Multivariate Chebyshev Inequality With Estimated Mean and Variance

@article{Stellato2015MultivariateCI,
  title={Multivariate Chebyshev Inequality With Estimated Mean and Variance},
  author={Bartolomeo Stellato and Bart Van Parys and Paul J. 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. 
View on Taylor & Francis
arxiv.org
36 Citations
Highly Influential Citations
5
Background Citations
13
Methods Citations
10
Results Citations
1

36 Citations

The multivariate Markov and multiple Chebyshev inequalities

  • H. Ogasawara
  • Mathematics
    Communications in Statistics - Theory and Methods
  • 2019
Abstract A sharp probability inequality named the multivariate Markov inequality is derived for the intersection of the survival functions for non-negative random variables as an extension of the…

A Multivariate Chebyshev Bound of the Selberg Form

The least upper bound for the probability that a random vector with fixed mean and covariance will be outside the ball is found. This probability bound is determined by solving a scalar equation and,…

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

ABSTRACT We obtain a generalization of the Chebyshev's inequality for random elements taking values in a separable Hilbert space with estimated mean and covariance.

Probabilistic reachable and invariant sets for linear systems with correlated disturbance

Fréchet inequalities via convex optimization

Quantifying the risk carried by an aggregate position Sd := ∑d i=1Xi comprising many risk factors Xi is fundamental to both insurance and financial risk management. Fréchet inequalities quantify the…

Cost-Sensitive Approach to Batch Size Adaptation for Gradient Descent

A novel approach to automatically determine the batch size in stochastic gradient descent methods by optimizing the ratio between a lower bound to a linear or quadratic Taylor approximation of the expected improvement and the number of samples used to estimate the gradient.

Robust propagation of probability boxes by interval predictor models

Cost-Sensitive Approach for Batch Size Optimization

A novel approach to automatically determine the batch size in stochastic gradient descent methods by optimizing the ratio between a lower bound to the first-order Taylor approximation of the expected improvement and the number of samples used to estimate the gradient.

Probabilistic Interval Predictor Based on Dissimilarity Functions

It is shown that the proposed method constitutes a generalization of classical estimation methods, and a cross-validation scheme is used to tune the two parameters on which the methodology relies.

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

A new set-membership observer design method for discrete-time linear parameter varying systems that provides a deterministic state interval that is build as the sum of the estimated system states and its corresponding estimation errors bounds.

References

SHOWING 1-10 OF 18 REFERENCES

Chebyshev Inequality With Estimated Mean and Variance

Abstract Chebyshev's inequality is investigated when the population mean and variance are estimated from a sample. The necessary modification to the inequality is simple and is actually valid when…

Can the bounds in the multivariate Chebyshev inequality be attained

A very simple proof of the multivariate Chebyshev's inequality

Abstract In this short note, a very simple proof of the Chebyshev's inequality for random vectors is given. This inequality provides a lower bound for the percentage of the population of an arbitrary…

Generalized Chebyshev Bounds via Semidefinite Programming

A sharp lower bound on the probability of a set defined by quadratic inequalities, given the first two moments of the distribution, can be efficiently computed using convex optimization. This result…

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

Chebyshev’s inequality was recently extended to the multivariate case. In this paper this new inequality is used to obtain distribution-free confidence regions for an arbitrary bivariate random…

A New Generalization of Chebyshev Inequality for Random Vectors

It is demonstrated that the new generalization of Chebyshev inequality for random vectors is much less conservative than the classical generalization.

The Distribution of Robust Distances

Mahalanobis-type distances in which the shape matrix is derived from a consistent, high-breakdown robust multivariate location and scale estimator have an asymptotic chi-squared distribution as is…

Convex Optimization

A comprehensive introduction to the subject of convex optimization shows in detail how such problems can be solved numerically with great efficiency.

A Robust Optimization Perspective on Stochastic Programming

An approach for constructing uncertainty sets for robust optimization using new deviation measures for random variables termed the forward and backward deviations is introduced, which converts the original model into a second-order cone program, which is computationally tractable both in theory and in practice.

Note on a Method for Calculating Corrected Sums of Squares and Products

In many problems the "corrected sum of squares" of a set of values must be calculated i.e. the sum of squares of the deviations of the values about their mean. The most usual way is to calculate the…