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

We develop here several goodness-of-fit tests for testing the k-monotonicity of a discrete density, based on the empirical distribution of the observations. Our tests are non-parametric, easy to implement and are proved to be asymptotically of the desired level and consistent. We propose an estimator of the degree of k-monotonicity of the distribution based on the non-parametric goodness-of-fit tests. We apply our work to the estimation of the total number of classes in a population. A large simulation study allows to assess the performances of our procedures.