3.556866e-05 3.459097e-05 3.375233e-05
#> [96] 3.219839e-05 3.091766e-05 2.968018e-05 2.835209e-05 2.651873e-05
#> [101] 2.481757e-05 2.456116e-05 2.192151e-05 2.122607e-05 2.000946e-05
#> [106] 1.944903e-05 1.851434e-05 1.754881e-05 1.650787e-05 1.632736e-05
#> [111] 1.512632e-05 1.430343e-05 1.309713e-05 1.304663e-05 1.267424e-05
#> [116] 1.230003e-05 1.191153e-05 1.081167e-05 1.051888e-05 1.009886e-05
#> [121] 9.784421e-06 9.446237e-06 9.119859e-06 8.555828e-06 8.307256e-06
#> [126] 7.999610e-06 7.523006e-06 7.378027e-06 7.147094e-06 6.919628e-06
#> [131] 6.505670e-06 6.297241e-06 5.884740e-06 5.732735e-06 5.654350e-06
#> [136] 5.477679e-06 5.062538e-06 4.889882e-06 4.726518e-06 4.486673e-06
#> [141] 4.199925e-06 4.124125e-06 3.970441e-06 3.755511e-06 3.530481e-06
#> [146] 3.375259e-06 3.274698e-06 3.115624e-06 3.107262e-06 2.944510e-06
#> [151] 2.914535e-06 2.842486e-06 2.746895e-06 2.669125e-06 2.607098e-06
#> [156] 2.431512e-06 2.295001e-06 2.263556e-06 2.189459e-06 1.984165e-06
#> [161] 1.945662e-06 1.934820e-06 1.844129e-06 1.807692e-06 1.754454e-06
#> [166] 1.723385e-06 1.691814e-06 1.665232e-06 1.592284e-06 1.551705e-06
#> [171] 1.503367e-06 1.475808e-06 1.433096e-06 1.402096e-06 1.395393e-06
#> [176] 1.372546e-06 1.333633e-06 1.306296e-06 1.287466e-06 1.249523e-06
#> [181] 1.237971e-06 1.212792e-06 1.197709e-06 1.191649e-06 1.162927e-06
#> [186] 1.150653e-06 1.134476e-06 1.111107e-06 1.105313e-06 1.099739e-06
#> [191] 1.081828e-06 1.070663e-06 1.060832e-06 1.054051e-06 1.047258e-06
#> [196] 1.040328e-06 1.037812e-06 1.026300e-06 1.022764e-06 1.018354e-06
#>
#> $gamma.U
#> [1] 1.639823
#>
#> $gamma.Theta
#> [1] 0.1711223
## test under an alternative distribution
X=matrix(rchisq(d*n,2),nrow=n,ncol=d)
EllKEPT(X,kerU="Gaussian",kerTheta="Gaussian")
#> $stat
#> [1] 6.752065
#>
#> $pval
#> [1] 2.090237e-08
#>
#> $lambda
#> [1] 1.986481e-01 6.305016e-02 3.838634e-02 3.137967e-02 2.171311e-02
#> [6] 1.993660e-02 1.815107e-02 1.431834e-02 1.225698e-02 1.196938e-02
#> [11] 9.407515e-03 8.142329e-03 6.738778e-03 5.901827e-03 4.933235e-03
#> [16] 4.255830e-03 4.046241e-03 3.094990e-03 3.033703e-03 2.378113e-03
#> [21] 2.187035e-03 1.955007e-03 1.888815e-03 1.612870e-03 1.447161e-03
#> [26] 1.183244e-03 1.122242e-03 1.077942e-03 9.163241e-04 8.180345e-04
#> [31] 6.652330e-04 5.849722e-04 5.371874e-04 5.054401e-04 4.133024e-04
#> [36] 3.585258e-04 3.508391e-04 3.293313e-04 3.182007e-04 2.651407e-04
#> [41] 2.475293e-04 2.170687e-04 2.066795e-04 1.770060e-04 1.622819e-04
#> [46] 1.398318e-04 1.245260e-04 1.120208e-04 1.070121e-04 9.978310e-05
#> [51] 8.942821e-05 8.105804e-05 7.570996e-05 6.851905e-05 5.867991e-05
#> [56] 5.649918e-05 4.846919e-05 4.275588e-05 4.082093e-05 3.699943e-05
#> [61] 3.587698e-05 3.194061e-05 2.880280e-05 2.554064e-05 2.294628e-05
#> [66] 2.112798e-05 2.079770e-05 1.742206e-05 1.635621e-05 1.481902e-05
#> [71] 1.450897e-05 1.291135e-05 1.156915e-05 1.025757e-05 9.755990e-06
#> [76] 9.076175e-06 7.955211e-06 7.489201e-06 7.106387e-06 6.583437e-06
#> [81] 6.271078e-06 6.091655e-06 5.619293e-06 4.794746e-06 4.440952e-06
#> [86] 4.291134e-06 4.047241e-06 3.956218e-06 3.527688e-06 3.318637e-06
#> [91] 3.131008e-06 2.993648e-06 2.855329e-06 2.611661e-06 2.523954e-06
#> [96] 2.366527e-06 2.284466e-06 2.143272e-06 1.996843e-06 1.985993e-06
#> [101] 1.900831e-06 1.806238e-06 1.683718e-06 1.677376e-06 1.601065e-06
#> [106] 1.524649e-06 1.483150e-06 1.461147e-06 1.418083e-06 1.386128e-06
#> [111] 1.334231e-06 1.282992e-06 1.267332e-06 1.234027e-06 1.219317e-06
#> [116] 1.202001e-06 1.196340e-06 1.180536e-06 1.146674e-06 1.140252e-06
#> [121] 1.128661e-06 1.117254e-06 1.100898e-06 1.094297e-06 1.085539e-06
#> [126] 1.076237e-06 1.069811e-06 1.058975e-06 1.052867e-06 1.051144e-06
#> [131] 1.044906e-06 1.042514e-06 1.037613e-06 1.035173e-06 1.031457e-06
#> [136] 1.030782e-06 1.027411e-06 1.024958e-06 1.023281e-06 1.021269e-06
#> [141] 1.020052e-06 1.018639e-06 1.018105e-06 1.016529e-06 1.014584e-06
#> [146] 1.013913e-06 1.012920e-06 1.012665e-06 1.012423e-06 1.011832e-06
#> [151] 1.011062e-06 1.010682e-06 1.010116e-06 1.009929e-06 1.009775e-06
#> [156] 1.009609e-06 1.009408e-06 1.009254e-06 1.009083e-06 1.008959e-06
#> [161] 1.008844e-06 1.008738e-06 1.008594e-06 1.008505e-06 1.008149e-06
#> [166] 1.008113e-06 1.007838e-06 1.007699e-06 1.007672e-06 1.007618e-06
#> [171] 1.007523e-06 1.007511e-06 1.007430e-06 1.007386e-06 1.007378e-06
#> [176] 1.007339e-06 1.007309e-06 1.007296e-06 1.007242e-06 1.007226e-06
#> [181] 1.007139e-06 1.007090e-06 1.007081e-06 1.007056e-06 1.007015e-06
#> [186] 1.007004e-06 1.006983e-06 1.006942e-06 1.006932e-06 1.006907e-06
#> [191] 1.006893e-06 1.006859e-06 1.006837e-06 1.006811e-06 1.006779e-06
#> [196] 1.006727e-06 1.006699e-06 1.006615e-06 1.006609e-06 1.006590e-06
#>
#> $gamma.U
#> [1] 1.363939
#>
#> $gamma.Theta
#> [1] 0.1753525
EllKEPT(X,kerU="PIQ",kerTheta="PIQ")
#> $stat
#> [1] 5.248494
#>
#> $pval
#> [1] 7.965881e-08
#>
#> $lambda
#> [1] 1.681253e-01 4.276903e-02 2.799635e-02 2.315884e-02 1.657691e-02
#> [6] 1.496849e-02 1.337115e-02 1.083961e-02 9.502867e-03 8.581909e-03
#> [11] 7.502175e-03 6.793257e-03 6.122526e-03 4.925315e-03 4.293040e-03
#> [16] 3.755138e-03 3.414600e-03 2.703612e-03 2.561935e-03 2.223502e-03
#> [21] 2.129022e-03 2.062154e-03 1.758018e-03 1.664715e-03 1.593518e-03
#> [26] 1.389357e-03 1.258687e-03 1.163176e-03 1.023969e-03 9.286954e-04
#> [31] 8.426801e-04 7.763067e-04 7.451352e-04 7.015969e-04 5.870326e-04
#> [36] 5.525524e-04 5.434055e-04 4.920030e-04 4.593294e-04 4.265727e-04
#> [41] 4.001678e-04 3.930602e-04 3.537891e-04 3.281991e-04 2.940195e-04
#> [46] 2.595685e-04 2.492552e-04 2.326787e-04 2.111550e-04 1.991546e-04
#> [51] 1.950637e-04 1.798922e-04 1.726208e-04 1.602102e-04 1.480775e-04
#> [56] 1.388535e-04 1.281713e-04 1.161962e-04 1.089949e-04 1.049055e-04
#> [61] 1.003987e-04 9.141801e-05 8.760341e-05 8.295283e-05 7.760425e-05
#> [66] 6.821927e-05 6.325293e-05 6.162777e-05 6.080784e-05 5.389360e-05
#> [71] 5.043313e-05 4.889279e-05 4.271890e-05 4.142056e-05 3.825057e-05
#> [76] 3.608596e-05 3.416505e-05 3.354918e-05 3.197996e-05 2.965817e-05
#> [81] 2.938730e-05 2.526080e-05 2.516142e-05 2.269271e-05 2.198247e-05
#> [86] 2.094556e-05 1.885433e-05 1.777613e-05 1.693943e-05 1.604756e-05
#> [91] 1.449395e-05 1.439053e-05 1.343011e-05 1.290670e-05 1.258476e-05
#> [96] 1.185632e-05 1.072034e-05 1.033490e-05 9.932452e-06 9.216460e-06
#> [101] 9.016065e-06 8.686556e-06 8.227631e-06 7.996624e-06 7.546665e-06
#> [106] 7.133778e-06 6.929309e-06 6.407289e-06 5.994624e-06 5.425438e-06
#> [111] 5.222466e-06 5.105290e-06 4.985375e-06 4.880015e-06 4.256927e-06
#> [116] 3.950593e-06 3.875374e-06 3.769658e-06 3.538326e-06 3.348176e-06
#> [121] 3.092529e-06 2.987990e-06 2.898348e-06 2.717778e-06 2.529647e-06
#> [126] 2.493510e-06 2.411603e-06 2.359400e-06 2.234577e-06 2.195952e-06
#> [131] 2.033490e-06 1.976666e-06 1.921278e-06 1.864945e-06 1.854714e-06
#> [136] 1.769864e-06 1.731915e-06 1.660096e-06 1.601396e-06 1.567613e-06
#> [141] 1.528728e-06 1.482562e-06 1.457725e-06 1.417546e-06 1.395499e-06
#> [146] 1.379075e-06 1.353130e-06 1.287149e-06 1.270796e-06 1.247312e-06
#> [151] 1.205882e-06 1.196261e-06 1.192655e-06 1.166774e-06 1.149481e-06
#> [156] 1.144411e-06 1.123045e-06 1.120619e-06 1.114242e-06 1.102358e-06
#> [161] 1.080686e-06 1.071963e-06 1.064998e-06 1.060520e-06 1.052757e-06
#> [166] 1.050728e-06 1.049227e-06 1.043849e-06 1.040282e-06 1.036696e-06
#> [171] 1.035110e-06 1.033298e-06 1.028135e-06 1.024485e-06 1.023305e-06
#> [176] 1.017348e-06 1.015528e-06 1.015177e-06 1.014369e-06 1.013026e-06
#> [181] 1.012812e-06 1.012344e-06 1.011337e-06 1.010474e-06 1.010125e-06
#> [186] 1.009867e-06 1.009339e-06 1.008681e-06 1.008542e-06 1.008140e-06
#> [191] 1.007562e-06 1.007312e-06 1.007223e-06 1.006995e-06 1.006916e-06
#> [196] 1.006825e-06 1.006712e-06 1.006662e-06 1.006587e-06 1.006535e-06
#>
#> $gamma.U
#> [1] 1.363939
#>
#> $gamma.Theta
#> [1] 0.1753525