Shape constrained inference

While non-parametric methods like kernel density estimation are flexible and less restrictive on the underlying data-generating process, often they require tuning parameters. Shape constraint methods do not rely on tuning parameters and impose conditions mainly on the shape of certain functions. Also, shape restrictions like unimodality or log-concavity arise naturally and apply to a large class of continuous densities.

Location estimation

The traditional approach for estimating the location parameters in semiparametric models involves the use of various tuning parameters. In "Adaptive estimation in symmetric location model under log-concavity constraint", I show that shape restriction of log-concavity leads to adaptive estimation with virtually no tuning in symmetric location models. The method can be implemented using the R package log.location.

Bi-s*-concave distributions

The class of s-concave densities includes many common continuous densities, such as the class of log-concave densities (s=0) and t-densities. Although this is a rich class, s-concave densities are necessarily unimodal, thereby excluding many types of mixture densities which naturally arise in many fields like speech recognition, pattern recognition, climatology, to name a few. In our paper "Bi-s*-concave distributions", we introduce a new shape-constrained class of distribution functions, the bi-s*-concave class, which permits distributions with bimodal and multimodal densities. Below is the image of bi-s*-concave bands for the distribution function of a t-distribution.

HIV vaccine trial

The RV144 vaccine efficacy trial, conducted in Thailand, revealed that when IgG antibody bound to V1V2 region of HIV-1 envelope proteins, the HIV infection rate was lower. Although the RV144 regimen is the first HIV vaccine regimen to show any positive indication that the HIV pandemic can be controlled by vaccination, its efficacy (31.2%) is at best moderate. So a new regimen (HVTN 702) was developed by modification of the RV144 regimen. Two small-scale trials in South Africa, named HVTN097 and HVTN100, evaluated the immune responses invoked by RV144 and HVTN702 regimens, respectively.

In our paper, we  show that the immune response generated by IgG binding to V1V2 regions of clade C antigens is stochastically larger in HVTN097 (RV144 regimen) trial, which may be used to explain why HVTN702 regimen  has much smaller efficacy than that of RV144. This is a work in collaboration with Fred Hutchinson cancer Research Center. Since the data indicate that the densities of the aggregated immune responses are unimodal, we  developed novel shape-constrained tools  (tests of stochastic dominance and measure of discrepancies) to compare the trials. We showed that the shape-constrained methods empirically perform better than the non-parametric counterparts but enjoy the same asymptotic guarantees.  Our methods can be implemented using the R package SDNNtests. See also my presentation on this project  here.