Samuelson's inequality

If we let

: $\backslash overline\{x\}\; =\; \backslash frac\{x\_1+\backslash cdots+x\_n\}\{n\}$

be the sample mean and

: $s\; =\; \backslash sqrt\{\backslash frac\{1\}\{n\}\; \backslash sum\_\{i=1\}^n\; (x\_i\; -\; \backslash overline\{x\})^2\; \}$

be the standard deviation of the sample, then

: $\backslash overline\{x\}\; -\; s\backslash sqrt\{n-1\}\; \backslash le\; x\_j\; \backslash le\; \backslash overline\{x\}\; +\; s\backslash sqrt\{n-1\}\backslash qquad\; \backslash text\{for\; \}\; j\; =\; 1,\backslash dots,n.${{cite book |title=Advances in Inequalities from Probability Theory and Statistics |first1=Neil S. |last1=Barnett |first2=Sever Silvestru |last2=Dragomir |publisher=Nova Publishers |year=2008 |isbn=978-1-60021-943-6 |page=164 }}

Equality holds on the left (or right) for $x\_j$ if and only if all the n − 1 $x\_i$s other than $x\_j$ are equal to each other and greater (smaller) than $x\_j.$

If you instead define $s\; =\; \backslash sqrt\{\backslash frac\{1\}\{n-1\}\; \backslash sum\_\{i=1\}^n\; (x\_i\; -\; \backslash overline\{x\})^2\; \}$ then the inequality $\backslash overline\{x\}\; -\; s\backslash sqrt\{n-1\}\; \backslash le\; x\_j\; \backslash le\; \backslash overline\{x\}\; +\; s\backslash sqrt\{n-1\}$ still applies and can be slightly tightened to $\backslash overline\{x\}\; -\; s\backslash tfrac\{n-1\}\{\backslash sqrt\{n\}\}\; \backslash le\; x\_j\; \backslash le\; \backslash overline\{x\}\; +\; s\backslash tfrac\{n-1\}\{\backslash sqrt\{n\}\}.$

{{Main|Chebyshev's inequality#Samuelson's inequality}}

Chebyshev's inequality locates a certain fraction of the data within certain bounds, while Samuelson's inequality locates all the data points within certain bounds.

The bounds given by Chebyshev's inequality are unaffected by the number of data points, while for Samuelson's inequality the bounds loosen as the sample size increases. Thus for large enough data sets, Chebyshev's inequality is more useful.

{{expand section|date=July 2017}}

Samuelson’s inequality has several applications in statistics and mathematics. It is useful in the studentization of residuals which shows a rationale for why this process should be done externally to better understand the spread of residuals in regression analysis.

In matrix theory, Samuelson’s inequality is used to locate the eigenvalues of certain matrices and tensors.

Furthermore, generalizations of this inequality apply to complex data and random variables in a probability space.{{Cite journal |last1=JIN |first1=HONGWEI |last2=BEN´ITEZ |first2=JULIO |title=Some generalizations and probability versions of Samuelson's inequality |url=https://files.ele-math.com/articles/mia-20-01.pdf |access-date=4 September 2024 |journal=Mathematical Inequalities & Applications |date=2017 |pages=1–12 |language=English |doi=10.7153/mia-20-01}}{{Cite book |title= Paul Samuelson|chapter=Samuelson's Approach to Revealed Preference Theory: Some Recent Advances |chapter-url=https://dipot.ulb.ac.be/dspace/bitstream/2013/314701/3/working_paper.pdf |series=Remaking Economics: Eminent Post-War Economists |date=2019 |doi=10.1057/978-1-137-56812-0_9 |last1=Demuynck |first1=Thomas |last2=Hjertstrand |first2=Per |pages=193–227 |isbn=978-1-137-56811-3 }}

Samuelson was not the first to describe this relationship: the first was probably Laguerre in 1880 while investigating the roots (zeros) of polynomials.Laguerre E. (1880) Mémoire pour obtenir par approximation les racines d'une équation algébrique qui a toutes les racines réelles. Nouv Ann Math 2^e série, 19, 161–172, 193–202

Consider a polynomial with all roots real:

: $a\_0x^n\; +\; a\_1x^\{n-1\}\; +\; \backslash cdots\; +\; a\_\{n-1\}x\; +\; a\_n\; =\; 0$

Without loss of generality let $a\_0\; =\; 1$ and let

: $t\_1\; =\; \backslash sum\; x\_i$ and $t\_2\; =\; \backslash sum\; x\_i^2$

Then

: $a\_1\; =\; -\; \backslash sum\; x\_i\; =\; -t\_1$

and

:

In terms of the coefficients

: $t\_2\; =\; a\_1^2\; -\; 2a\_2$

Laguerre showed that the roots of this polynomial were bounded by

: $-a\_1\; /\; n\; \backslash pm\; b\; \backslash sqrt\{n\; -\; 1\}$

where

: $b\; =\; \backslash frac\{\backslash sqrt\{nt\_2\; -\; t\_1^2\}\}\{n\}\; =\; \backslash frac\{\backslash sqrt\{na\_1^2\; -\; a\_1^2\; -\; 2na\_2\}\}\{n\}$

Inspection shows that $-\backslash tfrac\{a\_1\}\{n\}$ is the mean of the roots and that b is the standard deviation of the roots.

Laguerre failed to notice this relationship with the means and standard deviations of the roots, being more interested in the bounds themselves. This relationship permits a rapid estimate of the bounds of the roots and may be of use in their location.

When the coefficients $a\_1$ and $a\_2$ are both zero no information can be obtained about the location of the roots, because not all roots are real (as can be seen from Descartes' rule of signs) unless the constant term is also zero.

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Category:Statistical inequalities