Testing normality using kernel methods

Testing normality is one of the most studied areas in inference. Many methodologies have been proposed. Some are based on characterization of the normal variate, while most others are based on weaker properties of the normal. In this investigation, we propose a new procedure, which is based on the well-known characterization; if X-1 and X-2 are two independent copies of a variable with distribution F, then X-1 and X-2 are normal if and only if X-1 - X-2 and X-1 + X-2 are independent. If X-1,..., X-n is a random sample from F, we test that F is normal by testing nonparametrically that u(ii*) = X-i - X-i* and nu(ii*) = X-i + X-i* are independent, i not equal i* = 1, 2..... n. This procedure has several major advantages; it applies equally to one-dimensional or multi-dimensional cases, it does not require estimation of parameters, it does not require transformation to uniformity, it does not require use of special tables of coefficients, and it does have very good power requiring much less number of iterations to reach stable results.