Exploring the statistical applicability of the Poincaré inequality: a test of normality

A new test of normality based on Poincaré inequality is proposed and analyzed. It rests on the characterization of the normal distribution given by Borovkov and Utev, i. e., a r. v. is normal if and only if its Poincaré constant is equal to its variance. The test statistic is computed by estimating the Poincaré constant via orthonormal polynomials. In case of known expectation and variance, the asymptotic distribution of the test statistic is derived under the null hypothesis, and the consistency of the test is proved. An analysis of effects of the degree of polynomials on the procedure is sketched. A Monte Carlo method is used to approximate the distribution when population expectation and variance are replaced by their empirical counterparts. In this framework, we study the level and power of the test for finite samples by means of a simulation experiment including a comparison with other tests.