Feb 12

3:30 pm

## Robust Inference Using Higher Order Influence Function

### Lingling Li

Seminar

Harvard School of Public Health - Department of Biostatistics

Suppose we obtain $n$ i.i.d copies of a random vector $O$ with unknown distribution $F(\\theta)$, $\\theta \\in Theta$. Our goal is to construct honest $100 (1 - \\alpha)$% asymptotic confidence intervals (CI) (whose width shrinks to zero with increasing $n$ at the fastest possible rate), through higher order influence functions, for a functional $\\psi(\\theta)$ in a model that places no restrictions on $F$; other than, perhaps, bounds on both the $L_p$ norms and the roughness (more generally, the complexity) of certain density and conditional expectation functions. The theory of higher order influence functions extends the first order semiparametric theory, reproduces many previous results, produces new non-$\\sqrt{n}$ results, and opens up the ability to perform optimal non-$\\sqrt{n}$ inference in complex high dimensional models. We present novel rate-optimal point and intervals estimators for various functionals of central importance to biostatistics in settings in which estimation at the expected $\\sqrt{n}$-rate is not possible, owing to the curse of dimensionality.