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Wikipedia / Chebyshev's inequality

In probability theory, Chebyshev's inequality (also called the Bienaymé–Chebyshev inequality) provides an upper bound on the probability of deviation of a random variable is the standard deviation (the square root of the variance).

The rule is often called Chebyshev's theorem, about the range of standard deviations around the mean, in statistics. The inequality has great utility because it can be applied to any probability distribution in which the mean and variance are defined. For example, it can be used to prove the weak law of large numbers.

Its practical usage is similar to the 68–95–99.7 rule, which applies only to normal distributions. Chebyshev's inequality is more general, stating that a minimum of just 75% of values must lie within two standard deviations of the mean and 88.88% within three standard deviations for a broad range of different probability distributions.

The term Chebyshev's inequality may also refer to Markov's inequality, especially in the context of analysis. They are closely related, and some authors refer to Markov's inequality as "Chebyshev's First Inequality," and the similar one referred to on this page as "Chebyshev's Second Inequality."

Chebyshev's inequality is tight in the sense that for each chosen positive constant, there exists a random variable such that the inequality is in fact an equality.

History

The theorem is named after Russian mathematician Pafnuty Chebyshev, although it was first formulated by his friend and colleague Irénée-Jules Bienaymé. The theorem was first proved by Bienaymé in 1853 and more generally proved by Chebyshev in 1867. His student Andrey Markov provided another proof in his 1884 Ph.D. thesis.

Statement

Chebyshev's inequality is usually stated for random variables, but can be generalized to a statement about measure spaces.

Probabilistic statement

(integrable) be a random variable with finite non-zero variance (and thus finite expected value). Then for any real number,

and the inequality is trivial as all probabilities are at most 1.

. Equivalently, it implies that the probability of values lying within the interval (i.e. its "coverage") is at least.

Because it can be applied to completely arbitrary distributions provided they have a known finite mean and variance, the inequality generally gives a poor bound compared to what might be deduced if more aspects are known about the distribution involved.

Modification with strict inequality signs

respectively.

Measure-theoretic statement

be a measure space be an extended real-valued measurable function,

then:

This statement follows from the Markov inequality otherwise.

Example

chance to be outside that range, by Chebyshev's inequality. But if we additionally know that the distribution is normal, we can say there is a 75% chance the word count is between 770 and 1230 (which is an even tighter bound).

Sharpness of bounds

and

Chebyshev's inequality is an equality for precisely those distributions which are affine transformations of this example.

Proof

Markov's inequality:

It can also be proved directly using conditional expectation:

can be quite poor.

Chebyshev's inequality can also be obtained directly from a simple comparison of areas, starting from the representation of an expected value as the difference of two improper Riemann integrals (at the drawing in the definition of expected value for arbitrary real-valued random variables).

Extensions

Several extensions of Chebyshev's inequality have been developed.

Selberg's inequality

Selberg derived a generalization to arbitrary intervals.. Selberg's inequality states,

, this reduces to Chebyshev's inequality. These are known to be the best possible bounds.

Finite-dimensional vector

. Then the following inequality holds.

This is known as the Birnbaum–Raymond–Zuckerman inequality after the authors who proved it for two dimensions. This result can be rewritten in terms of vectors.

One can also get a similar infinite-dimensional Chebyshev's inequality. A second related inequality has also been derived by Chen. be the dimension be the covariance matrix. Then

is the transpose. The inequality can be written in terms of the Mahalanobis distance as

is defined by

Navarro.

Stellato et al. showed that this multivariate version of the Chebyshev inequality can be easily derived analytically as a special case of Vandenberghe et al. where the bound is computed by solving a semidefinite program (SDP).

Known correlation

If the variables are independent this inequality can be sharpened.

.. Then

This result can be sharpened to having different bounds for the two random variables and having asymmetric bounds, as in Selberg's inequality.

correlated variables.

variables and

Olkin and Pratt's inequality was subsequently generalised by Godwin.

Higher moments

Mitzenmacher and Upfal, one can get a family of tail bounds

th moment exists, this bound is tighter than Chebyshev's inequality. This strategy, called the method of moments, is often used to prove tail bounds.

Exponential moment

A related inequality sometimes known as the exponential Chebyshev's inequality is the inequality

be the cumulant generating function,

Taking the Legendre–Fenchel transformation and using the exponential Chebyshev's inequality we have

This inequality may be used to obtain exponential inequalities for unbounded variables.

Bounded variables

is the absolute value

Finite samples

Univariate case

Saw et al samples are to be employed to bound the expected value of a new drawing from the same distribution. The following simpler version of this inequality is given by Kabán.

is the sample standard deviation.

This inequality holds even when the population moments do not exist, and when the sample is only weakly exchangeably) has been determined by Konijn., the 95 percent confidence interval (this is 2.28 times larger than the value found on the assumption of normality showing the loss on precision resulting from ignorance of the precise nature of the distribution).

An equivalent inequality can be derived in terms of the sample mean instead,

) has been determined by Konijn.

the Saw–Yang–Mo inequality is approximately

Beasley et al have suggested a modification of this inequality

In empirical testing this modification is conservative but appears to have low statistical power. Its theoretical basis currently remains unexplored.

Dependence on sample size

. Chebyshev's inequality states that at most approximately 11.11% of the distribution will lie at least three standard deviations away from the mean. Kabán's version of the inequality for a finite sample states that at most approximately 12.05% of the sample lies outside these limits. The dependence of the confidence intervals on sample size is further illustrated below.

, the 95% confidence interval is approximately ±13.5789 standard deviations.

the 95% confidence interval is approximately ±4.9595 standard deviations; the 99% confidence interval is approximately ±14.00 standard deviations.

the 95% confidence interval is approximately ±4.5574 standard deviations; the 99% confidence interval is approximately ±11.1620 standard deviations.

the 95% and 99% confidence intervals are approximately ±4.5141 and approximately ±10.5330 standard deviations respectively.

The Chebyshev inequality for the distribution gives 95% and 99% confidence intervals of approximately ±4.472 standard deviations and ±10 standard deviations respectively.

Samuelson's inequality

Although Chebyshev's inequality is the best possible bound for an arbitrary distribution, this is not necessarily true for finite samples. Samuelson's inequality sample standard deviations of the mean.

standard deviations of the mean, which is worse than Samuelson's inequality. However, the benefit of Chebyshev's inequality is that it can be applied more generally to get confidence bounds for ranges of standard deviations that do not depend on the number of samples.

Semivariances

An alternative method of obtaining sharper bounds is through the use of semivariances) semivariances are defined as

is the arithmetic mean is the number of elements in the sample.

The variance of the sample is the sum of the two semivariances:

In terms of the lower semivariance Chebyshev's inequality can be written

Putting

Chebyshev's inequality can now be written

A similar result can also be derived for the upper semivariance.

If we put

Chebyshev's inequality can be written

, use of the semivariance sharpens the original inequality.

If the distribution is known to be symmetric, then

and

This result agrees with that derived using standardised variables.

Note: The inequality with the lower semivariance has been found to be of use in estimating downside risk in finance and agriculture.

Multivariate case

Stellato et al. simplified the notation and extended the empirical Chebyshev inequality from Saw et al. then

Remarks

, this inequality corresponds to the one from Saw et al. Moreover, the right-hand side can be simplified by upper bounding the floor function by its argument

which corresponds to the multivariate Chebyshev inequality.

Sharpened bounds

Chebyshev's inequality is important because of its applicability to any distribution. As a result of its generality it may not (and usually does not) provide as sharp a bound as alternative methods that can be used if the distribution of the random variable is known. To improve the sharpness of the bounds provided by Chebyshev's inequality a number of methods have been developed; for a review see eg.

Cantelli's inequality

Cantelli's inequality due to Francesco Paolo Cantelli)

that takes the values

An application: distance between the mean and the median

The one-sided variant can be used to prove the proposition that for probability distributions having an expected value and a median, the mean and the median can never differ from each other by more than one standard deviation be respectively the mean, the median, and the standard deviation. Then

There is no need to assume that the variance is finite because this inequality is trivially true if the variance is infinite.

in the statement for the one-sided inequality gives:

, we get

that satisfies the inequalities

this implies that the median lies within one standard deviation of the mean. A proof using Jensen's inequality also exists.

Bhattacharyya's inequality

Bhattacharyya extended Cantelli's inequality using the third and fourth moments of the distribution.

then

to be reasonably large.

this simplifies to

wins a factor 2 over Chebyshev's inequality.

Gauss's inequality

In 1823 Gauss showed that for a distribution with a unique mode at zero,

Vysochanskij–Petunin inequality

The Vysochanskij–Petunin inequality generalizes Gauss's inequality, which only holds for deviation from the mode of a unimodal distribution, to deviation from the mean, or more generally, any center. is a unimodal distribution, then the inequality states that

For symmetrical unimodal distributions, the median and the mode are equal, so both the Vysochanskij–Petunin inequality and Gauss's inequality apply to the same center. Further, for symmetrical distributions, one-sided bounds can be obtained by noticing that

present in these tail bounds lead to better confidence intervals than Chebyshev's inequality. For example, for any symmetrical unimodal distribution, the Vysochanskij–Petunin inequality states that 4/(9 × 3^2) = 4/81 ≈ 4.9% of the distribution lies outside 3 standard deviations of the mode.

Bounds for specific distributions

DasGupta has shown that if the distribution is known to be normal

From DasGupta's inequality it follows that for a normal distribution at least 95% lies within approximately 2.582 standard deviations of the mean. This is less sharp than the true figure (approximately 1.96 standard deviations of the mean).

DasGupta has determined a set of best possible bounds for a normal distribution for this inequality.
Steliga and Szynal have extended these bounds to the Pareto distribution.
Grechuk et al. developed a general method for deriving the best possible bounds in Chebyshev's inequality for any family of distributions, and any deviation risk measure in place of standard deviation. In particular, they derived Chebyshev inequality for distributions with log-concave densities.

Related inequalities

Several other related inequalities are also known.

Paley–Zygmund inequality

The Paley–Zygmund inequality gives a lower bound on tail probabilities, as opposed to Chebyshev's inequality which gives an upper bound. Applying it to the square of a random variable, we get

Haldane's transformation

One use of Chebyshev's inequality in applications is to create confidence intervals for variates with an unknown distribution. Haldane noted, using an equation derived by Kendall,) has a zero mean, unit variance and both finite skewness) and kurtosis) then the variate can be converted to a normally distributed standard score):

This transformation may be useful as an alternative to Chebyshev's inequality or as an adjunct to it for deriving confidence intervals for variates with unknown distributions.

While this transformation may be useful for moderately skewed and/or kurtotic distributions, it performs poorly when the distribution is markedly skewed and/or kurtotic.

He, Zhang and Zhang's inequality

For any collection of non-negative independent random variables with expectation 1

Integral Chebyshev inequality

There is a second (less well known) inequality also named after Chebyshev.

are two monotonic functionss of the same monotonicity, then

are of opposite monotonicity, then the above inequality works in the reverse way.

This inequality is related to Jensen's inequality, Kantorovich's inequality, the Hermite–Hadamard inequality and Walter's conjecture.

Other inequalities

There are also a number of other inequalities associated with Chebyshev:

Notes

The Environmental Protection Agency has suggested best practices for the use of Chebyshev's inequality for estimating confidence intervals.

References

External links

Formal proof in the Mizar system.