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ent .Wli sEeiu rfso of Professor Emeritus is Wallis F. Kenneth c ttsia Science article Statistical original the the of by reprint published electronic an is This ern iesfo h rgnli aiainand pagination in detail. original typographic the from differs reprint h w-ic omldsrbto aet pub- to came distribution normal two-piece The nttt fMteaia Mathematical of Institute 10.1214/13-STS417 lxadSli ( Sellin and Blix ; .INTRODUCTION 1. nttt fMteaia Statistics Mathematical of Institute , ae yOcrSenn ssont ewtotfoundation. phrases: without and be words to reiterate shown Key Pearson, is Sheynin, Karl Oscar by by generalisa originality later and Fechner’s rediscoveries of its denial to Fechner Theodor Gustav ip,FacsYir deot,Kr ero,FacsGal Francis Pearson, Sheynin. Karl Edgeworth, Ysidro Francis Lipps, uinfo t rgni h posthumous the in origin its from bution Abstract. 04 o.2,N.1 106–112 1, No. 29, Vol. 2014, 1973 1998 . 2014 , , eeec htpost- that reference a ),” )rfrraest John- to readers refer )) itiuin nStatis- in Distributions hspprtae h itr ftetopeenra distri normal two-piece the of the traces paper This 1994 o icsinof discussion for ) utvTedrFcnr oto Friedrich Gottlob Fechner, Theodor Gustav This . in 1 rbblt est ucin(PDF) function density probability where (1) in fFcnrsbscies sapeuet the introduc- distribution. to technical the prelude brief to a tion a follows As there ideas. exten-discussion, basic Fechner’s and of distribution record sions the and claims, of various rediscoveries the several of nature and source itiuinwt parameters with distribution fFcnrsoiiaiyhsrcnl enrepeated been ( recently Sheynin by has originality ( Fechner’s Pearson of originality by disputed to distribution” been claim Fechner had Fechner’s “the However as 378). it (page to Gaussian refers “two-sided and the law,” as translates which name, the contrary, Fechner’s the in On introduced Kollektivmasslehre name. originally other was any distribution un- or appearance, this no der made distribution normal piece tics omdb aigtelf afo omldistribu- normal ( a parameters of half with left tion the taking by formed itr fsaitc,Hl ( Hald statistics, of history Gesetz Gauss’sche Zweiseitige cln hmt ietecmo common the give ( to parameters them scaling with distribution normal a h , the admvariable random A f JhsnadKt ( Kotz and (Johnson Kollektivmasslehre ( x A = ) ( = µ  si ( in as , √ 2004 A A 2 π exp[ exp[ ( σ .I hspprw epriethe reappraise we paper this In ). 1 − − 1 ( + .Tesaigfco ple to applied factor scaling The ). 1897 ( ( century a d x x σ X o,Oscar ton, ,σ µ, in.The tions. 2 − − ) 19)of (1897) sthe as ) / 1970 µ µ a w-ic normal two-piece a has 2) 1 ) ) 1998 n h ih afof half right the and ) 2 2 − / / 1 ,σ µ, ) nwihtetwo- the which in )), nhsmonumental his In . 2 2 h itiuinis distribution The . σ σ 1905 rfr h latter the prefers ) 2 1 2 2 - 1 ] ] x , x , Zweispaltiges and ,woedenial whose ), f σ ,σ µ, ( 2 ≥ ≤ µ fi has it if = ) µ, µ, 2 ,and ), A or at 2 K. F. WALLIS

−1 −1 quantiles xp = F (p) and zp =Φ (p). Then in the left piece of the distribution we have xα = σ1zβ + µ, where β = α(σ1 + σ2)/2σ1. And in the right piece of the distribution, defining quantiles with reference to their upper tail probabilities, we have x1−α = σ2z1−δ + µ, where δ = α(σ1 + σ2)/2σ2. In particu- lar, with σ1 <σ2, as in Figure 1, the median of the −1 distribution is x0.5 = σ2Φ (1 (σ1 + σ2)/4σ2)+ µ. In this case the three central− values are ordered >median>mode; with negative this order is reversed. Although the two-piece normal PDF is continuous at µ, its first derivative is not and the second deriva- Fig. 1. The probability density function of the two-piece nor- tive has a break at µ, as first noted by Ranke and mal distribution. Dashed line: left half of N(µ, σ1) and right Greiner (1904). This has the disadvantage of making half of N(µ, σ2) distributions with µ = 2.5 and σ1 <σ2. Solid line: the two-piece normal distribution. standard asymptotic likelihood theory inapplicable, nevertheless standard asymptotic results are avail- able by direct proof for the specific example. the left half of the N(µ,σ1) PDF is 2σ1/(σ1 + σ2) The remainder of this paper is organised as fol- while that applied to the right half of N(µ,σ2) is lows. In Section 2 we revisit the distribution’s origin 2σ2/(σ1 + σ2), so the probability mass under the in Gustav Theodor Fechner’s Kollektivmasslehre, left or right piece is σ1/(σ1 + σ2) or σ2/(σ1 + σ2), edited by Gottlob Friedrich Lipps and published in respectively. An example with σ1 <σ2, in which the 1897, ten years after Fechner’s death. In Section 3 we two-piece is positively skewed, note an early rediscovery, two years later, by Fran- is shown in Figure 1. The skewness becomes extreme cis Ysidro Edgeworth. In Section 4 we turn to the as σ1 0 and the distribution collapses to the half- normal→ distribution, while the skewness is reduced first discussion in the English language of Fechner’s contribution, in a characteristically long and argu- as σ1 σ2, reaching zero when σ1 = σ2 and the dis- tribution→ is again the normal distribution. mentative article by (1905). Pearson The mean and of the distribution are derives some properties of “Fechner’s double Gaus- sian curve,” but asserts that it is “historically in- 2 (2) E(X)= µ + (σ2 σ1), correct to attribute [it] to Fechner.” We re-examine rπ − Pearson’s evidence in support of this position, in 2 2 particular having in mind its reappearance in Os- (3) var(X)= 1 (σ2 σ1) + σ1σ2.  − π  − car Sheynin’s (2004) appraisal of Fechner’s statis- tical work. Pearson also argues that “the curve is Expressions for the third and fourth moments about not general enough,” especially in comparison with the mean are increasingly complicated and unin- his family of curves. The overall result was that the formative. Skewness is more readily interpreted in Fechner distribution was overlooked for some time, terms of the ratio of the areas under the two pieces to the extent that there have been several indepen- of the PDF, which is σ1/σ2, or a monotone transfor- dent rediscoveries of the distribution in more recent mation thereof such as (σ2 σ1)/(σ2 + σ1), which is years; these are noted in Section 5, together with the value taken by the skewness− measure of Arnold some extensions. and Groeneveld (1995). With only three parameters there is a one-to-one relation between (the absolute 2. THE ORIGINATORS: FECHNER AND value of) skewness and . The conventional LIPPS -based measure of kurtosis, β2, ranges from 3 (symmetry) to 3.8692 (the half-normal extreme Gustav Theodor Fechner (1801–1887) is known as asymmetry), hence the distribution is leptokurtic. the founder of psychophysics, the study of the re- Quantiles of the distribution can be conveniently lation between psychological sensation and physi- obtained by scaling the appropriate standard nor- cal stimulus, through his 1860 book Elemente der mal quantiles. For the respective cumulative distri- Psychophysik. Stigler’s (1986, pages 242–254) as- bution functions (CDFs) F (x) and Φ(z) we define sessment of this “landmark” contribution concludes THE TWO-PIECE NORMAL DISTRIBUTION 3 that “at a stroke, Fechner had created a method- the use of the Gaussian distribution to calculate the ology for a new quantitative psychology.” However, probability of an observation falling in a given inter- his final work, Kollektivmasslehre, is devoted more val. The measure of location is the , generally to the study of mass phenomena and the A, and the measure of dispersion is the mean abso- search for empirical regularities therein, with ex- lute deviation, ε (related to the , amples of frequency distributions taken from many in the Gaussian distribution, by ε = σ 2/π). Ta- fields, including aesthetics, anthropology, astron- bles of the standard normal distributionp are not yet omy, botany, meteorology and zoology. In his Fore- available, and his calculations proceed via the error word, Fechner mentions the long gestation period function (see Stigler (1986), pages 246–248, e.g.), of the book, and states its main objective as the and prove to be remarkably accurate. establishment of a generalisation of the Gaussian In previous work Fechner had introduced other law of random errors, to overcome its limitations of “main values” of a , the Zen- symmetric probabilities and relatively small positive tralwert or “central value” C, and the Dichteste and negative deviations from the arithmetic mean. Wert or “densest value” D, subsequently known He also appeals to astronomical and statistical in- in English as the median and the mode. Arguing stitutes to use their mechanical calculation powers that the equality of A, C and D is the exception to produce accurate tables of the Gaussian distri- rather than the rule, he next introduces the Zweis- bution, which he had desperately missed during his paltiges Gauss’sche Gesetz to represent this asym- work on the book. But the book had not been com- metry. Calculating mean absolute deviations from pleted when Fechner died in November 1887. the mode separately for positive and negative devi- The eventual publication of Kollektivmasslehre ations from D, the “law of proportions” is invoked, in 1897 followed extensive work on the incomplete that these should be in the same ratio as the num- manuscript by Gottlob Friedrich Lipps (1865–1931). bers of observations on which they are based. On In his Editor’s Preface, Lipps says that he received converting from relative frequencies of observations the manuscript in early 1895 and that material he to probabilities, and from subset mean absolute de- has worked on is placed in square brackets in the viations to subset standard deviations, it is seen that published work. It is not clear how much unfin- this is exactly the requirement discussed above, that ished material was left behind by Fechner or to what the probabilities below and above the mode are in extent Lipps had to guess at Fechner’s intentions. the ratio σ1/σ2, to give a curve that is continuous It would appear that the overall structure of the at the mode. Fechner says that he first discovered book had already been set out by Fechner, since this law empirically, and warns that determination most of the later chapters have early paragraphs by of the mode from raw is not straightforward. Fechner, before square-bracketed paragraphs begin He goes on to show that, in this distribution, the to appear. Also, some earlier chapters by Fechner median lies between the mean and the mode. have forward references to later material that ap- The first mathematical expression of the two nor- pears in square brackets. In general, Lipps’ mate- mal curves with different precision soon appears in rial is more mathematical: he was more of a math- what is the first square-bracketed paragraph in the ematician than Fechner, who perhaps had set some book and the only such paragraph in Chapter 5. sections aside for attention later, only to run out More extensive workings by Lipps appear in Chap- of time. Lipps also has a lighter style: for example, ter 19, “The asymmetry laws,” where every para- Sheynin (2004, page 54) complains about some ear- graph is enclosed in square brackets. Here Lipps lier work that “Fechner’s style is troublesome. Very traces the development and properties of the dis- often his sentences occupy eight lines, and some- tribution more formally, including an expression for times much more—sentences of up to 16 lines are the density function [equation (6), page 297] which easy to find.” The same is true of the present work. corresponds to equation (1) on converting between The origin of the two-piece normal distribution measures of dispersion. Nevertheless, the key steps is in Chapter 5 of Kollektivmasslehre, titled “The in that development, in Chapter 5, were Fechner’s Gaussian law of random deviations and its general- alone. isations.” Here Fechner uses very little mathemat- We note that the second “generalisation” pre- ics, postponing more analytical treatment to later sented later in Chapter 5 of Kollektivmasslehre is chapters. He first presents a numerical example of a form of log-normal distribution, but this receives 4 K. F. WALLIS less emphasis and is not our present focus of atten- nal he had co-founded four years earlier. The arti- tion. cle is a response to a review of Pearson’s and Fech- ner’s works on skew variation by Ranke and Greiner 3. AN EARLY REDISCOVERY: EDGEWORTH (1904) in the leading German anthropology journal. Pearson’s title quotes most of the title of the Ger- In 1898–1900 Edgeworth contributed a five-part man article, omitting its reference to anthropology, article “On the representation of statistics by math- and adds the words “A rejoinder,” although the run- ematical formulae” to the Journal of the Royal Sta- ning head throughout his article is “Skew variation, tistical Society, each part appearing in a different is- a rejoinder.” He explains that the German journal sue of the journal. His objective was “to recommend had provisionally accepted a rejoinder, but when it formulae which have some affinity to the normal law arrived the editors did not “see fit to publish” his of error, as being specially suited to represent statis- reply, so he placed it in , of which he was, tics of frequency.” The first two parts deal with the in effect, managing editor. From a statistical point “method of translation,” or transformations to nor- of view this seems to have been a more appropriate mality, and the “method of separation,” or mixtures outcome, since his article contains much general sta- of normals, using modern terminology. tistical discussion and is most often cited for its in- In the third part Edgeworth considers the “method troduction of the terms platykurtic, leptokurtic and of composition,” in which he constructs “a composite mesokurtic. probability-curve, consisting of two half-probability However, Pearson’s article also contains extensive curves of different types, tacked together at the attacks on Ranke and Greiner, who had argued that, mode, or greatest ordinate, of each, so as to form for the anthropologist, only the Gaussian law is of a continuous whole, as in the accompanying figure” importance. In this respect the article is a good ex- (1899, page 373, emphasis in original; the figure is ample of his well-documented behaviour. For exam- very similar to the solid line in Figure 1 above). He ple, Stigler (1999, Chapter 1) opens by observing gives expressions for the two appropriately scaled that “Karl Pearson’s long life was punctuated by half-normal curves, as above, using the modulus, controversies, controversies he often instigated, usu- equal to √2 standard deviation, as his preferred ally pursued with a zealous energy bordering on ob- measure of spread. He says that this idea of two session;” he “was a fighter who vigorously reacted probability curves with different moduli is suggested against opinions that seemed to detract from his by Ludwig (1898); however, its development in the own theories. Instead of giving room for other meth- context of the normal distribution is Edgeworth’s ods and seeking cooperation, his aggressive style alone, since Ludwig’s comment comes in a discus- led to controversy” (Hald (1998), page 651); he was sion of frequency curves based on the binomial dis- ever “relentless in controversy” (Cox (2001), page 5) tribution. and “beyond question a fierce antagonist” (Porter To “determine the constants,” that is, estimate (2004), page 266). Some of this antagonism is di- the parameters, given a sample mean and second rected towards Fechner: although Pearson and Fech- and third sample moments, Edgeworth rearranges ner are on the same side of the debate with Ranke their definitions to obtain a cubic equation in the and Greiner about asymmetry, Pearson sees “Fech- distance between the mean and the mode; the re- ner’s double Gaussian curve” as a rival to his family quired parameter estimates follow from the real so- of curves, and criticises it on both statistical and lution to this equation. He gives a practical exam- historical grounds. ple and compares the method of composition to the Using the parameterisation in terms of σ1 and σ2 methods discussed earlier. In his opinion, the “essen- as in equation (1), Pearson presents expressions for tial attribute” of the new method is its “deficiency the first four moments of the distribution. Rather of a priori justification,” in contrast to the normal than “the rough process by which Fechner deter- distribution itself. mines the mode and obtains the constants of the distribution,” he shows that “fitting by my method 4. THE CRITICS: PEARSON AND SHEYNIN of moments is perfectly straightforward.” To do this, he obtains the cubic equation discussed above, and The first English-language discussion of Fechner’s says in a footnote (page 197) “This cubic was, I contribution appears in a 44-page article by Karl believe, first given by Edgeworth,” but there is no Pearson, published in 1905 in Biometrika, the jour- reference. He observes that the skewness and kur- THE TWO-PIECE NORMAL DISTRIBUTION 5 tosis are not independent of one another, so that information on his method, which was published “we cannot have any form of symmetry but the in the same issue of the Society’s journal (Galton mesokurtic.” He obtains the bounds on β2 given (1896)), together with a reply by Yule (1896b). Gal- above, but notes that many empirical distributions ton explains how his method of , in this with values outside this have been observed. example method of deciles, smooths the original fre- Hence, Pearson’s overall conclusion is that “the dou- quency table or “frequency polygon” of Yule by in- ble Gaussian curve fails us hopelessly.” Curiously, terpolating deciles and plotting them. He then men- having defined platykurtic as “more flat-topped” tions another approach, namely and leptokurtic as “less flat-topped” than the nor- . . . the extremely rude and scarcely defen- mal curve, as has become standard usage, he con- sible method, but still a sometimes ser- trarily describes Fechner’s double Gaussian curve as viceable one, of looking upon skew-curves platykurtic, despite having shown its positive excess as made up of the halves of two differ- kurtosis. Similarly, another curve, the symmetrical ent normal curves pieced together at the binomial, is said to be “essentially leptokurtic, that mode. . . . On trying it, again for curios- is, β2 < 3” (page 175). ity’s sake, with the present series for all Turning to questions of precedence, Pearson’s the five years, there was of course no er- counter claims appear in a footnote (page 196) at the ror for the 2nd, 5th, and 8th deciles, . . . start of the statistical discussion summarised above, which reads as follows: because he had inferred the spread or standard devi- ation of the lower half-normal distribution from the Here again it is historically incorrect to at- lower 20% point of the standard normal distribu- tribute these curves to Fechner. They had tion, and similarly for the upper part; he goes on to been proposed by De Vries in 1894, and discuss the errors of fit at the other deciles. But no termed “half-Galton curves,” and Galton “law of proportions” or scaling is applied, and the was certainly using them in 1897. See the resulting curve is discontinuous, like the initial two discussion in Yule’s memoir, R. Statist. halves of normal curves in Figure 1. Yule (1896b) Soc. Jour. Vol. LX, page 45 et seq. recognises this in his response, noting that, in con- Pearson was familiar with De Vries (1894), having trast, his skew-curve “presents a continuous distri- used two of his J-shaped botanical frequency distri- bution round the mode.” Galton was certainly not butions as Examples XI and XII in his 1895 arti- using Fechner’s curves. cle on skew variation. De Vries said that these de- The erroneous assertions in Pearson’s footnote served the name half-Galton (i.e., half-normal) sim- may be due to his combativeness. Several authors ply on the basis of the appearance of the empiri- also discuss the tremendous volume of work he un- cal distributions, and no fitting was attempted, nor dertook. For example, Cox (2001, page 6) observes did he make any proposal to place two such curves that he “wrote more than 90 papers in Biometrika together to give a more general asymmetric distri- in the period up to 1915, few of them brief, and ap- bution. Fechner’s curve had not been proposed by pears to have been the moving spirit behind many De Vries. [Edgeworth knew that his composite curve more.” He founded not only the journal but also the had not either, noting at the outset (1899, page 373) Laboratory at University College Lon- that “It will be observed that the following construc- don at this time. His son Egon remarks that the tion is not much indebted to the “half-Galtonian” volume of work “led inevitably to a certain hurry in curve employed by Professor De Vries.”] execution” (E. S. Pearson (1936), page 222). This Galton comes a little closer, but Pearson is again remark is made during discussion of one of Pear- incorrect. His citation is inaccurate, since he clearly son’s two well-known errors, recently reappraised by has in mind Yule’s paper read at the Royal Statis- Stigler (2008), but it perhaps also applies to the tical Society in January 1896, published with dis- mistakes discussed above, which are of a smaller or- cussion later that year (Yule (1896a)). Galton had der of magnitude. Nevertheless, Pearson’s assertions opened the discussion at the meeting and mentioned in the quoted footnote are mistaken, and his chal- his method of percentiles as an alternative to the lenge to Fechner’s claim to priority is unjustified, method of frequency curves developed by Pearson and thereby unjust. and applied by Yule. In response to a request at the Sheynin (2004), in his review of Fechner’s statis- meeting, he provided a memorandum giving fuller tical work, has a very brief discussion of the double- 6 K. F. WALLIS sided Gaussian law, quoting from sections of Kollek- mode ordering. He notes that Fechner had shown tivmasslehre that had been worked on by Lipps, and this Lagegesetz der Mittelwerte for the two-piece hence underestimating the role of Fechner’s law of normal distribution, and investigates more general proportions. In his discussion (page 68) he states conditions in which it holds. The second excep- that the double-sided Gaussian law was not origi- tional appearance of the Fechner distribution in the nal to Fechner, this having been pointed out, force- statistical literature pre-1998 is more substantial. fully, by Pearson (1905). As if quoting from Pearson, Barnard (1989), seeking a family of distributions and giving no citation for De Vries (1894), Sheynin “which may be expected to represent most of the states “De Vries, in 1894, had applied the double- types of skewness liable to arise in practice,” intro- sided law.” In this statement “applied” is somewhat duces the distribution stronger than Pearson’s “proposed,” hence is further 1 M(x µ) a from the truth, and Sheynin’s denial of Fechner’s K exp − − , x µ,  −2 σ   ≤ originality is similarly inaccurate and unjust. f(x)= a  1 x µ  K exp − , x µ, 5. LATER REDISCOVERIES AND −2 σ   ≥  EXTENSIONS which reparameterises and generalises the two-piece The result of Pearson’s critique appears to have normal distribution in equation (1). He calls it the been that, with two exceptions discussed below, the Fechner family, because by allowing the skewness Fechner distribution, with this attribution, disap- parameter M (M > 0) to differ from 1 it embod- Kollektivmasslehre peared from the statistical literature until its reap- ies Fechner’s idea in of having different scales for positive and negative deviations pearance in Hald’s (1998) history. Meanwhile, three independent rediscoveries occurred. from the mode, µ. It also allows for nonnormal kur- tosis by allowing a (1 a< ) to differ from 2. The First, in the physics literature, is Gibbons and σ is equal≤ to∞ the standard deviation Mylroie’s (1973) “joined half-Gaussian” distribu- if (M, a) = (1, 2) but not otherwise, in general. This tion, cited by Johnson, Kotz and Balakrishnan “Fechner family of unimodal densities” also appears (1994), as noted above; the distribution is fitted by in a later article (Barnard (1995)), which is cited by what recognise as the method of mo- Hald (1998, page 380). We note that the case a = 1, ments. Second, in the statistics literature, is the the asymmetric , has a consider- “three-parameter two-piece normal” distribution of able life of its own, beginning before Barnard’s work: John (1982), also cited by Johnson, Kotz and Bal- see, for example, Kotz, Kozubowski and Podgorski akrishnan (1994); John compares estimation by the (2001, Chapter 3) and the references therein. method of moments and maximum likelihood. In the Two further extensions of note, independent of same journal Kimber (1985) notes that John (1982) Fechner, can be found in Bayesian statistics. For is a rediscovery, with reference to Gibbons and Myl- the application of Monte Carlo integration with im- roie (1973); he proves the asymptotic normality of portance to , Geweke ML estimators and provides a likelihood ratio test (1989) uses “split” (i.e., two-piece) multivariate nor- of symmetry. Finally, in the meteorology literature, mal and Student-t distributions as importance sam- Toth and Szentimrey (1990) introduce the “binor- pling densities. The generalisation by Fernandez and mal” distribution, again fitted by ML, with a test Steel (1998) is also cast in a Bayesian setting: as in of symmetry. The same name is used by Garvin and Barnard’s Fechner family, there is a single skewness McClean (1997), who nevertheless again attribute parameter, which is convenient whenever it is de- the distribution to Gibbons and Mylroie. In all these sired to assign priors to skewness; nevertheless, it articles the distribution is parameterised in terms of has general applicability. For any univariate PDF the mode, using various symbols, and the standard f(x) which is unimodal and symmetric around 0, deviations σ1 and σ2, as in (1) above. An alternative Fernandez and Steel’s class of two-piece or split dis- parameterisation, with a single explicit skewness pa- tributions p(x γ), indexed by a skewness parameter rameter, is given by Mudholkar and Hutson (2000), γ (γ> 0), is | who do acknowledge Fechner’s priority. A modern, but pre-Hald (1998) attribution to Kf(γx), x 0, ≤ Kollektivmasslehre occurs at the start of an explo- (4) p(x γ)=  x | Kf , x 0, ration by Runnenburg (1978) of the mean, median,  γ  ≥  THE TWO-PIECE NORMAL DISTRIBUTION 7

−1 −1 where K = 2(γ + γ ) . If γ> 1 there is positive Garvin, J. S. and McClean, S. I. (1997). Convolution and skewness, and inverting γ produces the mirror image sampling theory of the binormal distribution as a prereq- of the density function around 0. Unlike Barnard’s uisite to its application in statistical process control. The 46 Fechner family there is no explicit kurtosis parame- 33–47. Geweke, J. (1989). Bayesian inference in econometric mod- ter; kurtosis is introduced, if desired, by the choice els using Monte Carlo integration. Econometrica 57 1317– of f(x), most commonly as Student-t. An extension 1339. MR1035115 with two tail parameters to allow different tail be- Gibbons, J. F. and Mylroie, S. (1973). Estimation of im- haviour in an asymmetric two-piece t-distribution is purity profiles in ion-implanted amorphous targets using developed by Zhu and Galbraith (2010). joined half-Gaussian distributions. Appl. Phys. Lett. 22 568–569. Hald, A. 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