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. 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".

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


 * $$\frac{C_1}{C_2} = \frac{{G_1}/{G_2}}{{G_3}/{G_4}} = \frac{G_1 G_4}{G_2 G_3} = \frac{G_1}{G_2} \times \frac{G_4}{G_3} = C_1 \times C_2,$$

and 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 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(μX, σX2) and Y = N(μY, σY2) ratio Z = X/Y is given by the following expression:


 * $$ p_Z(z)= \frac{b(z) \cdot c(z)}{a^3(z)} \frac{1}{\sqrt{2 \pi} \sigma_x \sigma_y} \left[2 \Phi \left( \frac{b(z)}{a(z)} \right) - 1 \right] + \frac{1}{a^2(z) \cdot \pi \sigma_x \sigma_y } e^{- \frac{1}{2} \left( \frac{\mu_x^2}{\sigma_x^2} + \frac{\mu_y^2}{\sigma_y^2} \right)} $$

where


 * $$ a(z)= \sqrt{\frac{1}{\sigma_x^2} z^2 + \frac{1}{\sigma_y^2}} $$


 * $$ b(z)= \frac{\mu_x }{\sigma_x^2} z + \frac{\mu_y}{\sigma_y^2} $$


 * $$ c(z)= e^{\frac {1}{2} \frac{b^2(z)}{a^2(z)} - \frac{1}{2} \left( \frac{\mu_x^2}{\sigma_x^2} + \frac{\mu_y^2}{\sigma_y^2} \right)} $$


 * $$ \Phi(z)= \int_{-\infty}^{z}\, \frac{1}{\sqrt{2 \pi}} e^{- \frac{1}{2} u^2}\ du\ $$

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 μX = μY = 0, and σX = σY = 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.

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 :
 * $$t = \frac{\mu_y z - \mu_x}{\sqrt{\sigma_y^2 z^2 - 2\rho \sigma_x \sigma_y z + \sigma_x^2}}$$

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.,
 * $$p_X(x) = \begin{cases} 1 \qquad 0 < x < 1 \\ 0 \qquad \mbox{otherwise}\end{cases}$$

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 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


 * $$p_Z(z|a) = \frac{a^2}{\pi^2(z^2-a^4)} \ln \left(\frac{z^2}{a^4}\right).$$

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
 * $$\Lambda = {|\mathbf{X}|/|\mathbf{X}+\mathbf{Y}|} $$

has a Wilks' lambda distribution.