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Last edited on
Oct 18, 2021 by JJ
.
Thesis:
Bachelor in Mathematics

Author:
Daniel Fridljand

Title:
Online-Schätzung des geometrischen Medians in Hilberträumen

Supervisor:
Jan JOHANNES

Abstract:
In this thesis we consider a new algorithm estimating the geometric median. The geometric median is one possible generalization of the real median to Hilbert spaces. This new estimator is based on the stochastic gradient method and is calculated recursivly. In comparison to the standard algorithm for this problem so far, this new approach is more efficient and faster, especially in case of high dimensional data. Furthermore the estimator can be simply updated when the data arrives sequentially. One most of the most important results of this thesis is the almost sure consistency of the estimatior. We derive a rate of convergence in L2, show that our estimator is asymptotically normal and determine the exact asymptotic distribution. Lastly, we illustrate the asymptotic normality with a simulation.

References:
H. Cardot, P. Cénac, and A. Godichon-Baggioni. Online estimation of the geometric median in Hilbert spaces: nonasymptotic confidence balls. The Annals of Statistics, 45(2):591–614, 2017.
H. Cardot, P. Cénac, and P.-A. Zitt. Efficient and fast estimation of the geometric median in Hilbert spaces with an averaged stochastic gradient algorithm. Bernoulli, 19(1):18-–43, 2013.