Cantelli's inequality

In probability theory, Cantelli's inequality is a generalization of Chebyshev's inequality in the case of a single "tail".[1][2][3] The inequality states that

${\displaystyle \Pr(X-\mathbb {E} [X]\geq \lambda )\quad {\begin{cases}\leq {\frac {\sigma ^{2}}{\sigma ^{2}+\lambda ^{2}}}&{\text{if }}\lambda >0,\\[8pt]\geq 1-{\frac {\sigma ^{2}}{\sigma ^{2}+\lambda ^{2}}}&{\text{if }}\lambda <0.\end{cases}}}$

where

${\displaystyle X}$ is a real-valued random variable,

${\displaystyle \Pr }$ is the probability measure,

${\displaystyle \mathbb {E} [X]}$ is the expected value of ${\displaystyle X}$,

${\displaystyle \sigma ^{2}}$ is the variance of ${\displaystyle X}$.

Combining the cases of ${\displaystyle \lambda >0}$ and ${\displaystyle \lambda <0}$ gives, for ${\displaystyle \delta >0,}$

${\displaystyle \Pr(|X-\mathbb {E} [X]|\geq \delta )\leq {\frac {2\sigma ^{2}}{\sigma ^{2}+\delta ^{2}}}.}$

A weaker version of Chebyshev's inequality.

While the inequality is often attributed to Francesco Paolo Cantelli who published it in 1928,[4] it originates in Chebyshev's work of 1874.[5] The Chebyshev inequality implies that in any data sample or probability distribution, "nearly all" values are close to the mean in terms of the absolute value of the difference between the points of the data sample and the weighted average of the data sample. The Cantelli inequality (sometimes called the "Chebyshev–Cantelli inequality" or the "one-sided Chebyshev inequality") gives a way of estimating how the points of the data sample are bigger than or smaller than their weighted average without the two tails of the absolute value estimate. The Chebyshev inequality has "higher moments versions" and "vector versions", and so does the Cantelli inequality.

Proof

Case ${\displaystyle \lambda >0}$

Let ${\displaystyle X}$ be a real-valued random variable with finite variance ${\displaystyle \sigma ^{2}}$ and expectation ${\displaystyle \mu }$, and define ${\displaystyle Y=X-\mathbb {E} [X]}$ (so that ${\displaystyle \mathbb {E} [Y]=0}$ and ${\displaystyle \operatorname {Var} (Y)=\sigma ^{2}}$).

Then, for any ${\displaystyle u\geq 0}$, we have

${\displaystyle \Pr(X-\mathbb {E} [X]\geq \lambda )=\Pr(Y\geq \lambda )=\Pr(Y+u\geq \lambda +u)\leq \Pr((Y+u)^{2}\geq (\lambda +u)^{2})\leq {\frac {\mathbb {E} [(Y+u)^{2}]}{(\lambda +u)^{2}}}={\frac {\sigma ^{2}+u^{2}}{(\lambda +u)^{2}}}.}$

the last inequality being a consequence of Markov's inequality. As the above holds for any choice of ${\displaystyle u\in \mathbb {R} }$, we can choose to apply it with the value that minimizes the function ${\displaystyle u\geq 0\mapsto {\frac {\sigma ^{2}+u^{2}}{(\lambda +u)^{2}}}}$. By differentiating, this can be seen to be ${\displaystyle u_{\ast }={\frac {\sigma ^{2}}{\lambda }}}$, leading to

${\displaystyle \Pr(X-\mathbb {E} [X]\geq \lambda )\leq {\frac {\sigma ^{2}+u_{\ast }^{2}}{(\lambda +u_{\ast })^{2}}}={\frac {\sigma ^{2}}{\lambda ^{2}+\sigma ^{2}}}}$ if ${\displaystyle \lambda >0}$

Case ${\displaystyle \lambda <0}$

We proceed as before, writing ${\displaystyle \alpha =-\lambda >0}$ and for any ${\displaystyle u\geq 0}$

${\displaystyle \Pr(X-\mathbb {E} [X]<\lambda )=\Pr(-Y>\alpha )\leq {\frac {\sigma ^{2}}{\alpha ^{2}+\sigma ^{2}}}={\frac {\sigma ^{2}}{\lambda ^{2}+\sigma ^{2}}}}$

using the previous derivation on ${\displaystyle -Y}$. By taking the complement of the left-hand side, we obtain

${\displaystyle \Pr(X-\mathbb {E} [X]\geq \lambda )\geq 1-{\frac {\sigma ^{2}}{\lambda ^{2}+\sigma ^{2}}}}$ if ${\displaystyle \lambda <0}$

Generalizations

Using more moments, various stronger inequalities can be shown. He, Zhang and Zhang and showed,[6] when ${\displaystyle \mathbb {E} [X]=0}$ and ${\displaystyle \mathbb {E} [X^{2}]=1}$:

${\displaystyle \Pr(X\geq \lambda )\leq 1-(2{\sqrt {3}}-3){\frac {(1+\lambda ^{2})^{2}}{\mathbb {E} [X^{4}]+6\lambda ^{2}+\lambda ^{4}}}}$

See also

• Paley–Zygmund inequality

References

1. Research and practice in multiple criteria decision making: proceedings of the XIVth International Conference on Multiple Criteria Decision Making (MCDM), Charlottesville, Virginia, USA, June 8–12, 1998, edited by Y.Y. Haimes and R.E. Steuer, Springer, 2000, ISBN 3540672664.
2. "Tail and Concentration Inequalities" by Hung Q. Ngo
3. "Concentration-of-measure inequalities" by Gábor Lugosi
4. Cantelli, F. P. (1928), "Sui confini della probabilita," Atti del Congresso Inter- nazional del Matematici, Bologna, 6, 47-5
5. Ghosh, B.K., 2002. Probability inequalities related to Markov's theorem. The American Statistician, 56(3), pp.186-190
6. He, S.; Zhang, J.; Zhang, S. (2010). "Bounding probability of small deviation: A fourth moment approach". Mathematics of Operations Research. 35 (1): 208–232. doi:10.1287/moor.1090.0438. S2CID 11298475.