Ratio distribution

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.

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 t-distribution 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.[1][2][3][4][5][6][7][8][9]

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]

  1. ^ Geary, R. C. (1930). "The Frequency Distribution of the Quotient of Two Normal Variates". Journal of the Royal Statistical Society. 93 (3): 442–446. doi:10.2307/2342070. JSTOR 2342070.
  2. ^ Fieller, E. C. (November 1932). "The Distribution of the Index in a Normal Bivariate Population". Biometrika. 24 (3/4): 428–440. doi:10.2307/2331976. JSTOR 2331976.
  3. ^ Curtiss, J. H. (December 1941). "On the Distribution of the Quotient of Two Chance Variables". The Annals of Mathematical Statistics. 12 (4): 409–421. doi:10.1214/aoms/1177731679. JSTOR 2235953.
  4. ^ George Marsaglia (April 1964). Ratios of Normal Variables and Ratios of Sums of Uniform Variables. Defense Technical Information Center.
  5. ^ Marsaglia, George (March 1965). "Ratios of Normal Variables and Ratios of Sums of Uniform Variables". Journal of the American Statistical Association. 60 (309): 193–204. doi:10.2307/2283145. JSTOR 2283145. Archived from the original on September 23, 2017.
  6. ^ Hinkley, D. V. (December 1969). "On the Ratio of Two Correlated Normal Random Variables". Biometrika. 56 (3): 635–639. doi:10.2307/2334671. JSTOR 2334671.
  7. ^ Hayya, Jack; Armstrong, Donald; Gressis, Nicolas (July 1975). "A Note on the Ratio of Two Normally Distributed Variables". Management Science. 21 (11): 1338–1341. doi:10.1287/mnsc.21.11.1338. JSTOR 2629897.
  8. ^ Springer, Melvin Dale (1979). The Algebra of Random Variables. Wiley. ISBN 0-471-01406-0.
  9. ^ Pham-Gia, T.; Turkkan, N.; Marchand, E. (2006). "Density of the Ratio of Two Normal Random Variables and Applications". Communications in Statistics – Theory and Methods. 35 (9). Taylor & Francis: 1569–1591. doi:10.1080/03610920600683689. S2CID 120891296.
  10. ^ Brody, James P.; Williams, Brian A.; Wold, Barbara J.; Quake, Stephen R. (October 2002). "Significance and statistical errors in the analysis of DNA microarray data" (PDF). Proc Natl Acad Sci U S A. 99 (20): 12975–12978. Bibcode:2002PNAS...9912975B. doi:10.1073/pnas.162468199. PMC 130571. PMID 12235357.