A ratio distribution (also known as a 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.
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".
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 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.
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 y extends from x = 0 to x = Ry and has incremental probability .
Secondly, integrating the horizontal strips upward over all y yields the volume of probability above the line
Finally, differentiate with respect to to get the pdf .
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 pdf 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 pdf . 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.
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
When X and Y 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 r then is uniformly distributed on with density so the problem reduces to finding the probability distribution of Z under the mapping
We have, by conservation of probability
and setting we get
There is a spurious factor of 2 here. Actually, two values of spaced by map onto the same value of z, 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 Student t in which the density depends only on radius . It does not extend to the ratio of two independent Student t distributions which give the Cauchy ratio shown in a section below for one degree of freedom.
In the absence of correlation , the probability density function of the two normal variables X = N(μX, σX2) and Y = N(μY, σY2) ratio Z = X/Y is given exactly by the following expression, derived in several sources:
Approximations to correlated noncentral normal ratio
A transformation to the log domain was suggested by Katz(1978) (see binomial section below). Let the ratio be
Take logs to get
Since then asymptotically
Alternatively, Geary (1930) suggested that
has approximately a standard Gaussian distribution:
This transformation has been called the Geary–Hinkley transformation; the approximation is good if Y is unlikely to assume negative values, basically .
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 used with modern math packages and similar problems may occur in some of Marsaglia's equations. Pham-Ghia has exhaustively discussed these methods. Hinkley's correlated results are exact but it is shown below that the correlated ratio condition can be transformed simply 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
is invariant under this transformation and retains the same pdf.
The term in the numerator is made separable by expanding:
in which and z has now become a ratio of uncorrelated non-central normal samples with an invariant z-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 .
Contours of the correlated bivariate Gaussian distribution (not to scale) giving ratio x/y
pdf of the Gaussian ratio z and a simulation (points) for
The figures above show an example of a positively correlated ratio with 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 easily understood that for a ratio the wedge almost bypasses the distribution mass altogether and this coincides with a near-zero region in the theoretical pdf. Conversely as reduces toward zero the line collects a higher probability.
This transformation will be recognized as being the same as that used by Geary (1932) as a partial result in his eqn viii but whose derivation and limitations were hardly explained. Thus the first part of Geary's transformation to approximate Gaussianity in the previous section is actually exact and not dependent on the positivity of Y. The offset result is also consistent with the "Cauchy" correlated zero-mean Gaussian ratio distribution in the first section. Marsaglia has applied the same result but using a nonlinear method to achieve it.
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.[note 1]
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.
If two independent random variables, X and Y 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 (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 :
defines , Fisher's F density distribution, the PDF of the ratio of two Chi-squares with m, n degrees of freedom.
The CDF of the Fisher density, found in F-tables is defined in the beta prime distribution article.
If we enter an F-test table with m = 3, n = 4 and 5% probability in the right tail, the critical value is found to be 6.59. This coincides with the integral
which includes the regular gamma, chi, chi-squared, exponential, Rayleigh, Nakagami and Weibull distributions involving fractional powers. Note that here a is a scale parameter, rather than a rate parameter; d is a shape parameter.
This result was first derived by Katz et al. in 1978.
Suppose X ~ Binomial(n,p1) and Y ~ Binomial(m,p2) and X, Y are independent. Let T = (X/n)/(Y/m).
Then log(T) is approximately normally distributed with mean log(p1/p2) and variance ((1/p1) − 1)/n + ((1/p2) − 1)/m.
The binomial ratio distribution is of significance in clinical trials: if the distribution of T is known as above, the probability of a given ratio arising purely by chance can be estimated, i.e. a false positive trial. A number of papers compare the robustness of different approximations for the binomial ratio.
In the ratio of Poisson variables R = X/Y there is a problem that Y is zero with finite probability so R is undefined. To counter this, we consider the truncated, or censored, ratio R' = X/Y' where zero sample of Y are discounted. Moreover, in many medical-type surveys, there are systematic problems with the reliability of the zero samples of both X and Y and it may be good practice to ignore the zero samples anyway.
The probability of a null Poisson sample being , the generic pdf of a left truncated Poisson distribution is
which sums to unity. Following Cohen, for n independent trials, the multidimensional truncated pdf is
and the log likelihood becomes
On differentiation we get
and setting to zero gives the maximum likelihood estimate
Note that as then so the truncated maximum likelihood estimate, though correct for both truncated and untruncated distributions, gives a truncated mean value which is highly biassed relative to the untruncated one. Nevertheless it appears that is a sufficient statistic for since depends on the data only through the sample mean in the previous equation which is consistent with the methodology of the conventional Poisson distribution.
Absent any closed form solutions, the following approximate reversion for truncated is valid over the whole range .
which compares with the non-truncated version which is simply . Taking the ratio is a valid operation even though may use a non-truncated model while has a left-truncated one.
Then substituting from the equation above, we get Cohen's variance estimate
The variance of the point estimate of the mean , on the basis of n trials, decreases asymptotically to zero as n increases to infinity. For small it diverges from the truncated pdf variance in Springael for example, who quotes a variance of
for n samples in the left-truncated pdf shown at the top of this section. Cohen showed that the variance of the estimate relative to the variance of the pdf, , ranges from 1 for large (100% efficient) up to 2 as approaches zero (50% efficient).
These mean and variance parameter estimates, together with parallel estimates for X, can be applied to Normal or Binomial approximations for the Poisson ratio. Samples from trials may not be a good fit for the Poisson process; a further discussion of Poisson truncation is by Dietz and Bohning and there is a Zero-truncated Poisson distribution Wikipedia entry.
Ratios of Quadratic Forms involving Wishart Matrices
Probability distribution can be derived from random quadratic forms
where and/or are random. If A is the inverse of another matrix B then is a random ratio in some sense, frequently arising in Least Squares estimation problems.
In the Gaussian case if A is a matrix drawn from a complex Wishart distribution of dimensionality p x p and k degrees of freedom with while is an arbitrary complex vector with Hermitian (conjugate) transpose , the ratio
follows the Gamma distribution
The result arises in least squares adaptive Wiener filtering - see eqn(A13) of. Note that the original article contends that the distribution is .
Similarly, for full-rank ( zero-mean real-valued Wishart matrix samples
, and V a random vector independent of W, the ratio
This result is usually attributed to Muirhead (1982).
Given complex Wishart matrix , the ratio
follows the Beta distribution (see eqn(47) of)
The result arises in the performance analysis of constrained least squares filtering and derives from a more complex but ultimately equivalent ratio that if then
In its simplest form, if and then the ratio of the (1,1) inverse element squared to the sum of modulus squares of the whole top row elements has distribution
^Note, however, that and can be individually log-normally distributed without having a bivariate log-normal distribution. As of 2022-06-08 the Wikipedia article on "Copula (probability theory)" includes a density and contour plot of two Normal marginals joint with a Gumbel copula, where the joint distribution is not bivariate normal.
^Díaz-Francés, Eloísa; Rubio, Francisco J. (2012-01-24). "On the existence of a normal approximation to the distribution of the ratio of two independent normal random variables". Statistical Papers. Springer Science and Business Media LLC. 54 (2): 309–323. doi:10.1007/s00362-012-0429-2. ISSN0932-5026. S2CID122038290.
^Of course, any invocation of a central limit theorem assumes suitable, commonly met regularity conditions, e.g., finite variance.
^ abcKermond, John (2010). "An Introduction to the Algebra of Random Variables". Mathematical Association of Victoria 47th Annual Conference Proceedings – New Curriculum. New Opportunities. The Mathematical Association of Victoria: 1–16. ISBN978-1-876949-50-1.
^"SLAPPF". Statistical Engineering Division, National Institute of Science and Technology. Retrieved 2009-07-02.
^Hamedani, G. G. (Oct 2013). "Characterizations of Distribution of Ratio of Rayleigh Random Variables". Pakistan Journal of Statistics. 29 (4): 369–376.