Karl Pearson and the Origin of Kurtosis

Fiori, Anna Maria; Zenga, M

doi:10.1111/j.1751-5823.2009.00076.x

Cited by 39 publications

(21 citation statements)

References 16 publications

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“…If the values are about the same as the normal distribution, it is mesokurtic. These phrases are frequently confused (see Fiori & Zenga, 2009, for a historical analysis; see also Đorić, Nikolić-Đorić, Jevremović, & Mališić, 2009). "Student" [Gosset] (1927, p. 160) offered a handy memory aid, depicted in Fig.…”

mentioning

confidence: 99%

Problematic standard errors and confidence intervals for skewness and kurtosis

Wright

Herrington

2011

Behav Res

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Many statistics packages print skewness and kurtosis statistics with estimates of their standard errors. The function most often used for the standard errors (e.g., in SPSS) assumes that the data are drawn from a normal distribution, an unlikely situation. Some textbooks suggest that if the statistic is more than about 2 standard errors from the hypothesized value (i.e., an approximate value for the critical value from the t distribution for moderate or large sample sizes when α = 5%), the hypothesized value can be rejected. This is an inappropriate practice unless the standard error estimate is accurate and the sampling distribution is approximately normal. We show distributions where the traditional standard errors provided by the function underestimate the actual values, often being 5 times too small, and distributions where the function overestimates the true values. Bootstrap standard errors and confidence intervals are more accurate than the traditional approach, although still imperfect. The reasons for this are discussed. We recommend that if you are using skewness and kurtosis statistics based on the 3 rd and 4 th moments, bootstrapping should be used to calculate standard errors and confidence intervals, rather than using the traditional standard. Software in the freeware R for this article provides these estimates.

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confidence: 99%

Problematic standard errors and confidence intervals for skewness and kurtosis

Wright

Herrington

2011

Behav Res

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“…(19). The probability density functions of PD-system are solutions of the following differential equation (Pearson 1920;Elderton 1953;Hald 2007;Nagahara 2007Nagahara , 2011Friori and Zenga 2009;Dzhun' 2011;Xi et al 2012)…”

Section: Pearson Distributionsmentioning

confidence: 99%

“…We showed previously that one should not expect such large anomalies for geodetic observation sets. Thus, but also on the basis of the previous papers of the author (Wiśniewski 1985(Wiśniewski , 1987(Wiśniewski , 1991, we will assume that the system of the Pearson distributions (PD-system) which contains types I ÷ XII (Pearson 1920;Elderton 1953;Friori and Zenga 2009;Xi et al 2012) is the set of theoretical distributions which are steered by the four first moments.…”

Section: Introductionmentioning

confidence: 99%

M-estimation with probabilistic models of geodetic observations

Wiśniewski

2014

J Geod

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The paper concerns M-estimation with probabilistic models of geodetic observations that is called M P estimation. The special attention is paid to M P estimation that includes the asymmetry and the excess kurtosis, which are basic anomalies of empiric distributions of errors of geodetic or astrometric observations (in comparison to the Gaussian errors). It is assumed that the influence function of M P estimation is equal to the differential equation that defines the system of the Pearson distributions. The central moments μ k , k = 2, 3, 4, are the parameters of that system and thus, they are also the parameters of the chosen influence function. The M P estimation that includes the Pearson type IV and VII distributions (M PD(l) method) is analyzed in great detail from a theoretical point of view as well as by applying numerical tests. The chosen distributions are leptokurtic with asymmetry which refers to the general characteristic of empirical distributions. Considering M-estimation with probabilistic models, the Gram-Charlier series are also applied to approximate the models in question (M G−C method). The paper shows that M P estimation with the application of probabilistic models belongs to the class of robust estimations; M PD(l) method is especially effective in that case. It is suggested that even in the absence of significant anomalies the method in question should be regarded as robust against gross errors while its robustness is controlled by the pseudo-kurtosis.

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“…More detail on various aspects of the Pearson system is provided by the paper of Fiori & Zenga (2009) in this issue of the International Statistical Review .…”

Section: Karl Pearson's System Of Frequency Curvesmentioning

confidence: 99%

Karl Pearson in Russian Contexts

Seneta¹

2009

Int Statistical Rev

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The confluence of statistics and probability into mathematical statistics in the Russian Empire through the interaction, 1910-1917, of A.A. Chuprov and A.A. Markov was influenced by the writings of the English Biometric School, especially those of Karl Pearson. The appearance of the Russian-language exposition of Pearsonian ideas by E. E. Slutsky in 1912 was instrumental in this confluence. Slutsky's predecessors in such writings (Lakhtin, Orzhentskii, and Leontovich) were variously of mathematical, political economy, and biological backgrounds. Work emanating from the interpolational nature of Pearson's system of frequency curves was continued subsequently through the work of Markov, Bernstein, Romanovsky, and Kravchuk (Krawtchouk), who laid a solid probabilistic foundation. The correlational nature in the interpolational early work of Chebyshev, and work of the English Biometric School in the guise of linear least-squares fitting exposited as the main component of Slutsky's book, was developed in population as well as sample context by Chuprov. He also championed the expectation operation in providing exact relations between sample and population moments, in direct interaction with Karl Pearson. Romanovsky emerges as the most adaptive and modern mathematical statistician. Copyright (c) 2009 The Author. Journal compilation (c) 2009 International Statistical Institute.

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Karl Pearson and the Origin of Kurtosis

Cited by 39 publications

References 16 publications

Problematic standard errors and confidence intervals for skewness and kurtosis

Problematic standard errors and confidence intervals for skewness and kurtosis

M-estimation with probabilistic models of geodetic observations

Karl Pearson in Russian Contexts

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