Convex estimation

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

“…Most real observed abundance distributions are at least decreasing and appear to be k-monotone for some k 2. Several examples were already studied when considering convexity [15,14]. Let us consider three examples in order to illustrate how our procedure applies when we aim at estimating the total number of classes taking into account the hollowed shape of the abundance distribution.…”

“…For a given integer k, assuming that the distribution p is a k-abundance distribution, we propose two estimators of N , N k , see (15), and N k , see (17). In practical cases, we do not know the degree of monotonicity of p. Because we observe X 1 , .…”

“…Such approach may be well adapted in some situations because it combines the straightforwardness of parametric models (no choice of parameter is left to the user) and the great flexibility of nonparametric estimation. Moreover shape constraint arises naturally in many frameworks such as insurance [24], reliability studies [29], epidemiology [3] or ecology [14,15].…”

“…See Bunge and Fitzpatrick [10] for a review of the different approaches to deal with that problem. Durot et al [14,15] proposed an estimator of the total number of classes based on an estimator of the abundance distribution under the constraint of convexity. Giguelay [19] generalises their work to k-monotonicity.…”

“…In order to validate the chosen model before estimating the number of classes, we propose a goodnessof-fit test for testing k-monotonicity. To the best of our knowledge, very few works are available for testing a shape constraint on a discrete density: Akakpo et al [1] proposed a procedure for testing monotonicity (k = 1), while Durot et al [15] and Balabdaoui et al [7] considered the problem of testing convexity (k = 2). The testing procedures they proposed rely on the asymptotic distribution of some distance between the empirical distribution and the estimation of the density under the shape constraint.…”