Impact of Karl Pearson's Work on Statistical Developments in India

Karl Pearson's work greatly inspired P. C. Mahalanobis's interest in statistics, who was at the centre of modern statistical developments in India. Mahalanobis learned statistics on his own, reading Pearson's articles and his journal, and encouraged others to study Pearson's papers. Pearson was a Honorary Fellow of the Indian Statistical Institute, which has been a leading statistical training and research center in India. The statistical revolution that Pearson brought also facilitated the training and research of many Indian statisticians.     

Karl Pearson—The Scientific Life in a Statistical Age by Theodore M. Porter: A Review

Porter presents an excellent account of the young Karl Pearson and his extraordinarily varied activities. These ranged from the Cambridge Mathematical Tripos Exams to German history and folklore, and included free thought, socialism, the woman's question, and the law. Returning to science, Pearson produced the famous Grammar of Science . He decided on a career in statistics only at age 35. Porter emphasizes Pearson's often acrimonious but largely successful battles to show the wide applicability and importance of statistics in many areas of science and public affairs. Eugenics became a passion for Pearson. Avoiding all formulas Porter fails to give any concrete ideas of even Pearson's most important contributions to statistical theory. We try to sketch these here.     

Karl Pearson and the Establishment of Mathematical Statistics

At the end of the nineteenth century, the content and practice of statistics underwent a series of transitions that led to its emergence as a highly specialised mathematical discipline. These intellectual and later institutional changes were, in part, brought about by a mathematical-statistical translation of Charles Darwin's redefinition of the biological species as something that could be viewed in terms of populations. Karl Pearson and W.F.R. Weldon's mathematical reconceptualisation of Darwinian biological variation and "statistical" population of species in the 1890s provided the framework within which a major paradigmatic shift occurred in statistical techniques and theory. Weldon's work on the shore crab in Naples and Plymouth from 1892 to 1895 not only brought them into the forefront of ideas of speciation and provided the impetus to Pearson's earliest statistical innovations, but it also led to Pearson shifting his professional interests from having had an established career as a mathematical physicist to developing one as a biometrician. The innovative statistical work Pearson undertook with Weldon in 1892 and later with Francis Galton in 1894 enabled him to lay the foundations of modern mathematical statistics. While Pearson's diverse publications, his establishment of four laboratories and the creation of new academic departments underscore the plurality of his work, the main focus of his life-long career was in the establishment and promulgation of his statistical methodology.     

Karl Pearson's Influence in the United States

Karl Pearson, the founder of mathematical statistics, was the leading statistical researcher from the 1890s up to about 1920. His interests were wide-ranging and so his impact on statistics in the United States was also wide-ranging. Many American researchers came to University College London to study with him. Others studied his work from afar. In the United States, Pearsonian statistics first penetrated the academic landscape in biology. This was soon followed by the fields of economics and psychology. It was not until relatively late in Pearson's career that several American mathematicians took up statistics as a serious research topic.     

Two families of kurtosis measures

Two families of kurtosis measures are defined as K ₁(b)=E[ab ^−|z|] and K ₂(b)=E[a(1−|z|^b)] where z denotes the standardized variable and a is a normalizing constant chosen such that the kurtosis is equal to 3 for normal distributions. K ₂(b) is an extension of Stavig's robust kurtosis. As with Pearson's measure of kurtosis β₂=E[z ⁴], both measures are expected values of continuous functions of z that are even, convex or linear and strictly monotonic in ℜ⁻ and in ℜ⁺. In contrast to β₂, our proposed kurtosis measures give more importance to the central part of the distribution instead of the tails. Tests of normality based on these new measures are more sensitive with respect to the peak of the distribution. K ₁(b) and K ₂(b) satisfy Van Zwet's ordering and correlate highly with other kurtosis measures such as L-kurtosis and quantile kurtosis. RID="*" ID="*"  The authors thank the referees for their insightful comments that significantly improved the clarity of the article.     

On measuring skewness and kurtosis

The paper considers some properties of measures of asymmetry and peakedness of one dimensional distributions. It points to some misconceptions of the first and the second Pearson coefficients, the measures of asymetry and shape, that frequently occur in introductory textbooks. Also it presents different ways for obtaining the estimated values for the coefficients of skewness and kurtosis and statistical tests which include them.     

Simulating non‐normal distributions with specified L‐moments and L‐correlations

This paper derives a procedure for simulating continuous non‐normal distributions with specified L‐moments and L‐correlations in the context of power method polynomials of order three. It is demonstrated that the proposed procedure has computational advantages over the traditional product‐moment procedure in terms of solving for intermediate correlations. Simulation results also demonstrate that the proposed L‐moment‐based procedure is an attractive alternative to the traditional procedure when distributions with more severe departures from normality are considered. Specifically, estimates of L‐skew and L‐kurtosis are superior to the conventional estimates of skew and kurtosis in terms of both relative bias and relative standard error. Further, the L‐correlation also demonstrated to be less biased and more stable than the Pearson correlation. It is also shown how the proposed L‐moment‐based procedure can be extended to the larger class of power method distributions associated with polynomials of order five.     

Characterizing systems of distributions by quantile measures

Modelling an empirical distribution by means of a simple theoretical distribution is an interesting issue in applied statistics. A reasonable first step in this modelling process is to demand that measures for location, dispersion, skewness and kurtosis for the two distributions coincide. Up to now, the four measures used hereby were based on moments.
In this paper measures are considered which are based on quantiles. Of course, the four values of these quantile measures do not uniquely determine the modelling distribution. They do, however, within specific systems of distributions, like Pearson's or Johnson's; they share this property with the four moment-based measures.
This opens the possibility of modelling an empirical distribution—within a specific system—by means of quantile measures. Since moment-based measures are sensitive to outliers, this approach may lead to a better fit. Further, tests of fit—e.g. a test for normality—may be constructed based on quantile measures. In view of the robustness property, these tests may achieve higher power than the classical moment-based tests.
For both applications the limiting joint distribution of quantile measures will be needed; they are derived here as well.     

Karl Pearson in Russian Contexts

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.     

Impact of diversification on the distribution of stock returns: International evidence

In the past, stock returns are often assumed to be normally distributed. Potential gains from international portfolio diversification are thus based on a mean-variance framework. However, numerous empirical results reveal that stock returns are actually not normally distributed. Although previous studies found that both skewness and kurtosis can be rapidly diversified away, these results are only valid for a random sample of a given portfolio size. This paper studies the joint effect of diversification and intervaling on the skewness and kurtosis of eleven international stock market indexes with a holding period spanning from one to six months. A complete set of all possible combinations of portfolios is used. It is found that diversification does not reduce either skewness or kurtosis. As the portfolio size increases, portfolio returns become more negatively skewed and more leptokurtic. As a result, a rational investor may not gain from international diversification.     

Entropy densities with an application to autoregressive conditional skewness and kurtosis

The entropy principle yields, for a given set of moments, a density that involves the smallest amount of prior information. We first show how entropy densities may be constructed in a numerically efficient way as the minimization of a potential. Next, for the case where the first four moments are given, we characterize the skewness–kurtosis domain for which densities are defined. This domain is found to be much larger than for Hermite or Edgeworth expansions. Last, we show how this technique can be used to estimate a GARCH model where skewness and kurtosis are time varying. We find that there is little predictability of skewness and kurtosis for weekly data.     

Forecasting in GARCH models with polynomially modified innovations

Orthogonal polynomials can be used to modify the moments of the distribution of a random variable. In this paper, polynomially adjusted distributions are employed to model the skewness and kurtosis of the conditional distributions of GARCH models. To flexibly capture the skewness and kurtosis of data, the distributions of the innovations that are polynomially reshaped include, besides the Gaussian, also leptokurtic laws such as the logistic and the hyperbolic secant. Modeling GARCH innovations with polynomially adjusted distributions can effectively improve the precision of the forecasts. This strategy is analyzed in GARCH models with different specifications for the conditional variance, such as the APARCH, the EGARCH, the Realized GARCH, and APARCH with time-varying skewness and kurtosis. An empirical application on different types of asset returns shows the good performance of these models in providing accurate forecasts according to several criteria based on density forecasting, downside risk, and volatility prediction.     

Multilinear estimation of skewness and kutosis in linear models

Summary Estimation of the coefficients of skewness and kurtosis in a classical linear model situation is presented as an application of multilinear algebra and standard theory of mean estimation. The resulting estimators have optimality properties among all estimators that are invariant under mean translations, polynomials of degree three (skewness) or four (kurtosis) in the observations, and unbiased.     

Maximum entropy autoregressive conditional heteroskedasticity model

In many applications, it has been found that the autoregressive conditional heteroskedasticity (ARCH) model under the conditional normal or Student’s t distributions are not general enough to account for the excess kurtosis in the data. Moreover, asymmetry in the financial data is rarely modeled in a systematic way. In this paper, we suggest a general density function based on the maximum entropy (ME) approach that takes account of asymmetry, excess kurtosis and also of high peakedness. The ME principle is based on the efficient use of available information, and as is well known, many of the standard family of distributions can be derived from the ME approach. We demonstrate how we can extract information functional from the data in the form of moment functions. We also propose a test procedure for selecting appropriate moment functions. Our procedure is illustrated with an application to the NYSE stock returns. The empirical results reveal that the ME approach with a fewer moment functions leads to a model that captures the stylized facts quite effectively.     

Retrieving the implicit risk neutral density of WTI options with a semi-nonparametric approach

This paper contributes to the literature on the estimation of the Risk Neutral Density (RND) function by proposing a log-semi-nonparametric (log-SNP) distribution as the implicit RND when the Gram-Charlier model is used for option pricing. The performance of the model is compared to the lognormal (Black Scholes) benchmark for a sample of option prices for West Texas Intermediate (WTI) crude oil that were traded in the period between January 2016 and December 2017. Results show that the lognormal specification tends to systematically undervalue option prices and that the proposed log-SNP distribution, which explicitly adjusts for negative skewness and excess kurtosis, results in markedly improved accuracy, especially in periods of market instability. As a result, the implied skewness and excess kurtosis are relevant sources of information on market expectations that should be used for hedging and risk management purposes.     

Kurtosis of GARCH and stochastic volatility models with non-normal innovations

Both volatility clustering and conditional non-normality can induce the leptokurtosis typically observed in financial data. In this paper, the exact representation of kurtosis is derived for both GARCH and stochastic volatility models when innovations may be conditionally non-normal. We find that, for both models, the volatility clustering and non-normality contribute interactively and symmetrically to the overall kurtosis of the series.     

Autoregresive conditional volatility, skewness and kurtosis

This paper proposes a GARCH-type model allowing for time-varying volatility, skewness and kurtosis. The model is estimated assuming a Gram–Charlier (GC) series expansion of the normal density function for the error term, which is easier to estimate than the non-central t distribution proposed by [Harvey, C. R. & Siddique, A. (1999). Autorregresive Conditional Skewness. Journal of Financial and Quantitative Analysis 34, 465–487). Moreover, this approach accounts for time-varying skewness and kurtosis while the approach by Harvey and Siddique [Harvey, C. R. & Siddique, A. (1999). Autorregresive Conditional Skewness. Journal of Financial and Quantitative Analysis 34, 465–487] only accounts for non-normal skewness. We apply this method to daily returns of a variety of stock indices and exchange rates. Our results indicate a significant presence of conditional skewness and kurtosis. It is also found that specifications allowing for time-varying skewness and kurtosis outperform specifications with constant third and fourth moments.     

Simple alternatives for Box–Cox transformations

Simple transformations are given for reducing/stabilizing bias, skewness and kurtosis, including the first such transformations for kurtosis. The transformations are based on cumulant expansions and the effect of transformations on their main coefficients. The proposed transformations are compared to the most traditional Box–Cox transformations. They are shown to be more efficient.     

高阶矩风险与金融投资决策

针对传统投资组合理论没有考虑高阶矩风险这一缺陷,总结近期金融领域中有关偏度和峰度的研究成果,基于"均值-方差"效用函数的Taylor展开,讨论了投资者对高阶矩风险(偏度风险和峰度风险)的偏好特征。