6533b826fe1ef96bd128479c

RESEARCH PRODUCT

Recursive estimation of the conditional geometric median in Hilbert spaces

Peggy CénacHervé CardotPierre-andré Zitt

subject

Statistics and ProbabilityMallows-Wasserstein distanceRobbins-Monroasymptotic normalityCLTcentral limit theoremAsymptotic distributionMathematics - Statistics TheoryStatistics Theory (math.ST)01 natural sciencesMallows–Wasserstein distanceonline data010104 statistics & probability[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST]60F05FOS: MathematicsApplied mathematics[ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST]0101 mathematics62L20MathematicsaveragingSequential estimation010102 general mathematicsEstimatorRobbins–MonroConditional probability distribution[STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]Geometric medianstochastic gradient[ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH]robust estimatorRate of convergenceConvergence of random variablesStochastic gradient.kernel regressionsequential estimationKernel regressionStatistics Probability and Uncertainty

description

International audience; A recursive estimator of the conditional geometric median in Hilbert spaces is studied. It is based on a stochastic gradient algorithm whose aim is to minimize a weighted L1 criterion and is consequently well adapted for robust online estimation. The weights are controlled by a kernel function and an associated bandwidth. Almost sure convergence and L2 rates of convergence are proved under general conditions on the conditional distribution as well as the sequence of descent steps of the algorithm and the sequence of bandwidths. Asymptotic normality is also proved for the averaged version of the algorithm with an optimal rate of convergence. A simulation study confirms the interest of this new and fast algorithm when the sample sizes are large. Finally, the ability of these recursive algorithms to deal with very high-dimensional data is illustrated on the robust estimation of television audience profiles conditional on the total time spent watching television over a period of 24 hours.

yearjournalcountryeditionlanguage
2012-04-14
10.1214/12-ejs759https://projecteuclid.org/euclid.ejs/1357307949