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Mean estimation with sub-Gaussian rates in polynomial time. (English) Zbl 1454.62162
The author studies polynomial time algorithms for estimating the mean of a heavy-tailed multivariate random vector. The only assumption is that the random vector $$X$$ has finite mean and covariance. In this setting, the radius of confidence intervals achieved by the empirical mean are large compared to the case that $$X$$ is Gaussian or sub-Gaussian. A polynomial time algorithm is proposed to estimate the mean with sub-Gaussian-size confidence intervals under these assumptions. The algorithm is based on a new semidefinite programming relaxation of a high-dimensional median. Previous estimators which assumed only existence of finitely many moments of $$X$$ either sacrifice sub-Gaussian performance or are only known to be computable via brute-force search procedures requiring time exponential in the dimension. The algorithm runs in polynomial time, but it is not close to practical for any substantially high-dimensional data set.

MSC:
 62H12 Estimation in multivariate analysis 62G32 Statistics of extreme values; tail inference 68W01 General topics in the theory of algorithms 90C22 Semidefinite programming
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References:
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