## Department Colloquium

### Monday, January 27th, 2020

**Time:** 2:30 p.m. **Place:** Jeffery Hall 234

**Speaker:** Hok Kan Ling (Columbia University)

**Title:** Shape-constrained Estimation and Testing.

**Abstract: ** Shape-constrained inference has been gaining more attention recently. Such constraints are sometimes the direct consequence of the problem under investigation. In other times, they are used to replace parametric models while retaining qualitative shape properties that exist in problems from diverse disciplines. In this talk, I will first discuss the estimation of a monotone density in s-sample biased sampling models, which has been long missing in the literature due to certain non-standard nature of the problem. We established the asymptotic distribution of the maximum likelihood estimator (MLE) and the connection between this MLE and a Grenander-type estimator. In the second part of the talk, a nonparametric likelihood ratio test for the hypothesis testing problem on whether a random sample follows a distribution with a decreasing, k-monotone or log-concave density is proposed. The obtained test statistic has a surprisingly simple and universal asymptotic null distribution, which is Gaussian, instead of the well-known chi-square for generic likelihood ratio tests. We also established rates of convergence of the maximum likelihood estimator under weaker conditions than the existing literature that are of independent interest.

**Hok Kan (Brian) Ling** is a Ph.D. candidate in the Department of Statistics at Columbia University, working under the supervision of Dr. Zhiliang Ying. His research interests primarily lie in the areas of multivariate statistics, latent variable models, event history analysis, nonparametric estimation, semiparametric models and shape-restricted statistical inference.