- Series
- Stochastics Seminar
- Time
- Thursday, January 31, 2019 - 3:05pm for 1 hour (actually 50 minutes)
- Location
- Skiles 006
- Speaker
- V. Koltchinskii – SOM, GaTech
- Organizer
- Christian Houdré
We discuss a problem of asymptotically efficient (that is, asymptotically normal with minimax optimal limit variance) estimation of functionals of the form ⟨f(Σ),B⟩ of unknown covariance Σ based on i.i.d.mean zero Gaussian observations X1,…,Xn∈Rd with covariance $$\Sigma.Undertheassumptionsthatthedimensiond\leq n^{\alpha}forsome\alpha\in (0,1)andf:{\mathbb R}\mapsto {\mathbb R}isofsmoothnesss>\frac{1}{1-\alpha},weshowhowtoconstructanasymptoticallyefficientestimatorofsuchfunctionals(thesmoothnessthreshold\frac{1}{1-\alpha}$ is known to be optimal for a simpler problem of estimation of smooth functionals of unknown mean of normal distribution).
The proof of this result relies on a variety of probabilistic and analytic tools including Gaussian concentration, bounds on the remainders of Taylor expansions of operator functions and bounds on finite differences of smooth functions along certain Markov chains in the spaces of positively semi-definite matrices.