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

TL;DR

This paper introduces non-parametric goodness-of-fit tests for k-monotonicity in discrete distributions, along with an estimator for the degree of k-monotonicity, applied to estimating the number of classes in a population.

Contribution

It provides new, easy-to-implement tests for k-monotonicity and an estimator for its degree, with proven asymptotic properties and practical application to population class estimation.

Findings

Tests are asymptotically correct and consistent.

Estimator effectively assesses the degree of k-monotonicity.

Procedures perform well in simulation studies.

Abstract

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.