Global convergence of the log-concave MLE when the true distribution is geometric

摘要

Let X_1,...,X_n be i.i.d. from a discrete probability mass function (pmf) p. In Balabdaoui et al. [(2013), 'Asymptotic Distribution of the Discrete Log-Concave mle and Some Applications', JRSS-B, in press], the pointwise limit distribution of the log-concave maximum-likelihood estimator (MLE) was derived in both the well- and misspecified settings. In the well-specified setting, the geometric distribution was excluded, classified as being degenerate. In this article, we establish the global asymptotic theory of the log-concave MLE of a geometric pmf in all ℓ_q distances for q ∈ {1,2,...}( U ){∞}. We also show how these asymptotic results could be used in testing whether a pmf is geometric.