KEPTED

library(KEPTED)

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] 0.1678986
#> 
#> $pval
#> [1] 0.9744247
#> 
#> $lambda
#>   [1] 1.083185e-01 5.390656e-02 4.056266e-02 2.615402e-02 2.397304e-02
#>   [6] 1.678542e-02 1.552961e-02 1.304413e-02 1.230727e-02 1.136987e-02
#>  [11] 9.971014e-03 8.543179e-03 7.361673e-03 6.436971e-03 5.517840e-03
#>  [16] 5.082263e-03 4.624382e-03 4.052220e-03 3.739287e-03 3.224162e-03
#>  [21] 3.089185e-03 2.492850e-03 2.414644e-03 2.347269e-03 1.997503e-03
#>  [26] 1.846115e-03 1.568011e-03 1.510024e-03 1.278213e-03 1.182650e-03
#>  [31] 1.111139e-03 1.056183e-03 9.535575e-04 9.178953e-04 8.495422e-04
#>  [36] 7.729220e-04 6.636268e-04 6.217777e-04 5.605341e-04 5.322503e-04
#>  [41] 4.876541e-04 4.729021e-04 4.455285e-04 3.700578e-04 3.656361e-04
#>  [46] 3.495837e-04 3.108027e-04 2.585114e-04 2.514386e-04 2.466301e-04
#>  [51] 2.345982e-04 2.113775e-04 1.874199e-04 1.755225e-04 1.686672e-04
#>  [56] 1.512902e-04 1.412815e-04 1.213095e-04 1.175182e-04 1.086449e-04
#>  [61] 1.017939e-04 9.375189e-05 8.946093e-05 8.447743e-05 7.674666e-05
#>  [66] 7.334655e-05 6.281512e-05 6.123337e-05 5.947872e-05 5.290072e-05
#>  [71] 4.857779e-05 4.457709e-05 4.301431e-05 3.988720e-05 3.816786e-05
#>  [76] 3.614950e-05 3.301569e-05 2.961962e-05 2.854101e-05 2.590896e-05
#>  [81] 2.440504e-05 2.355928e-05 2.180757e-05 1.947096e-05 1.884867e-05
#>  [86] 1.698426e-05 1.613908e-05 1.543548e-05 1.456427e-05 1.297578e-05
#>  [91] 1.179732e-05 1.145114e-05 1.074038e-05 1.000424e-05 9.499516e-06
#>  [96] 8.863528e-06 8.453200e-06 8.321884e-06 7.506667e-06 7.033474e-06
#> [101] 6.730976e-06 6.077607e-06 5.898691e-06 5.292334e-06 5.067795e-06
#> [106] 4.962676e-06 4.405522e-06 4.078090e-06 3.836229e-06 3.634557e-06
#> [111] 3.500816e-06 3.428489e-06 3.174166e-06 2.914515e-06 2.779692e-06
#> [116] 2.554305e-06 2.493417e-06 2.388584e-06 2.304432e-06 2.231046e-06
#> [121] 2.159650e-06 2.072940e-06 1.927626e-06 1.874414e-06 1.799337e-06
#> [126] 1.694284e-06 1.654916e-06 1.635643e-06 1.595952e-06 1.515097e-06
#> [131] 1.474398e-06 1.431700e-06 1.410427e-06 1.363227e-06 1.347514e-06
#> [136] 1.320238e-06 1.288477e-06 1.274617e-06 1.253416e-06 1.232910e-06
#> [141] 1.208701e-06 1.201496e-06 1.184967e-06 1.178474e-06 1.161600e-06
#> [146] 1.142822e-06 1.141907e-06 1.122929e-06 1.109556e-06 1.105262e-06
#> [151] 1.094491e-06 1.087727e-06 1.078455e-06 1.072738e-06 1.066912e-06
#> [156] 1.058566e-06 1.054685e-06 1.046457e-06 1.043527e-06 1.040463e-06
#> [161] 1.035262e-06 1.034214e-06 1.032109e-06 1.029122e-06 1.027364e-06
#> [166] 1.023588e-06 1.021806e-06 1.020432e-06 1.019973e-06 1.019335e-06
#> [171] 1.018087e-06 1.015526e-06 1.015053e-06 1.014556e-06 1.013650e-06
#> [176] 1.012953e-06 1.012124e-06 1.011507e-06 1.011095e-06 1.010530e-06
#> [181] 1.009931e-06 1.009451e-06 1.009239e-06 1.008895e-06 1.008638e-06
#> [186] 1.008483e-06 1.008103e-06 1.008065e-06 1.007947e-06 1.007737e-06
#> [191] 1.007456e-06 1.007426e-06 1.007254e-06 1.007155e-06 1.007036e-06
#> [196] 1.006867e-06 1.006767e-06 1.006712e-06 1.006640e-06 1.006563e-06
#> 
#> $gamma.U
#> [1] 1.848109
#> 
#> $gamma.Theta
#> [1] 0.1778428
EllKEPT(X,kerU="PIQ",kerTheta="PIQ")
#> $stat
#> [1] 0.1503235
#> 
#> $pval
#> [1] 0.9650482
#> 
#> $lambda
#>   [1] 9.484795e-02 3.561610e-02 2.669564e-02 1.950793e-02 1.790316e-02
#>   [6] 1.131528e-02 1.055895e-02 9.870617e-03 9.207948e-03 8.684482e-03
#>  [11] 7.565306e-03 6.078789e-03 5.714858e-03 5.247969e-03 4.736065e-03
#>  [16] 4.218432e-03 4.105399e-03 3.632535e-03 3.289499e-03 3.021455e-03
#>  [21] 2.747719e-03 2.641269e-03 2.346379e-03 2.234228e-03 1.925483e-03
#>  [26] 1.833362e-03 1.751724e-03 1.494597e-03 1.315436e-03 1.273748e-03
#>  [31] 1.264748e-03 1.171063e-03 1.110554e-03 1.041019e-03 1.009482e-03
#>  [36] 9.696549e-04 8.866083e-04 8.345721e-04 7.793604e-04 7.497690e-04
#>  [41] 6.794692e-04 6.475261e-04 6.462045e-04 5.751280e-04 5.410628e-04
#>  [46] 5.215706e-04 4.658643e-04 4.593570e-04 4.425821e-04 4.253853e-04
#>  [51] 4.024009e-04 3.876845e-04 3.660065e-04 3.444262e-04 3.214553e-04
#>  [56] 3.105142e-04 2.958324e-04 2.729900e-04 2.625119e-04 2.461519e-04
#>  [61] 2.404277e-04 2.212422e-04 2.058213e-04 1.911617e-04 1.862934e-04
#>  [66] 1.839827e-04 1.782051e-04 1.634863e-04 1.585317e-04 1.443885e-04
#>  [71] 1.410117e-04 1.315955e-04 1.289836e-04 1.248029e-04 1.183916e-04
#>  [76] 1.123950e-04 1.073440e-04 1.040440e-04 1.003870e-04 9.055464e-05
#>  [81] 8.819229e-05 8.150879e-05 7.910840e-05 7.706104e-05 6.937992e-05
#>  [86] 6.721267e-05 6.436965e-05 5.901716e-05 5.788015e-05 5.236940e-05
#>  [91] 5.021582e-05 4.932436e-05 4.760109e-05 4.605967e-05 4.492889e-05
#>  [96] 4.149111e-05 3.903026e-05 3.767900e-05 3.664043e-05 3.421398e-05
#> [101] 3.279307e-05 3.117957e-05 2.914729e-05 2.884277e-05 2.778149e-05
#> [106] 2.728872e-05 2.630226e-05 2.545566e-05 2.398518e-05 2.249073e-05
#> [111] 2.147819e-05 2.121872e-05 1.955103e-05 1.876934e-05 1.793636e-05
#> [116] 1.722772e-05 1.655264e-05 1.613572e-05 1.509385e-05 1.392452e-05
#> [121] 1.334906e-05 1.268514e-05 1.253628e-05 1.174815e-05 1.136248e-05
#> [126] 1.062303e-05 9.888994e-06 9.744278e-06 9.066721e-06 8.934773e-06
#> [131] 8.787139e-06 8.246965e-06 7.669836e-06 7.281496e-06 6.685921e-06
#> [136] 6.588381e-06 6.335071e-06 6.109974e-06 5.799953e-06 5.524646e-06
#> [141] 5.458421e-06 5.046885e-06 4.692672e-06 4.567471e-06 4.450786e-06
#> [146] 4.323303e-06 4.165674e-06 3.967148e-06 3.850256e-06 3.760109e-06
#> [151] 3.674297e-06 3.598966e-06 3.381203e-06 3.250883e-06 3.174972e-06
#> [156] 2.974227e-06 2.872635e-06 2.712105e-06 2.642509e-06 2.432329e-06
#> [161] 2.407636e-06 2.306336e-06 2.249606e-06 2.190046e-06 2.074787e-06
#> [166] 1.925506e-06 1.885381e-06 1.822314e-06 1.747626e-06 1.725496e-06
#> [171] 1.669954e-06 1.579292e-06 1.539292e-06 1.508963e-06 1.452285e-06
#> [176] 1.419973e-06 1.413538e-06 1.377707e-06 1.355145e-06 1.335675e-06
#> [181] 1.331425e-06 1.302395e-06 1.270499e-06 1.225289e-06 1.205829e-06
#> [186] 1.171387e-06 1.155803e-06 1.144282e-06 1.130620e-06 1.118648e-06
#> [191] 1.108285e-06 1.091960e-06 1.071140e-06 1.069570e-06 1.062423e-06
#> [196] 1.055209e-06 1.041399e-06 1.030498e-06 1.026858e-06 1.012902e-06
#> 
#> $gamma.U
#> [1] 1.848109
#> 
#> $gamma.Theta
#> [1] 0.1778428

## test under an alternative distribution
X=matrix(rchisq(d*n,2),nrow=n,ncol=d)
EllKEPT(X,kerU="Gaussian",kerTheta="Gaussian")
#> $stat
#> [1] 5.824265
#> 
#> $pval
#> [1] 8.861197e-08
#> 
#> $lambda
#>   [1] 1.927915e-01 5.873504e-02 3.823065e-02 3.221209e-02 2.610856e-02
#>   [6] 2.087661e-02 1.974292e-02 1.411963e-02 1.077688e-02 8.397257e-03
#>  [11] 7.643907e-03 6.923854e-03 5.644479e-03 5.242526e-03 4.575622e-03
#>  [16] 3.855130e-03 3.511931e-03 3.325897e-03 2.932099e-03 2.481081e-03
#>  [21] 2.381883e-03 2.011032e-03 1.659369e-03 1.632388e-03 1.478255e-03
#>  [26] 1.183571e-03 1.113807e-03 9.866928e-04 9.169988e-04 8.123973e-04
#>  [31] 7.547914e-04 6.787842e-04 6.067348e-04 5.034679e-04 4.880360e-04
#>  [36] 4.245527e-04 3.775821e-04 3.547872e-04 3.347970e-04 2.978522e-04
#>  [41] 2.815555e-04 2.634804e-04 2.242460e-04 2.089340e-04 1.901784e-04
#>  [46] 1.712219e-04 1.410829e-04 1.346283e-04 1.185017e-04 9.886833e-05
#>  [51] 8.782255e-05 8.426498e-05 7.665894e-05 7.011435e-05 6.441106e-05
#>  [56] 5.916186e-05 5.203184e-05 4.831025e-05 4.599046e-05 4.231330e-05
#>  [61] 3.783512e-05 3.452000e-05 3.313868e-05 3.022687e-05 2.664902e-05
#>  [66] 2.227106e-05 1.989966e-05 1.767311e-05 1.687191e-05 1.612107e-05
#>  [71] 1.506046e-05 1.396138e-05 1.210153e-05 1.127126e-05 9.553097e-06
#>  [76] 9.345740e-06 8.530440e-06 8.044799e-06 7.828668e-06 7.539044e-06
#>  [81] 6.227406e-06 5.888178e-06 5.451610e-06 4.955405e-06 4.742398e-06
#>  [86] 4.553983e-06 4.418100e-06 4.291005e-06 3.984371e-06 3.464732e-06
#>  [91] 3.266889e-06 3.023045e-06 2.809593e-06 2.474113e-06 2.384086e-06
#>  [96] 2.299346e-06 2.143555e-06 2.093939e-06 2.028233e-06 1.946739e-06
#> [101] 1.824235e-06 1.734612e-06 1.716483e-06 1.592446e-06 1.572069e-06
#> [106] 1.506228e-06 1.457145e-06 1.413275e-06 1.376961e-06 1.336437e-06
#> [111] 1.296902e-06 1.279546e-06 1.244650e-06 1.234452e-06 1.206742e-06
#> [116] 1.179838e-06 1.160661e-06 1.154608e-06 1.138159e-06 1.131227e-06
#> [121] 1.122022e-06 1.103863e-06 1.095979e-06 1.081669e-06 1.076598e-06
#> [126] 1.069509e-06 1.063789e-06 1.051232e-06 1.049324e-06 1.042065e-06
#> [131] 1.040006e-06 1.037517e-06 1.034886e-06 1.029664e-06 1.024284e-06
#> [136] 1.023034e-06 1.022003e-06 1.020296e-06 1.019067e-06 1.018327e-06
#> [141] 1.017082e-06 1.016197e-06 1.015125e-06 1.014227e-06 1.013430e-06
#> [146] 1.012262e-06 1.011677e-06 1.011015e-06 1.010830e-06 1.010461e-06
#> [151] 1.010312e-06 1.010100e-06 1.009557e-06 1.009479e-06 1.009407e-06
#> [156] 1.009318e-06 1.009164e-06 1.008935e-06 1.008705e-06 1.008594e-06
#> [161] 1.008346e-06 1.008217e-06 1.008077e-06 1.007961e-06 1.007770e-06
#> [166] 1.007735e-06 1.007674e-06 1.007642e-06 1.007595e-06 1.007572e-06
#> [171] 1.007540e-06 1.007533e-06 1.007529e-06 1.007512e-06 1.007498e-06
#> [176] 1.007455e-06 1.007444e-06 1.007435e-06 1.007375e-06 1.007346e-06
#> [181] 1.007321e-06 1.007300e-06 1.007289e-06 1.007216e-06 1.007189e-06
#> [186] 1.007158e-06 1.007096e-06 1.007072e-06 1.007060e-06 1.007028e-06
#> [191] 1.007018e-06 1.006973e-06 1.006930e-06 1.006907e-06 1.006890e-06
#> [196] 1.006865e-06 1.006741e-06 1.006670e-06 1.006627e-06 1.006572e-06
#> 
#> $gamma.U
#> [1] 1.473997
#> 
#> $gamma.Theta
#> [1] 0.1604577
EllKEPT(X,kerU="PIQ",kerTheta="PIQ")
#> $stat
#> [1] 4.426728
#> 
#> $pval
#> [1] 4.765418e-07
#> 
#> $lambda
#>   [1] 1.646927e-01 3.737424e-02 2.732228e-02 2.501739e-02 2.326216e-02
#>   [6] 1.767231e-02 1.366167e-02 1.137133e-02 8.896104e-03 6.086912e-03
#>  [11] 5.672798e-03 5.493414e-03 4.948331e-03 4.696390e-03 4.088014e-03
#>  [16] 3.583218e-03 2.983184e-03 2.769883e-03 2.574814e-03 2.267945e-03
#>  [21] 2.158971e-03 1.754301e-03 1.656940e-03 1.575126e-03 1.487786e-03
#>  [26] 1.212008e-03 1.192868e-03 1.045950e-03 9.950876e-04 9.282261e-04
#>  [31] 8.876219e-04 8.382806e-04 7.405832e-04 6.702246e-04 6.309618e-04
#>  [36] 6.107121e-04 5.842754e-04 5.401185e-04 5.055844e-04 4.465680e-04
#>  [41] 4.266652e-04 4.156612e-04 3.974379e-04 3.319859e-04 3.029870e-04
#>  [46] 2.916491e-04 2.734473e-04 2.465599e-04 2.408118e-04 2.266227e-04
#>  [51] 2.164394e-04 1.931864e-04 1.869942e-04 1.690429e-04 1.552360e-04
#>  [56] 1.435750e-04 1.213272e-04 1.140580e-04 1.100192e-04 1.060931e-04
#>  [61] 1.032872e-04 9.373116e-05 9.195594e-05 8.513171e-05 8.113227e-05
#>  [66] 7.775030e-05 7.483478e-05 6.806239e-05 6.391047e-05 5.916341e-05
#>  [71] 5.406257e-05 5.149294e-05 4.774569e-05 4.634647e-05 4.407713e-05
#>  [76] 4.155996e-05 3.824695e-05 3.504332e-05 3.321416e-05 3.274004e-05
#>  [81] 3.090641e-05 2.854689e-05 2.680167e-05 2.489154e-05 2.305263e-05
#>  [86] 2.196689e-05 2.007703e-05 1.896019e-05 1.707490e-05 1.686278e-05
#>  [91] 1.498746e-05 1.390720e-05 1.341891e-05 1.324903e-05 1.196903e-05
#>  [96] 1.160769e-05 1.130520e-05 1.039602e-05 9.588650e-06 8.752201e-06
#> [101] 8.004470e-06 7.966420e-06 7.597211e-06 7.300246e-06 7.201040e-06
#> [106] 6.302628e-06 6.162509e-06 5.904935e-06 5.808608e-06 5.197134e-06
#> [111] 4.982071e-06 4.798298e-06 4.588215e-06 4.450646e-06 4.237827e-06
#> [116] 3.801555e-06 3.744177e-06 3.672724e-06 3.494031e-06 3.410424e-06
#> [121] 3.094765e-06 2.949380e-06 2.814830e-06 2.613163e-06 2.532212e-06
#> [126] 2.432266e-06 2.407259e-06 2.128700e-06 2.036591e-06 1.968435e-06
#> [131] 1.918702e-06 1.870490e-06 1.823156e-06 1.751050e-06 1.648840e-06
#> [136] 1.646505e-06 1.581296e-06 1.522441e-06 1.504556e-06 1.475580e-06
#> [141] 1.429871e-06 1.415576e-06 1.363400e-06 1.342504e-06 1.326256e-06
#> [146] 1.289035e-06 1.244445e-06 1.223586e-06 1.213199e-06 1.202074e-06
#> [151] 1.167207e-06 1.155458e-06 1.149660e-06 1.130122e-06 1.111508e-06
#> [156] 1.102203e-06 1.099642e-06 1.090302e-06 1.081971e-06 1.074569e-06
#> [161] 1.068449e-06 1.065425e-06 1.063802e-06 1.055475e-06 1.054668e-06
#> [166] 1.050670e-06 1.046495e-06 1.041232e-06 1.037201e-06 1.036650e-06
#> [171] 1.033112e-06 1.029596e-06 1.026862e-06 1.022747e-06 1.022547e-06
#> [176] 1.021323e-06 1.018499e-06 1.016635e-06 1.015807e-06 1.014720e-06
#> [181] 1.014248e-06 1.013457e-06 1.012781e-06 1.010894e-06 1.010254e-06
#> [186] 1.010005e-06 1.009240e-06 1.008920e-06 1.008776e-06 1.008171e-06
#> [191] 1.008127e-06 1.007908e-06 1.007451e-06 1.007315e-06 1.007227e-06
#> [196] 1.007188e-06 1.007157e-06 1.007121e-06 1.007057e-06 1.006842e-06
#> 
#> $gamma.U
#> [1] 1.473997
#> 
#> $gamma.Theta
#> [1] 0.1604577