The two-piece normal, binormal, or double Gaussian distribution: its origin and rediscoveries. (English) Zbl 1332.60009

Wallis, Kenneth F.

Stat. Sci. 29, No. 1, 106-112 (2014).

Summary: This paper traces the history of the two-piece normal distribution from its origin in the posthumous Collektivmasslehre of Gustav Theodor Fechner [Leipzig: W. Engelmann (1897; JFM 28.0208.01)] to its rediscoveries and generalisations. The denial of Fechner’s originality by Karl Pearson, reiterated a century later by Oscar Sheynin, is shown to be without foundation.

MSC:

60-03	History of probability theory
60E05	Probability distributions: general theory
01A55	History of mathematics in the 19th century
62E15	Exact distribution theory in statistics

Keywords:

Gustav Theodor Fechner; Gottlob Friedrich Lipps; Francis Ysidro Edgeworth; Karl Pearson; Francis Galton; Oscar Sheynin

Citations:

JFM 28.0208.01

References:

[1]	Arnold, B. C. and Groeneveld, R. A. (1995). Measuring skewness with respect to the mode. Amer. Statist. 49 34-38.
[2]	Barnard, G. A. (1989). Sophisticated theory and practice in quality improvement. Philos. Trans. R. Soc. Lond. A 327 581-589.
[3]	Barnard, G. A. (1995). Pivotal models and the fiducial argument. Internat. Statist. Rev. 63 309-323. · Zbl 0836.62002 · doi:10.2307/1403482
[4]	Blix, M. and Sellin, P. (1998). Uncertainty bands for inflation forecasts. Working Paper 65, Sveriges Riksbank, Stockholm.
[5]	Britton, E., Fisher, P. G. and Whitley, J. D. (1998). The Inflation Report projections: Understanding the fan chart. Bank of England Quarterly Bulletin 38 30-37.
[6]	Cox, D. R. (2001). Biometrika: The first 100 years. Biometrika 88 3-11. · Zbl 1073.62500 · doi:10.1093/biomet/88.1.3
[7]	De Vries, H. (1894). Ueber halbe Galton-Curven als Zeichen discontinuirlicher Variation. Berichte der Deutschen Botanischen Gesellschaft 12 197-207.
[8]	Edgeworth, F. Y. (1899). On the representation of statistics by mathematical formulae (Part III). J. Roy. Statist. Soc. 62 373-385.
[9]	Fechner, G. T. (1860). Elemente der Psychophysik . Breitkopf and Hartel, Leipzig.
[10]	Fechner, G. T. (ed. Lipps, G. F.) (1897). Kollektivmasslehre Engelmann, Leipzig.
[11]	Fernández, C. and Steel, M. F. J. (1998). On Bayesian modeling of fat tails and skewness. J. Amer. Statist. Assoc. 93 359-371. · Zbl 0910.62024 · doi:10.2307/2669632
[12]	Galton, F. (1896). Application of the method of percentiles to Mr. Yule’s data on the distribution of pauperism. J. Roy. Statist. Soc. 59 392-396.
[13]	Garvin, J. S. and McClean, S. I. (1997). Convolution and sampling theory of the binormal distribution as a prerequisite to its application in statistical process control. The Statistician 46 33-47.
[14]	Geweke, J. (1989). Bayesian inference in econometric models using Monte Carlo integration. Econometrica 57 1317-1339. · Zbl 0683.62068 · doi:10.2307/1913710
[15]	Gibbons, J. F. and Mylroie, S. (1973). Estimation of impurity profiles in ion-implanted amorphous targets using joined half-Gaussian distributions. Appl. Phys. Lett. 22 568-569.
[16]	Hald, A. (1998). A History of Mathematical Statistics from 1750 to 1930. Wiley, New York. · Zbl 0979.01012
[17]	John, S. (1982). The three-parameter two-piece normal family of distributions and its fitting. Comm. Statist. A-Theory Methods 11 879-885. · doi:10.1080/03610928208828279
[18]	Johnson, N. L. and Kotz, S. (1970). Distributions in Statistics . Wiley, New York. · Zbl 0213.21101
[19]	Johnson, N. L., Kotz, S. and Balakrishnan, N. (1994). Continuous Univariate Distributions , Vol. 1 , 2nd ed. Wiley, New York. · Zbl 0811.62001
[20]	Kimber, A. C. (1985). Methods for the two-piece normal distribution. Comm. Statist. A-Theory Methods 14 235-245. · Zbl 0562.62024 · doi:10.1080/03610928508828907
[21]	Kotz, S., Kozubowski, T. J. and Podgórski, K. (2001). The Laplace Distribution and Generalizations : A Revisit With Applications to Communications , Economics , Engineering , and Finance . Birkhäuser, Boston, MA. · Zbl 0977.62003
[22]	Ludwig, F. (1898). Die pflanzlichen Variationscurven und die Gauss’sche Wahrscheinlichkeitscurve (continuation). Botanisches Centralblatt 73 343-349.
[23]	Mudholkar, G. S. and Hutson, A. D. (2000). The epsilon-skew-normal distribution for analyzing near-normal data. J. Statist. Plann. Inference 83 291-309. · Zbl 0943.62012 · doi:10.1016/S0378-3758(99)00096-8
[24]	Pearson, K. (1895). Contributions to the mathematical theory of evolution.-II. Skew variation in homogeneous material. Phil. Trans. R. Soc. Lond. A 186 343-414.
[25]	Pearson, K. (1905). Das fehlergesetz und seine verallgemeinerungen durch Fechner und Pearson. A rejoinder. Biometrika 4 169-212. · JFM 36.0313.10
[26]	Pearson, E. S. (1936). Karl Pearson: An appreciation of some aspects of his life and work. Part I: 1857-1906. Biometrika 28 193-257. · JFM 62.1041.07
[27]	Porter, T. M. (2004). Karl Pearson : The Scientific Life in a Statistical Age . Princeton Univ. Press, Princeton, NJ. · Zbl 1069.62001
[28]	Ranke, K. E. and Greiner, A. (1904). Das Fehlergesetz und seine Verallgemeinerungen durch Fechner und Pearson in ihrer Tragweite für die Anthropologie. Archiv fur Anthropologie 2 295-332.
[29]	Runnenburg, J. T. (1978). Mean, median, mode. Stat. Neerl. 32 73-79. · Zbl 0437.62024 · doi:10.1111/j.1467-9574.1978.tb01386.x
[30]	Sheynin, O. (2004). Fechner as a statistician. British J. Math. Statist. Psych. 57 53-72. · doi:10.1348/000711004849196
[31]	Stigler, S. M. (1986). The History of Statistics : The Measurement of Uncertainty Before 1900. The Belknap Press of Harvard Univ. Press, Cambridge, MA. · Zbl 0656.62005
[32]	Stigler, S. M. (1999). Statistics on the Table : The History of Statistical Concepts and Methods . Harvard Univ. Press, Cambridge, MA. · Zbl 0997.62506
[33]	Stigler, S. M. (2008). Karl Pearson’s theoretical errors and the advances they inspired. Statist. Sci. 23 261-271. · Zbl 1327.62013 · doi:10.1214/08-STS256
[34]	Toth, Z. and Szentimrey, T. (1990). The binormal distribution: A distribution for representing asymmetrical but normal-like weather elements. J. Climate 3 128-136.
[35]	Yule, G. U. (1896a). Notes on the history of pauperism in England and Wales from 1850, treated by the method of frequency-curves; with an introduction on the method (with discussion). J. Roy. Statist. Soc. 59 318-357.
[36]	Yule, G. U. (1896b). Remarks on Mr. Galton’s note. J. Roy. Statist. Soc. 59 396-398.
[37]	Zhu, D. and Galbraith, J. W. (2010). A generalized asymmetric Student-\(t\) distribution with application to financial econometrics. J. Econometrics 157 297-305. · Zbl 1431.62222 · doi:10.1016/j.jeconom.2010.01.013