Testing k-monotonicity of a discrete distribution. Application to the estimation of the number of classes in a population

The development of nonparametric procedures for testing shape constraint (monotonicity, convexity, unimodality, etc.) has received increasing interest. Nevertheless, testing the k-monotonicity of a discrete density for k larger than 2 has received little attention. To deal with this issue, several testing procedures based on the empirical distribution of the observations have been developed. They are non-parametric, easy to implement and proven to be asymptotically of the desired level and consistent. An estimator of the degree of k-monotonicity of the distribution is presented. An application to the estimation of the total number of classes in a population is proposed. A large simulation study makes it possible to assess the performances of the various procedures.