Ratio distribution

A ratio distribution (or quotient distribution) is a probability distribution constructed as the distribution of the ratio of random variables having two other known distributions. Given two (usually independent) random variables X and Y, the distribution of the random 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 (normal) 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.^[1]^[2]^[3]^[4]^[5]^[6]^[7]^[8]^[9] Two distributions 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 (i.e., the square root of a chi-squared distribution), while the F-distributed random variable is the ratio of two independent chi-squared 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".^[10]

Algebra of random variables

Main article: 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 generally, one may talk of combinations of sums, differences, products and ratios. Many of these distributions are described in Melvin D. Springer's book from 1979 The Algebra of Random Variables.^[8]

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.^[8] 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_{2}}}{{G_{3}}/{G_{4}}}}={\frac {G_{1}G_{4}}{G_{2}G_{3}}}={\frac {G_{1}}{G_{2}}}\times {\frac {G_{4}}{G_{3}}}=C_{1}\times 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 random variables, X and Y, is by integration of the following form^[3]

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.^[8]

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 means then the form for the distribution of the ratio is much more complicated. In 1932 Fieller^[2] found a form for this distribution; here it is given in the more succinct form presented by David Hinkley.^[6] In the absence of correlation (cor(X,Y) = 0), the probability density function of the two normal variable X = N(μ_X, σ_X²) and Y = N(μ_Y, σ_Y²) ratio Z = X/Y is given by the following expression:

p_{Z}(z)={\frac {b(z)\cdot d(z)}{a^{3}(z)}}{\frac {1}{{\sqrt {2\pi }}\sigma _{x}\sigma _{y}}}\left[\Phi \left({\frac {b(z)}{a(z)}}\right)-\Phi \left(-{\frac {b(z)}{a(z)}}\right)\right]+{\frac {1}{a^{2}(z)\cdot \pi \sigma _{x}\sigma _{y}}}e^{{-{\frac {c}{2}}}}

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={\frac {\mu _{x}^{2}}{\sigma _{x}^{2}}}+{\frac {\mu _{y}^{2}}{\sigma _{y}^{2}}}

d(z)=e^{{{\frac {b^{2}(z)-ca^{2}(z)}{2a^{2}(z)}}}}

And $\Phi$ is the cumulative distribution function of the Normal distribution

\Phi (t)=\int _{{-\infty }}^{{t}}\,{\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 $\sigma _{X}\neq 1$ , $\sigma _{Y}\neq 1$ or $\rho \neq 0$ the more general Cauchy distribution is obtained

p_{Z}(z)={\frac {1}{\pi }}{\frac {\beta }{(z-\alpha )^{2}+\beta ^{2}}},

where ρ is the correlation coefficient between X and Y and

\alpha =\rho {\frac {\sigma _{x}}{\sigma _{y}}},

\beta ={\frac {\sigma _{x}}{\sigma _{y}}}{\sqrt {1-\rho ^{2}}}.

The complex distribution has also been expressed with Kummer's confluent hypergeometric function or the Hermite function.^[9]

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:^[1]

t={\frac {\mu _{y}z-\mu _{x}}{{\sqrt {\sigma _{y}^{2}z^{2}-2\rho \sigma _{x}\sigma _{y}z+\sigma _{x}^{2}}}}}

The transformation has been called the Geary–Hinkley transformation,^[7] and the approximation is good if Y is unlikely to assume negative values.

Uniform ratio distribution

With two independent random variables following a uniform distribution, e.g.,

p_{X}(x)={\begin{cases}1\qquad 0<x<1\\0\qquad {\mbox{otherwise}}\end{cases}}

the ratio distribution becomes

p_{Z}(z)={\begin{cases}1/2\qquad &0<z<1\\{\frac {1}{2z^{2}}}\qquad &z\geq 1\\0\qquad &{\mbox{otherwise}}\end{cases}}

Cauchy ratio distribution

If two independent random variables, X and Y each follow a Cauchy distribution with median equal to zero and shape factor $a$

p_{X}(x|a)={\frac {a}{\pi (a^{2}+x^{2})}}

then the ratio distribution for the random variable $Z=X/Y$ is ^[11]

p_{Z}(z|a)={\frac {1}{\pi ^{2}(z^{2}-1)}}\ln \left(z^{2}\right).

This distribution does not depend on $a$ and the result stated by Springer ^[8] (p158 Question 4.6) is not correct. The ratio distribution is similar to but not the same as the product distribution of the random variable $W=XY$ :

p_{W}(w|a)={\frac {a^{2}}{\pi ^{2}(w^{2}-a^{4})}}\ln \left({\frac {w^{2}}{a^{4}}}\right).

^[8]

More generally, if two independent random variables X and Y each follow a Cauchy distribution with median equal to zero and shape factor $a$ and $b$ respectively, then:

1. The ratio distribution for the random variable $Z=X/Y$ is ^[11]

p_{Z}(z|a,b)={\frac {ab}{\pi ^{2}(b^{2}z^{2}-a^{2})}}\ln \left({\frac {b^{2}z^{2}}{a^{2}}}\right).

2. The product distribution for the random variable $W=XY$ is ^[11]

p_{W}(w|a,b)={\frac {ab}{\pi ^{2}(w^{2}-a^{2}b^{2})}}\ln \left({\frac {w^{2}}{a^{2}b^{2}}}\right).

The result for the ratio distribution can be obtained from the product distribution by replacing $b$ with ${\frac {1}{b}}.$

Ratio of standard normal to standard uniform

Main article: Slash distribution

If X has a standard normal distribution and Y has a standard uniform distribution, then Z = X / Y has a distribution known as the slash distribution, with probability density function

p_{Z}(z)={\begin{cases}\left[\phi (0)-\phi (z)\right]/z^{2}\quad &z\neq 0\\\phi (0)/2\quad &z=0,\\\end{cases}}

where φ(z) is the probability density function of the standard normal distribution.^[12]

Other ratio distributions

Let X be a normal(0,1) distribution, Y and Z be a chi square distributions with m and n degrees of freedom respectively. Then

{\frac {X}{{\sqrt {Y/m}}}}=t_{m}

{\frac {Y/m}{Z/n}}=F_{{m,n}}

{\frac {Y}{Y+Z}}=\beta (m/2,n/2)

where t_m is Student's t distribution, $F$ is the F distribution and $\beta$ is the beta distribution.