A Modified Chebyshev Inequality and Its Appropriateness for Nonparametric Testing

Sample sizes might be small in a variety of applications. Many classical and often-applied statistical methods rely on asymptotic approximations and, therefore, should not be used in the case of small sample sizes. Approaches that can be applied to small data sets often involve computer-intensive methods such as permutation tests and bootstrapping. As an alternative, a modified Chebyshev inequality was proposed by Beasley et al. (Applied Statistics 2004; 53, 95–108) and suggested for nonparametric testing. Here, it is shown that this modified Chebyshev inequality does not hold in general and, therefore, it should not be used to construct a statistical test. Available nonparametric tests that can be recommended for small sample sizes are discussed.