Multivariate Chebyshev Inequality with Estimated Mean and Variance

B. Stellato, B. Van Parys and P. J. Goulart

The American Statistician, vol. 71, no. 2, 2017.
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@article{SPG:2017,
  author = {B. Stellato and B. Van Parys and P. J. Goulart},
  title = {Multivariate Chebyshev Inequality with Estimated Mean and Variance},
  journal = {The American Statistician},
  year = {2017},
  volume = {71},
  number = {2},
  url = {http://www.tandfonline.com/doi/full/10.1080/00031305.2016.1186559},
  doi = {10.1080/00031305.2016.1186559}
}

A variant of the well-known Chebyshev inequality for scalar random variables can be formulated in the case where the mean and variance are estimated from samples. In this paper we present a generalization of this result to multiple dimensions where the only requirement on the population is that the samples are independent and identically distributed. Furthermore, we show that as the number of samples tends to infinity our inequality converges to the theoretical multi-dimensional Chebyshev bound.