# Multivariate Chebyshev Inequality With Estimated Mean and Variance

@article{Stellato2015MultivariateCI, title={Multivariate Chebyshev Inequality With Estimated Mean and Variance}, author={B. Stellato and Bart P. G. Van Parys and P. Goulart}, journal={The American Statistician}, year={2015}, volume={71}, pages={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.

#### 25 Citations

Chebyshev's inequality for Hilbert-space valued random elements with estimated mean and covariance

- 2017

Chebyshev's inequality for Hilbert-space valued random elements with estimated mean and covariance

- Mathematics
- 2017

Probabilistic reachable and invariant sets for linear systems with correlated disturbance

- Mathematics, Computer Science
- ArXiv
- 2020

Cost-Sensitive Approach to Batch Size Adaptation for Gradient Descent

- Computer Science, Mathematics
- ArXiv
- 2017

Robust propagation of probability boxes by Interval Predictor Models

- Computer Science
- 2020

Probabilistic interval predictor based on dissimilarity functions

- Mathematics, Engineering
- ArXiv
- 2020

H-infinity set-membership observer design for discrete-time LPV systems

- Mathematics, Computer Science
- Int. J. Control
- 2020

#### References

SHOWING 1-10 OF 22 REFERENCES

Generalized Chebyshev Bounds via Semidefinite Programming

- Computer Science, Mathematics
- SIAM Rev.
- 2007

A NOTE ON CONFIDENCE REGIONS BASED ON THE BIVARIATE CHEBYSHEV INEQUALITY. APPLICATIONS TO ORDER STATISTICS AND DATA SETS.

- Mathematics
- 2014

A New Generalization of Chebyshev Inequality for Random Vectors

- Mathematics, Computer Science
- ArXiv
- 2007

Convex Optimization

- Mathematics, Computer Science
- IEEE Transactions on Automatic Control
- 2006

A Robust Optimization Perspective on Stochastic Programming

- Mathematics, Computer Science
- Oper. Res.
- 2007