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Gauss's inequality

{{distinguish|text=the Gaussian correlation inequality or the Gaussian isoperimetric inequality}}

In probability theory, Gauss's inequality (or the Gauss inequality) gives an upper bound on the probability that a unimodal random variable lies more than any given distance from its mode.

Let X be a unimodal random variable with mode m, and let τ 2 be the expected value of (X − m)2. (τ 2 can also be expressed as (μ − m)2 + σ 2, where μ and σ are the mean and standard deviation of X.) Then for any positive value of k,

:

\Pr(|X - m| > k) \leq \begin{cases}

\left( \frac{2\tau}{3k} \right)^2 & \text{if } k \geq \frac{2\tau}{\sqrt{3}} \\[6pt]

1 - \frac{k}{\tau\sqrt{3}} & \text{if } 0 \leq k \leq \frac{2\tau}{\sqrt{3}}.

\end{cases}

The theorem was first proved by Carl Friedrich Gauss in 1823.

Extensions to higher-order moments

Winkler in 1866 extended Gauss's inequality to rth moments Winkler A. (1886) Math-Natur theorie Kl. Akad. Wiss Wien Zweite Abt 53, 6–41 where r > 0 and the distribution is unimodal with a mode of zero. This is sometimes called Camp–Meidell's inequality.{{Cite journal|last=Pukelsheim|first=Friedrich|date=May 1994|title=The Three Sigma Rule|url=http://www.tandfonline.com/doi/abs/10.1080/00031305.1994.10476030|journal=The American Statistician|language=en|volume=48|issue=2|pages=88–91|doi=10.1080/00031305.1994.10476030|issn=0003-1305}}{{cite journal |last1=Bickel |first1=Peter J. |author-link1=Peter J. Bickel |last2=Krieger |first2=Abba M. |title=Extensions of Chebyshev's Inequality with Applications |journal=Probability and Mathematical Statistics |year=1992 |volume=13 |issue=2 |pages=293–310 |url=http://www.math.uni.wroc.pl/~pms/files/13.2/Article/13.2.11.pdf |access-date=6 October 2012 |issn=0208-4147}}

: P( | X | \ge k ) \le \left( \frac{ r } { r + 1 } \right)^r \frac{ \operatorname{ E }( | X | )^r } { k^r } \quad \text{if} \quad k^r \ge \frac{ r^r } { ( r + 1 )^{ r + 1 } } \operatorname{ E }( | X |^r ),

: P( | X | \ge k) \le \left( 1 - \left[ \frac{ k^r }{ ( r + 1 ) \operatorname{ E }( | X | )^r } \right]^{ 1 / r } \right) \quad \text{if} \quad k^r \le \frac{r^r} { (r + 1)^{r + 1} } \operatorname{E}( | X |^r ).

Gauss's bound has been subsequently sharpened and extended to apply to departures from the mean rather than the mode due to the Vysochanskiï–Petunin inequality. The latter has been extended by Dharmadhikari and Joag-Dev{{cite journal | last1 = Dharmadhikari | first1 = S. W. | last2 = Joag-Dev | first2 = K. | year = 1985 | title = The Gauss–Tchebyshev inequality for unimodal distributions | url = http://www.mathnet.ru/links/1043b08ee307d37ab64be90bd9332b88/tvp1919.pdf| journal = Teoriya Veroyatnostei i ee Primeneniya | volume = 30 | issue = 4| pages = 817–820 }}

: P( | X | > k ) \le \max\left( \left[ \frac r {( r + 1 ) k } \right]^r E| X^r |, \frac s {( s - 1 ) k^r } E| X^r | - \frac 1 { s - 1 } \right)

where s is a constant satisfying both s > r + 1 and s(s − r − 1) = rr and r > 0.

It can be shown that these inequalities are the best possible and that further sharpening of the bounds requires that additional restrictions be placed on the distributions.

See also

References

{{Reflist}}

  • {{cite journal|last=Gauss|first=C. F.|authorlink=Carl Friedrich Gauss|date=1823|title=Theoria Combinationis Observationum Erroribus Minimis Obnoxiae, Pars Prior|journal=Commentationes Societatis Regiae Scientiarum Gottingensis Recentiores|volume=5}}
  • {{cite book|title=A Dictionary of Statistics| publisher=Oxford University Press| last= Upton | first = Graham |author2=Cook, Ian| year=2008|chapter=Gauss inequality| url=http://www.answers.com/topic/gauss-inequality}}
  • {{Cite journal

| doi = 10.2307/2684690

| title = Chebyshev inequalities for unimodal distributions

| year = 1997

| journal = American Statistician

| volume = 51

| issue = 1

| pages = 34–40

| last1 = Sellke | first1 = T.M.

| last2 = Sellke | first2 = S.H.

| publisher = American Statistical Association

| jstor = 2684690

}}

  • {{Cite journal

| doi = 10.2307/2684253

| title = The Three Sigma Rule

| year = 1994

| author = Pukelsheim, F.

| journal = American Statistician

| volume = 48

| issue = 2

| pages = 88–91

| publisher = American Statistical Association

| jstor = 2684253

}}

Category:Probabilistic inequalities