Test of Normality based on Nonlinear Principal Components

A new test of normality based on Nonlinear Principal Components (NLPC) is introduced. Our testing procedure is based on the fact that among the distributions admitting NLPC, the Gaussian is the unique for which the variance of the first NLPC is equal to the variance of the distribution. Thus testing normality is equivalent to test the equality between the first NLPC's variance and the variance. The test statistic depends on a preliminary estimation of NLPC by orthonormal polynomials. The asymptotic distribution of this statistic is derived in case of known expectation and variance for i.i.d. sequences of observations, while a Monte Carlo method is used to approximate the distribution when these parameters are estimated. We study the level and the power of the test for finite samples by means of a simulation experiment including a comparison with several other tests