Ratio distribution

A ratio distribution (or "quotient distribution") is a statistical distribution constructed as the distribution of the ratio of random variables having two other distributions.Given two stochastic variables "X" and "Y", the distribution of the stochastic variable "Z" that is formed as the ratio

: $Z\; =\; X/Y,$

is a "ratio distribution".

The Cauchy distribution is an example of a ratio distribution. The random variable associated with this distribution comes about as the ratio of two Gaussian distributed variables with zero mean. Thus the Cauchy distribution is also called the "normal ratio distribution".A number of researchers have considered more general ratio distributions.Cite journal
title = The Frequency Distribution of the Quotient of Two Normal Variates
author = R. C. Geary
journal = Journal of the Royal Statistical Society
volume = 93
issue = 3
year = 1930
pages = 442–446
doi = 10.2307/2342070
url = http://links.jstor.org/sici?sici=0952-8385(1930)93%3A3%3C442%3ATFDOTQ%3E2.0.CO%3B2-V] [Cite journal
title = The Distribution of the Index in a Normal Bivariate Population
author = E. C. Fieller
journal = Biometrika
volume = 24
issue = 3/4
month = November
year = 1932
pages = 428–440
doi = 10.2307/2331976
url = http://biomet.oxfordjournals.org/cgi/content/citation/24/3-4/428 ] Cite journal
author = J. H. Curtiss
title = On the Distribution of the Quotient of Two Chance Variables
journal = The Annals of Mathematical Statistics
volume = 12
issue = 4
month = December
year = 1941
pages = 409–421
url = http://links.jstor.org/sici?sici=0003-4851(194112)12%3A4%3C409%3AOTDOTQ%3E2.0.CO%3B2-O
doi = 10.1214/aoms/1177731679 ] [George Marsaglia (April 1964). " [http://stinet.dtic.mil/oai/oai?&verb=getRecord&metadataPrefix=html&identifier=AD0600972 Ratios of Normal Variables and Ratios of Sums of Uniform Variables] ". Defense Technical Information Center.] [Cite journal
author = George Marsaglia
title = Ratios of Normal Variables and Ratios of Sums of Uniform Variables
journal = Journal of the American Statistical Association
volume = 60
issue = 309
month = March
year = 1965
pages = 193–204
doi = 10.2307/2283145
url = http://links.jstor.org/sici?sici=0162-1459(196503)60%3A309%3C193%3ARONVAR%3E2.0.CO%3B2-2] Cite journal
author = D. V. Hinkley
title = On the Ratio of Two Correlated Normal Random Variables
journal = Biometrika
volume = 56
issue = 3
month = December
year = 1969
pages = 635–639
doi = 10.2307/2334671
url = http://links.jstor.org/sici?sici=0006-3444(196912)56%3A3%3C635%3AOTROTC%3E2.0.CO%3B2-2] Cite journal
author = Jack Hayya, Donald Armstrong and Nicolas Gressis
title = A Note on the Ratio of Two Normally Distributed Variables
journal = Management Science
year = 1975
volume = 21
issue = 11
pages = 1338–1341
month = July
url = http://links.jstor.org/sici?sici=0025-1909(197507)21%3A11%3C1338%3AANOTRO%3E2.0.CO%3B2-2] Cite book
author = Melvin Dale Springer
title = The Algebra of Random Variables
publisher = Wiley
year = 1979
isbn = 0-471-01406-0] Cite journal
journal =
publisher = Taylor & Francis
volume = 35
issue = 9
year = 2006
pages = 1569–1591
doi = 10.1080/03610920600683689
title = Density of the Ratio of Two Normal Random Variables and Applications
author = T. Pham-Gia, N. Turkkan and E. Marchand] Two distribution often used in test-statistics, the "t"-distribution and the "F"-distribution, are also ratio distributions: The "t"-distributed random variable is the ratio of a Gaussian random variable divided by an independent chi-distributed random variable, while the "F"-distributed random variable is the ratio of two independent chi-square distributed random variables.

Often the ratio distributions are heavy-tailed, and it may be difficult to work with such distributions and develop an associated statistical test.A method based on the median has been suggested as a "work-around" [Cite journal
title = Significance and statistical errors in the analysis of DNA microarray data
author = James P. Brody, Brian A. Williams, Barbara J. Wold, and Stephen R. Quake
journal = Proc Natl Acad Sci U S A
year = 2002
month = October
volume = 99
issue = 20
pages = 12975–12978
doi = 10.1073/pnas.162468199] .

Algebra of random variables

The ratio is one type of algebra for random variables:Related to the ratio distribution are the product distribution, sum distribution and difference distribution. More general, one may talk of combinations of sum, differences, products and ratios.Many of these distributions are described in Melvin D. Springer's book from 1979 "The Algebra of Random Variables".

The algebraic rules known with ordinary numbers do not apply for the algebra of random variables.For example, if a product is "C = AB" and a ratio is "D=C/A" it does not necessarily mean that the distributions of "D" and "B" are the same. Indeed, a peculiar effect is seen for the Cauchy distribution: The product and the ratio of two independent Cauchy distributions (with the same scale parameter and the location parameter set to zero) will give the same distribution.This becomes evident when regarding the Cauchy distribution as itself a ratio distribution of two Gaussian distributions: Consider two Cauchy random variables, $C\_1$ and $C\_2$ each constructed from two Gaussian distributions $C\_1=G\_1/G\_2$ and $C\_2\; =\; G\_3/G\_4$ then

: $frac\{C\_1\}\{C\_2\}\; =\; frac$G_1}/{G_2G_3}/{G_4 = frac{G_1 G_4}{G_2 G_3} = frac{G_1}{G_2} imes frac{G_4}{G_3} = C_1 imes C_3,

where $C\_3\; =\; G\_4/G\_3$. The first term is the ratio of two Cauchy distributions while the last term is the product of two such distributions.

Derivation

A way of deriving the ratio distribution of "Z" from the joint distribution of the two other stochastic variables, "X" and "Y", is by integration of the following form

: $p\_Z(z)\; =\; int^\{+infty\}\_\{-infty\}\; |y|,\; p\_\{X,Y\}(zy,\; y)\; ,\; dy.$

This is not always straightforward.

The Mellin transform has also been suggested for derivation of ratio distributions.

Gaussian ratio distribution

When "X" and "Y" are independent and have a Gaussian distribution with zero mean the form of their ratio distribution is fairly simple: It is a Cauchy distribution.However, when the two distributions have non-zero mean then the form for the distribution of the ratio is much more complicated. In 1969 David Hinkley found a form for this distribution. In the absence of correlation (cor("X","Y") = 0), the probability density function of the two normal variable "X" = "N"("μ_X", "σ_X"²) and "Y" = "N"("μ_Y", "σ_Y"²) ratio "Z" = "X"/"Y" is given by the following expression:

: $p\_Z(z)=\; frac\{b(z)\; cdot\; c(z)\}\{a^3(z)\}\; frac\{1\}\{sqrt\{2\; pi\}\; sigma\_x\; sigma\_y\}\; left\; [2\; Phi\; left(\; frac\{b(z)\}\{a(z)\}\; ight)\; -\; 1\; ight]\; +\; frac\{1\}\{a^2(z)\; cdot\; pi\; sigma\_x\; sigma\_y\; \}\; e^\{-\; frac\{1\}\{2\}\; left(\; frac\{mu\_x^2\}\{sigma\_x^2\}\; +\; frac\{mu\_y^2\}\{sigma\_y^2\}\; ight)\}$

where

: $a(z)=\; sqrt\{frac\{1\}\{sigma\_x^2\}\; z^2\; +\; frac\{1\}\{sigma\_y^2$

: $b(z)=\; frac\{mu\_x\; \}\{sigma\_x^2\}\; z\; +\; frac\{mu\_y\}\{sigma\_y^2\}$

: $c(z)=\; e^\{frac\; \{1\}\{2\}\; frac\{b^2(z)\}\{a^2(z)\}\; -\; frac\{1\}\{2\}\; left(\; frac\{mu\_x^2\}\{sigma\_x^2\}\; +\; frac\{mu\_y^2\}\{sigma\_y^2\}\; ight)\}$

: $Phi(z)=\; int\_\{-infty\}^\{z\},\; frac\{1\}\{sqrt\{2\; pi\; e^\{-\; frac\{1\}\{2\}\; u^2\}\; du$

The above expression becomes even more complicated if the variables "X" and "Y" are correlated.It can also be shown that "p"("z") is a standard Cauchy distribution if "μ_X" = "μ_Y" = 0, and "σ_X" = "σ_Y" = 1. In such case "b"("z") = 0, and : $p(z)=\; frac\{1\}\{pi\}\; frac\{1\}\{1\; +\; z^2\}$

If $sigma\_X\; eq\; 1$, $sigma\_Y\; eq\; 1$ or $ho\; eq\; 0$ the more general Cauchy distribution is obtained

:

where ρ is the correlation coefficient between "X" and "Y" and

: $alpha\; =\; ho\; frac\{sigma\_x\}\{sigma\_y\},$

:

The complex distribution has also been expressed with Kummer's confluent hypergeometric function or the Hermite function.

A transformation to Gaussianity

A transformation has been suggested so that, under certain assumptions, the transformed variable "T" would approximately have a standard Gaussian distribution:: $t\; =\; frac\{mu\_y\; z\; -\; mu\_x\}\{sqrt\{sigma\_y^2\; z^2\; -\; 2\; ho\; sigma\_x\; sigma\_y\; z\; +\; sigma\_x^2$The transformation has been called the Geary-Hinkley transformation, and the approximation is good if "Y" is unlikely to assume negative values.

Uniform ratio distribution

With two random variables following a uniform distribution, e.g., : the ratio distribution becomes:

Cauchy ratio distribution

If two random variables, "X" and "Y" follows a Cauchy distribution: $p\_X(x|a)\; =\; frac\{a\}\{pi\; (a^2\; +\; x^2)\}$then the ratio distribution for the random variable $Z\; =\; X/Y$ is

: $p\_Z(z|a)\; =\; frac\{a^2\}\{pi^2(z^2-a^4)\}\; ln\; left(frac\{z^2\}\{a^4\}\; ight).$

This is also the product distribution of the random variable $W=XY.$

Ratio distributions in multivariate analysis

Ratio distributions also appear in multivariate analysis. If the random matrices X and Y follow a Wishart distribution then the ratio of the determinants

: $phi\; =\; |mathbf\{X\}|/|mathbf\{Y\}|$

is proportional to the product of independent F random variables. In the case where X and Y are from independent standardized Wishart distributions then the ratio : $Lambda\; =$has a Wilks' lambda distribution.

References

* Eric Weisstein and others, [http://mathworld.wolfram.com/RatioDistribution.html Ratio Distribution] , MathWorld.
* Eric Weisstein and others, [http://mathworld.wolfram.com/NormalRatioDistribution.html Normal Ratio Distribution] , MathWorld.
* [http://www.mathpages.com/home/kmath042/kmath042.htm Ratio Distributions] at MathPages

Notes