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] 0.3764998
#>
#> $pval
#> [1] 0.680362
#>
#> $lambda
#> [1] 1.819678e-01 8.859639e-02 4.311198e-02 3.573950e-02 2.496945e-02
#> [6] 2.092285e-02 2.015604e-02 1.584026e-02 1.345295e-02 1.148667e-02
#> [11] 9.925853e-03 7.997193e-03 6.641176e-03 6.078209e-03 5.481532e-03
#> [16] 5.088321e-03 4.646323e-03 4.298722e-03 3.890892e-03 3.534692e-03
#> [21] 3.066614e-03 2.896420e-03 2.597612e-03 2.298584e-03 2.120927e-03
#> [26] 1.805220e-03 1.774317e-03 1.629733e-03 1.430353e-03 1.369936e-03
#> [31] 1.125288e-03 1.051021e-03 9.011795e-04 8.670821e-04 8.321789e-04
#> [36] 7.539847e-04 7.083105e-04 6.776681e-04 5.760635e-04 5.316387e-04
#> [41] 5.099645e-04 4.458813e-04 3.862310e-04 3.592264e-04 3.571517e-04
#> [46] 3.254164e-04 2.980700e-04 2.754665e-04 2.625843e-04 2.563173e-04
#> [51] 2.338052e-04 2.128504e-04 1.969418e-04 1.698421e-04 1.670343e-04
#> [56] 1.528940e-04 1.406219e-04 1.297983e-04 1.227427e-04 1.199654e-04
#> [61] 1.014496e-04 1.002983e-04 9.136821e-05 8.097468e-05 7.572040e-05
#> [66] 7.236464e-05 6.854316e-05 6.273478e-05 6.031530e-05 5.384295e-05
#> [71] 5.052860e-05 4.827937e-05 4.164258e-05 3.861283e-05 3.752123e-05
#> [76] 3.343350e-05 3.167658e-05 3.131875e-05 2.803571e-05 2.698283e-05
#> [81] 2.621932e-05 2.298439e-05 2.114936e-05 2.042926e-05 1.840765e-05
#> [86] 1.678688e-05 1.627989e-05 1.587713e-05 1.476144e-05 1.355310e-05
#> [91] 1.243200e-05 1.151029e-05 1.099520e-05 1.007340e-05 9.504988e-06
#> [96] 9.327902e-06 8.022083e-06 7.333971e-06 7.124983e-06 6.497922e-06
#> [101] 6.070890e-06 5.893379e-06 5.525891e-06 4.878416e-06 4.723788e-06
#> [106] 4.184999e-06 4.042652e-06 3.837088e-06 3.802513e-06 3.481854e-06
#> [111] 3.366262e-06 3.161514e-06 2.989860e-06 2.842040e-06 2.633505e-06
#> [116] 2.540966e-06 2.353524e-06 2.318986e-06 2.274100e-06 2.074689e-06
#> [121] 2.055366e-06 1.951917e-06 1.853630e-06 1.806746e-06 1.717862e-06
#> [126] 1.637110e-06 1.599115e-06 1.563407e-06 1.509402e-06 1.468904e-06
#> [131] 1.428510e-06 1.387058e-06 1.350403e-06 1.339686e-06 1.331911e-06
#> [136] 1.314591e-06 1.247235e-06 1.244579e-06 1.224819e-06 1.214516e-06
#> [141] 1.198240e-06 1.175656e-06 1.148695e-06 1.147188e-06 1.135552e-06
#> [146] 1.121828e-06 1.111691e-06 1.097105e-06 1.089403e-06 1.083147e-06
#> [151] 1.076465e-06 1.072229e-06 1.068591e-06 1.059781e-06 1.051491e-06
#> [156] 1.048380e-06 1.046975e-06 1.043512e-06 1.041819e-06 1.037596e-06
#> [161] 1.033220e-06 1.031394e-06 1.025980e-06 1.025493e-06 1.022754e-06
#> [166] 1.021883e-06 1.020563e-06 1.018547e-06 1.018305e-06 1.016948e-06
#> [171] 1.016414e-06 1.015694e-06 1.014555e-06 1.013278e-06 1.012625e-06
#> [176] 1.012510e-06 1.011141e-06 1.010818e-06 1.010127e-06 1.009900e-06
#> [181] 1.009348e-06 1.008823e-06 1.008684e-06 1.008561e-06 1.008350e-06
#> [186] 1.008188e-06 1.008121e-06 1.008003e-06 1.007906e-06 1.007604e-06
#> [191] 1.007460e-06 1.007436e-06 1.007370e-06 1.007293e-06 1.007172e-06
#> [196] 1.007004e-06 1.006877e-06 1.006809e-06 1.006656e-06 1.006570e-06
#>
#> $gamma.U
#> [1] 1.647239
#>
#> $gamma.Theta
#> [1] 0.1779187
EllKEPT(X,kerU="PIQ",kerTheta="PIQ")
#> $stat
#> [1] 0.2739666
#>
#> $pval
#> [1] 0.7741983
#>
#> $lambda
#> [1] 1.692583e-01 7.210401e-02 2.875603e-02 2.449653e-02 2.046420e-02
#> [6] 1.653361e-02 1.240950e-02 1.177546e-02 9.196839e-03 8.036295e-03
#> [11] 7.316418e-03 6.271949e-03 5.392579e-03 4.875141e-03 4.598875e-03
#> [16] 4.377681e-03 4.251153e-03 3.758009e-03 3.449927e-03 3.316255e-03
#> [21] 3.262372e-03 2.793120e-03 2.545212e-03 2.295188e-03 2.013642e-03
#> [26] 1.840438e-03 1.731809e-03 1.567588e-03 1.447064e-03 1.398426e-03
#> [31] 1.328827e-03 1.207243e-03 1.157847e-03 1.062160e-03 9.920254e-04
#> [36] 9.842446e-04 8.729641e-04 8.569099e-04 7.944976e-04 7.659054e-04
#> [41] 7.422224e-04 6.692730e-04 6.640449e-04 5.948304e-04 5.753479e-04
#> [46] 5.560703e-04 5.208364e-04 4.974141e-04 4.480182e-04 4.228881e-04
#> [51] 4.055076e-04 3.855558e-04 3.669754e-04 3.495040e-04 3.361526e-04
#> [56] 2.994801e-04 2.922629e-04 2.747843e-04 2.617460e-04 2.443880e-04
#> [61] 2.423603e-04 2.224680e-04 2.162257e-04 2.049304e-04 1.880803e-04
#> [66] 1.813330e-04 1.760745e-04 1.647155e-04 1.628233e-04 1.540425e-04
#> [71] 1.381904e-04 1.311431e-04 1.275146e-04 1.190629e-04 1.171990e-04
#> [76] 1.144218e-04 1.072514e-04 1.035634e-04 9.585063e-05 9.519066e-05
#> [81] 8.800650e-05 8.588029e-05 8.230914e-05 7.869944e-05 7.371268e-05
#> [86] 7.040539e-05 6.649088e-05 6.406785e-05 5.974636e-05 5.588585e-05
#> [91] 5.432946e-05 4.968628e-05 4.737612e-05 4.650206e-05 4.409302e-05
#> [96] 4.159529e-05 3.821161e-05 3.675693e-05 3.475765e-05 3.412642e-05
#> [101] 3.327703e-05 3.163866e-05 3.128810e-05 2.871197e-05 2.764771e-05
#> [106] 2.546442e-05 2.475085e-05 2.345919e-05 2.272484e-05 2.197757e-05
#> [111] 2.134873e-05 2.078774e-05 2.048082e-05 1.856412e-05 1.716656e-05
#> [116] 1.654508e-05 1.588934e-05 1.517091e-05 1.448585e-05 1.378776e-05
#> [121] 1.254964e-05 1.207902e-05 1.158289e-05 1.146400e-05 1.130087e-05
#> [126] 1.039271e-05 1.003035e-05 9.003831e-06 8.821090e-06 8.594679e-06
#> [131] 7.919648e-06 7.608858e-06 7.221278e-06 6.735347e-06 6.416098e-06
#> [136] 6.123271e-06 5.996181e-06 5.749146e-06 5.631612e-06 5.264578e-06
#> [141] 5.152943e-06 4.761763e-06 4.541517e-06 4.445937e-06 4.356193e-06
#> [146] 4.167659e-06 4.058523e-06 3.917721e-06 3.750446e-06 3.454469e-06
#> [151] 3.227278e-06 3.120220e-06 3.027287e-06 2.961300e-06 2.870249e-06
#> [156] 2.729545e-06 2.592736e-06 2.512555e-06 2.479633e-06 2.381087e-06
#> [161] 2.287081e-06 2.251077e-06 2.043508e-06 2.016440e-06 1.938797e-06
#> [166] 1.873516e-06 1.768061e-06 1.665270e-06 1.633410e-06 1.585826e-06
#> [171] 1.560544e-06 1.504851e-06 1.492560e-06 1.467257e-06 1.410087e-06
#> [176] 1.394581e-06 1.382281e-06 1.330084e-06 1.275907e-06 1.253344e-06
#> [181] 1.240057e-06 1.213440e-06 1.196957e-06 1.193291e-06 1.165892e-06
#> [186] 1.158715e-06 1.145436e-06 1.125872e-06 1.118762e-06 1.102773e-06
#> [191] 1.098264e-06 1.078841e-06 1.075271e-06 1.070437e-06 1.060145e-06
#> [196] 1.052644e-06 1.039910e-06 1.032961e-06 1.028541e-06 1.020732e-06
#>
#> $gamma.U
#> [1] 1.647239
#>
#> $gamma.Theta
#> [1] 0.1779187
## 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] 7.706394
#>
#> $pval
#> [1] 0
#>
#> $lambda
#> [1] 2.172896e-01 4.327907e-02 3.737429e-02 2.697266e-02 2.010041e-02
#> [6] 1.859594e-02 1.650557e-02 1.464813e-02 1.335625e-02 9.281665e-03
#> [11] 8.486144e-03 7.605659e-03 6.768523e-03 5.641675e-03 5.051870e-03
#> [16] 4.375521e-03 4.230136e-03 3.568814e-03 3.346071e-03 3.212226e-03
#> [21] 2.471269e-03 2.362615e-03 1.907496e-03 1.795785e-03 1.629322e-03
#> [26] 1.438447e-03 1.308893e-03 1.250765e-03 1.050098e-03 1.011473e-03
#> [31] 8.501476e-04 7.650124e-04 6.947302e-04 6.582059e-04 5.713462e-04
#> [36] 5.443006e-04 4.830611e-04 4.182759e-04 3.825068e-04 3.247626e-04
#> [41] 3.024651e-04 2.901548e-04 2.583408e-04 2.429902e-04 2.220493e-04
#> [46] 2.085067e-04 1.787841e-04 1.599485e-04 1.486446e-04 1.317417e-04
#> [51] 1.243676e-04 1.065379e-04 9.508227e-05 8.737639e-05 7.678316e-05
#> [56] 6.809033e-05 6.535631e-05 5.650594e-05 5.497675e-05 4.709536e-05
#> [61] 4.205226e-05 3.791184e-05 3.531072e-05 3.161782e-05 2.892730e-05
#> [66] 2.567464e-05 2.380599e-05 2.290636e-05 2.058238e-05 1.788982e-05
#> [71] 1.715542e-05 1.480667e-05 1.433681e-05 1.383789e-05 1.320902e-05
#> [76] 1.041941e-05 9.698043e-06 9.000028e-06 7.668401e-06 7.161355e-06
#> [81] 6.465927e-06 6.022729e-06 5.853875e-06 5.196830e-06 4.975589e-06
#> [86] 4.699145e-06 4.391192e-06 4.179254e-06 3.318934e-06 3.213359e-06
#> [91] 3.117403e-06 2.934936e-06 2.839912e-06 2.602962e-06 2.329896e-06
#> [96] 2.160692e-06 2.094482e-06 1.962286e-06 1.892245e-06 1.757580e-06
#> [101] 1.704236e-06 1.645955e-06 1.542281e-06 1.525692e-06 1.474027e-06
#> [106] 1.389727e-06 1.383770e-06 1.353560e-06 1.334608e-06 1.298677e-06
#> [111] 1.281683e-06 1.243691e-06 1.238373e-06 1.198160e-06 1.173856e-06
#> [116] 1.138868e-06 1.117449e-06 1.110570e-06 1.097931e-06 1.091000e-06
#> [121] 1.083604e-06 1.072821e-06 1.066636e-06 1.063063e-06 1.055921e-06
#> [126] 1.053410e-06 1.051306e-06 1.037880e-06 1.033789e-06 1.032893e-06
#> [131] 1.032704e-06 1.027136e-06 1.026420e-06 1.025557e-06 1.021633e-06
#> [136] 1.019201e-06 1.017599e-06 1.016415e-06 1.015476e-06 1.014779e-06
#> [141] 1.014437e-06 1.013077e-06 1.012287e-06 1.011999e-06 1.011640e-06
#> [146] 1.011130e-06 1.010670e-06 1.010245e-06 1.010120e-06 1.009956e-06
#> [151] 1.009698e-06 1.009357e-06 1.009236e-06 1.009139e-06 1.008988e-06
#> [156] 1.008965e-06 1.008712e-06 1.008511e-06 1.008306e-06 1.008267e-06
#> [161] 1.008103e-06 1.008013e-06 1.007944e-06 1.007876e-06 1.007855e-06
#> [166] 1.007701e-06 1.007663e-06 1.007639e-06 1.007572e-06 1.007551e-06
#> [171] 1.007477e-06 1.007423e-06 1.007391e-06 1.007325e-06 1.007263e-06
#> [176] 1.007234e-06 1.007209e-06 1.007194e-06 1.007128e-06 1.007090e-06
#> [181] 1.007056e-06 1.007049e-06 1.007045e-06 1.007032e-06 1.006968e-06
#> [186] 1.006944e-06 1.006925e-06 1.006915e-06 1.006897e-06 1.006883e-06
#> [191] 1.006847e-06 1.006818e-06 1.006813e-06 1.006741e-06 1.006721e-06
#> [196] 1.006690e-06 1.006639e-06 1.006613e-06 1.006545e-06 1.006515e-06
#>
#> $gamma.U
#> [1] 1.415863
#>
#> $gamma.Theta
#> [1] 0.1753465
EllKEPT(X,kerU="PIQ",kerTheta="PIQ")
#> Warning in imhof(test.stat.hs, lambda.hs): Note that Qq + abserr is positive.
#> $stat
#> [1] 6.086574
#>
#> $pval
#> [1] 0
#>
#> $lambda
#> [1] 1.779124e-01 3.111117e-02 2.708441e-02 1.966797e-02 1.612432e-02
#> [6] 1.543329e-02 1.322711e-02 1.225666e-02 9.874790e-03 8.840102e-03
#> [11] 8.290380e-03 6.123218e-03 5.309160e-03 4.752175e-03 4.374944e-03
#> [16] 3.833225e-03 3.118932e-03 3.028536e-03 2.888989e-03 2.518892e-03
#> [21] 2.259866e-03 2.179212e-03 2.012727e-03 1.887515e-03 1.583134e-03
#> [26] 1.464775e-03 1.357590e-03 1.266645e-03 1.205061e-03 1.162694e-03
#> [31] 9.253487e-04 8.764667e-04 7.856836e-04 7.567351e-04 7.135993e-04
#> [36] 6.847097e-04 6.299896e-04 5.608097e-04 5.364245e-04 5.148483e-04
#> [41] 4.655604e-04 4.166789e-04 4.126706e-04 3.935354e-04 3.659803e-04
#> [46] 3.373145e-04 3.260193e-04 2.976133e-04 2.809211e-04 2.472218e-04
#> [51] 2.347031e-04 2.061474e-04 1.897781e-04 1.784320e-04 1.665190e-04
#> [56] 1.619107e-04 1.486394e-04 1.425652e-04 1.292381e-04 1.185360e-04
#> [61] 1.139727e-04 1.044086e-04 1.027287e-04 9.482360e-05 8.764063e-05
#> [66] 8.339841e-05 7.573833e-05 7.038392e-05 6.146650e-05 6.068425e-05
#> [71] 5.849790e-05 5.531343e-05 4.883114e-05 4.557217e-05 4.065341e-05
#> [76] 3.846266e-05 3.716876e-05 3.513266e-05 3.339416e-05 3.180008e-05
#> [81] 3.014372e-05 2.806879e-05 2.524860e-05 2.358914e-05 2.321499e-05
#> [86] 2.193973e-05 2.112365e-05 1.902126e-05 1.849171e-05 1.653576e-05
#> [91] 1.568648e-05 1.524953e-05 1.442947e-05 1.333984e-05 1.314179e-05
#> [96] 1.217186e-05 1.156505e-05 1.038404e-05 1.014050e-05 9.340428e-06
#> [101] 8.072172e-06 7.959245e-06 7.746198e-06 6.718116e-06 6.214019e-06
#> [106] 5.866003e-06 5.645622e-06 5.352389e-06 5.113291e-06 5.004526e-06
#> [111] 4.674688e-06 4.238219e-06 4.132783e-06 3.818068e-06 3.557514e-06
#> [116] 3.307465e-06 3.159942e-06 2.997813e-06 2.960766e-06 2.786128e-06
#> [121] 2.690231e-06 2.520020e-06 2.345984e-06 2.238862e-06 2.211387e-06
#> [126] 2.054573e-06 2.011219e-06 1.930259e-06 1.913868e-06 1.797350e-06
#> [131] 1.758678e-06 1.654816e-06 1.607491e-06 1.563644e-06 1.530689e-06
#> [136] 1.485358e-06 1.423558e-06 1.388155e-06 1.365946e-06 1.347973e-06
#> [141] 1.331459e-06 1.307643e-06 1.276401e-06 1.238250e-06 1.210094e-06
#> [146] 1.178391e-06 1.168067e-06 1.156970e-06 1.131615e-06 1.128921e-06
#> [151] 1.120484e-06 1.114604e-06 1.097919e-06 1.090072e-06 1.084973e-06
#> [156] 1.076612e-06 1.066083e-06 1.060805e-06 1.058784e-06 1.056219e-06
#> [161] 1.049380e-06 1.048913e-06 1.045172e-06 1.040307e-06 1.036231e-06
#> [166] 1.035054e-06 1.031063e-06 1.029610e-06 1.025841e-06 1.025545e-06
#> [171] 1.023425e-06 1.020959e-06 1.019014e-06 1.018337e-06 1.017028e-06
#> [176] 1.014200e-06 1.013690e-06 1.013648e-06 1.012735e-06 1.012564e-06
#> [181] 1.010606e-06 1.010013e-06 1.009876e-06 1.008792e-06 1.008546e-06
#> [186] 1.008258e-06 1.007962e-06 1.007609e-06 1.007578e-06 1.007510e-06
#> [191] 1.007418e-06 1.007197e-06 1.007171e-06 1.007024e-06 1.006869e-06
#> [196] 1.006725e-06 1.006693e-06 1.006678e-06 1.006595e-06 1.006428e-06
#>
#> $gamma.U
#> [1] 1.415863
#>
#> $gamma.Theta
#> [1] 0.1753465