A ratio distribution (also known as a quotient distribution) is a probability distribution constructed as the distribution of the ratio (or quotient) of random variables having two other known distributions. Given two (usually independent) random variables and , the distribution of the random variable that is formed as the ratio is a ratio distribution.
An example is the Cauchy distribution (also called the normal ratio distribution), which comes about as the ratio of two normally distributed variables with zero mean. Two other distributions often used in test-statistics are also ratio distributions: the arises from a Gaussian random variable divided by an independent chi-distributed random variable, while the F-distribution originates from the ratio of two independent chi-squared distributed random variables. More general ratio distributions have been considered in the literature.; ; .
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 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.
The algebraic rules known with ordinary numbers do not apply for the algebra of random variables. For example, if a product is and a ratio is it does not necessarily mean that the distributions of and 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 of zero means: Consider two Cauchy random variables, and , each constructed from two Gaussian distributions and . Then
where . 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 from the joint distribution of the two other random variables , , with joint , is by integration of the following form
If the two variables are independent then and this becomes
This may not be straightforward. By way of example take the classical problem of the ratio of two standard Gaussian samples. The joint is
Defining we have
Using the known definite integral we get
which is the Cauchy distribution, or with .
The Mellin transform has also been suggested for derivation of ratio distributions.
In the case of positive independent variables, proceed as follows. The diagram shows a separable bivariate distribution which has support in the positive quadrant and we wish to find the pdf of the ratio . The hatched volume above the line represents the cumulative distribution of the function multiplied with the logical function . The density is first integrated in horizontal strips; the horizontal strip at height extends from to and has incremental probability .
Secondly, integrating the horizontal strips upward over all yields the volume of probability above the line
Finally, differentiate with respect to to get the .
Move the differentiation inside the integral:
and since
then
As an example, find the of the ratio when
We have
thus
Differentiation with respect to yields the of :
Moments of random ratios
From Mellin transform theory, for distributions existing only on the positive half-line , we have the product identity
provided , are independent. For the case of a ratio of samples like , in order to make use of this identity it is necessary to use moments of the inverse distribution. Set such that . Thus, if the moments of and can be determined separately, then the moments of can be found. The moments of are determined from the inverse of , often a tractable exercise. At simplest, .
To illustrate, let be sampled from a standard Gamma distribution whose -th moment is .
is sampled from an inverse Gamma distribution with parameter and has . The moments of this PDF are
Multiplying the corresponding moments gives
Independently, it is known that the ratio of the two Gamma samples follows the beta prime distribution:
whose moments are
Substituting we have
which is consistent with the product of moments above.
Means and variances of random ratios In the Product distribution section, and derived from Mellin transform theory (see section above), it is found that the mean of a product of independent variables is equal to the product of their means. In the case of ratios, we have
which, in terms of probability distributions, is equivalent to
Note that ; i.e., .
The variance of a ratio of independent variables is
Normal ratio distributions
Uncorrelated central normal ratio When and are independent and have a Gaussian distribution with zero mean, the form of their ratio distribution is a Cauchy distribution. This can be derived by setting , then showing that has circular symmetry. For a bivariate uncorrelated Gaussian distribution we have
If is a function only of , then is uniformly distributed on with density , so the problem reduces to finding the probability distribution of under the mapping
We have, by conservation of probability
and since ,
Setting
we get
There is a spurious factor of 2 here. Actually, two values of spaced by map onto the same value of , the density is doubled, and the final result is
When either of the two normal distributions is non-central, then the result for the distribution of the ratio is much more complicated and is given below in the succinct form presented by David Hinkley. The trigonometric method for a ratio does, however, extend to radial distributions like bivariate normals or a bivariate , in which the density depends only on radius . It does not extend to the ratio of two independent s, which give the Cauchy ratio shown in a section below for one degree of freedom.
Uncorrelated noncentral normal ratio
In the absence of correlation (), the probability density function of the ratio of two normal variables and is given exactly by the following expression, derived in several sources:
where
Under several assumptions (usually fulfilled in practical applications), it is possible to derive a highly accurate solid approximation to the . Main benefits include reduced formulae complexity, closed-form , simple defined median, and well defined error management. For the sake of simplicity introduce parameters: , , and . Then the so-called solid approximation to the uncorrelated noncentral normal ratio PDF is expressed by equation
Under certain conditions, a normal approximation is possible, with variance:
Correlated central normal ratio
The above expression becomes more complicated when the variables and are correlated. If but and , the more general Cauchy distribution is obtained,
where is the correlation coefficient between and , and
The complex distribution has also been expressed with Kummer's confluent hypergeometric function or the Hermite function.
Correlated noncentral normal ratio This was shown in Springer.
A transformation to the log domain was suggested by Katz . Let the ratio be
Take the logarithm of both sides to get
Observing that , then asymptotically
Alternatively, Geary suggested in 1930 that
has approximately a standard Gaussian distribution; This transformation has been called the Geary–Hinkley transformation. The approximation is good if is unlikely to assume negative values, i.e., .
Exact correlated noncentral normal ratio
This is developed by Springer and Hinkley. Geary showed how the correlated ratio could be transformed into a near-Gaussian form and developed an approximation for dependent on the probability of negative denominator values being vanishingly small. Fieller's later correlated ratio analysis is exact but care is needed when combining modern math packages with verbal conditions in the older literature. Pham-Ghia has exhaustively discussed these methods. Hinkley's correlated results are exact but it is shown below that the correlated ratio condition can also be transformed into an uncorrelated one so only the simplified Hinkley equations above are required, not the full correlated ratio version.
Let the ratio be:
in which are zero-mean correlated normal variables with variances and have means . Write such that become uncorrelated and has standard deviation . Then the ratio
is invariant under this transformation and retains the same .
The term in the numerator appears to be made separable by expanding
to get
in which and has now become a ratio of uncorrelated non-central normal samples with an invariant -offset.
Finally, to be explicit, the PDF of the ratio for correlated variables is found by inputting the modified parameters , and into the Hinkley equation above which returns the PDF for the correlated ratio with a constant offset on .
The figures above show an example of a positively correlated ratio with , , , and , in which the shaded wedges represent the increment of area selected by given ratio which accumulates probability where they overlap the distribution. The theoretical distribution, derived from the equations under discussion combined with Hinkley's equations, is highly consistent with a simulation result using 5,000 samples. In the top figure it is clear that for a ratio the wedge has almost bypassed the main distribution mass altogether and this explains the local minimum in the theoretical PDF . Conversely as moves either toward or away from one the wedge spans more of the central mass, accumulating a higher probability.
Complex normal ratio The ratio of correlated zero-mean circularly symmetric complex normal distributed variables was determined by Baxley et al. and has since been extended to the nonzero-mean and nonsymmetric case. In the correlated zero-mean case, the joint distribution of is
where
is an Hermitian transpose; and
The of is found to be
In the usual event that we get
Further closed-form results for the are also given.
The graph shows the pdf of the ratio of two complex normal variables with a correlation coefficient of . The PDF peak occurs at roughly the complex conjugate of a scaled down .
Ratio of log-normal
The ratio of independent or correlated log-normals is log-normal. This follows, because if and are log-normally distributed, then and are normally distributed. If they are independent or their logarithms follow a bivarate normal distribution, then the logarithm of their ratio is the difference of independent or correlated normally distributed random variables, which is normally distributed.
This is important for many applications requiring the ratio of random variables that must be positive, where joint distribution of and is adequately approximated by a log-normal. This is a common result of the multiplicative central limit theorem, also known as Gibrat's law, when is the result of an accumulation of many small percentage changes and must be positive and approximately log-normally distributed.
Uniform ratio distribution With two independent random variables following a uniform distribution, e.g.,
the ratio distribution becomes
Cauchy ratio distribution If two independent random variables, and each follow a Cauchy distribution with median equal to zero and shape factor ,
then the ratio distribution for the random variable is
This distribution does not depend on and the result stated by Springer is not correct.
The ratio distribution is similar to but not the same as the product distribution of the random variable :