# Bounding Probability of Small Deviation: A Fourth Moment Approach

@article{He2010BoundingPO, title={Bounding Probability of Small Deviation: A Fourth Moment Approach}, author={Simai He and Jiawei Zhang and Shuzhong Zhang}, journal={Math. Oper. Res.}, year={2010}, volume={35}, pages={208-232} }

In this paper we study the problem of upper bounding the probability that a random variable is above its expected value by a small amount (relative to the expected value), by means of the second and the fourth (central) moments of the random variable. In this particular context, many classical inequalities yield only trivial bounds. We obtain tight upper bounds by studying the primal-dual moments-generating conic optimization problems. As an application, we demonstrate that given the new…

## Topics from this paper

## 33 Citations

Probability Bounds for Polynomial Functions in Random Variables

- Mathematics, Computer ScienceMath. Oper. Res.
- 2014

This paper sets out to present a number of scenarios for f, S and S0 where certain probability bounds can be established, leading to a quality assurance of the procedure.

Bounding probability of small deviation on sum of independent random variables: Combination of moment approach and Berry-Esseen theorem

- Mathematics
- 2020

In the context of bounding probability of small deviation, there are limited general tools. However, such bounds have been widely applied in graph theory and inventory management. We introduce a…

Nonnegative k-sums, fractional covers, and probability of small deviations

- Computer Science, MathematicsJ. Comb. Theory, Ser. B
- 2012

More than twenty years ago, Manickam, Miklos, and Singhi conjectured that for any integers n, k satisfying n>=4k, every set of n real numbers with nonnegative sum has at least (n-1k-1)k-element…

The Discrete Moment Problem with Nonconvex Shape Constraints

- Computer Science, MathematicsOper. Res.
- 2021

This paper studies the discrete moment problems with additional "shape constraints" that guarantee the worst case distribution is either log-concave or has an increasing failure rate, and describes a computational approach to solving these low-dimensional problems, including numerical results for a representative set of instances.

A Unified Framework for Generalized Moment Problems: a Novel Primal-Dual Approach

- Mathematics
- 2022

Generalized moment problems optimize functional expectation over a class of distributions with generalized moment constraints, i.e., the function in the moment can be any measurable function. These…

Small Deviations of Sums of Random Variables by Brian Garnett Dissertation Director : Swastik Kopparty

- 2016

OF THE DISSERTATION Small Deviations of Sums of Random Variables by Brian Garnett Dissertation Director: Swastik Kopparty In this thesis, we study the probability of a small deviation from the mean…

Some Improvements on Markov's Theorem with Extensions

- MathematicsThe American Statistician
- 2019

ABSTRACT Markov's theorem for an upper bound of the probability related to a nonnegative random variable has been improved using additional information in almost the nontrivial entire range of the…

Various Problems in Extremal Combinatorics

- Mathematics
- 2012

Extremal combinatorics is a central theme of discrete mathematics. It deals with the problems of finding the maximum or minimum possible cardinality of a collection of finite objects satisfying…

Small deviations of sums of independent random variables

- Mathematics, Computer ScienceJ. Comb. Theory, Ser. A
- 2020

By a finer consideration of the first four moments, this work improves the bound to approximately $1/e \simeq .368$, so there is still (possibly) quite a gap left to fill.

Computing best bounds for nonlinear risk measures with partial information

- Mathematics
- 2013

Extreme events occur rarely, but these are often the circumstances where an insurance coverage is demanded. Given the first, say, n moments of the risk(s) of the events, one is able to compute or…

## References

SHOWING 1-10 OF 21 REFERENCES

The fourth moment method

- Mathematics, Computer ScienceSODA '91
- 1991

A new combinatorial method is introduced to lower bound the expectation of the absolute value of a random variable X by the expectations of a quartic in X, using only a fourth moment for the total discrepancy of a set system.

Optimal Inequalities in Probability Theory: A Convex Optimization Approach

- Computer Science, MathematicsSIAM J. Optim.
- 2005

It is shown that it is NP-hard to find tight bounds for k = 2 and $\Omega={\cal R}^n$, and an efficient algorithm for finding tight bounds when S is a union of convex sets, over which convex quadratic functions can be optimized efficiently.

A Semidefinite Programming Approach to Optimal-Moment Bounds for Convex Classes of Distributions

- Mathematics, Computer ScienceMath. Oper. Res.
- 2005

An optimization framework for computing optimal upper and lower bounds on functional expectations of distributions with special properties, given moment constraints is provided and generalizations of Chebyshev's inequality for symmetric and unimodal distributions are obtained.

Approximation and Intractability Results for the Maximum Cut Problem and its Variants

- Mathematics, Computer ScienceIEEE Trans. Computers
- 1991

It is shown that it is NP-complete to decide if a given graph has a normal maximum cut with at least a fraction 1/2+1/2n of its edges, where the positive constant epsilon can be taken smaller than any value chosen.

Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming

- Mathematics, Computer ScienceJACM
- 1995

This algorithm gives the first substantial progress in approximating MAX CUT in nearly twenty years, and represents the first use of semidefinite programming in the design of approximation algorithms.

Semidefnite Relaxation Bounds for Indefinite Homogeneous

- Mathematics
- 2008

In this paper we study the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic…

Generalized Chebychev Inequalities: Theory and Applications in Decision Analysis

- Mathematics, Computer ScienceOper. Res.
- 1995

A very general framework for analyzing these kinds of problems where, given certain "moments" of a distribution, the authors can compute bounds on the expected value of an arbitrary "objective" function.

The Probabilistic Method

- Computer ScienceSODA
- 1992

A particular set of problems - all dealing with “good” colorings of an underlying set of points relative to a given family of sets - is explored.

On the Relation Between Option and Stock Prices: A Convex Optimization Approach

- Economics, Computer ScienceOper. Res.
- 2002

Convex and semidefinite optimization methods, duality, and complexity theory are introduced to shed new light on the relation of option and stock prices based just on the no-arbitrage assumption, and it is shown that it is NP-hard to find best possible bounds in multiple dimensions.

Lectures on modern convex optimization - analysis, algorithms, and engineering applications

- Computer Science, MathematicsMPS-SIAM series on optimization
- 2001

The authors present the basic theory of state-of-the-art polynomial time interior point methods for linear, conic quadratic, and semidefinite programming as well as their numerous applications in engineering.