Multivariate Chebyshev Inequality With Estimated Mean and Variance

Bartolomeo Stellato, Bart P. G. Van Parys and Paul J. Goulart

The American Statistician, 2017, vol. 71, issue 2, 123-127

Abstract: 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 article, we present a generalization of this result to multiple dimensions where the only requirement 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.

Date: 2017
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DOI: 10.1080/00031305.2016.1186559

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