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The Edgeworth series distribution, introduced by Francis Ysidro Edgeworth in 1897. is a continuous probability distribution which approximates a non-normal probability distribution in terms of Hermite polynomials and cumulants. It relates the probability density function (PDF) to that of a standard normal distribution. The number of terms in the series can be increased to improve the accuracy of the approximation.
The Edgeworth series distribution has its roots in the study by Carl Friedrich Gauss in the early 19th century on the distribution of errors in a least squares regression, which he established to be approximately normal. Later, Edgeworth built on Gauss’s contribution, demonstrating that this distribution of errors can be estimated by a series composed of Hermite polynomials and cumulants.
Properties of the Edgeworth series distribution
The PDF for the bivariate Edgeworth series distribution (BVESD) is [1]
Where
- φ(z1, z2) = the standard bivariate normal density function,
- Aij’s = functions of the population cumulants,
- Dz1, Dz2 = partial derivative operators.
The characteristic function is
Generating samples from the Edgeworth series distribution PDF is challenging. Additionally, the conditional probability of misclassification for the ESD distribution is intractable due to the complexity of the expression.
In mathematics, an intractable problem is a problem that is difficult or impossible to solve using current methods. Intractable problems are often characterized by their size or complexity, and they may require a significant amount of time or resources to solve. Some other examples of intractable problems include the traveling salesman problem, the knapsack problem, and the Boolean satisfiability problem.
The Cornish-Fisher expansions are related to the Edgeworth form of distribution, but there’s no generalized theoretical superiority [2].
Uses of the Edgeworth series distribution
The Edgeworth series distribution is a powerful tool for analyzing non-normal data. It has a wide range of applications and has been used by many statisticians and economists, including Ronald Fisher, Jerzy Neyman, and John Tukey. The distribution is versatile and widely applicable; it facilitates the calculation of moments for non-normal distributions, the construction of confidence intervals and hypothesis tests, and even the simulation of non-normal data. In finance, the distribution is commonly used to model asset price distribution in markets.
The Edgeworth series distribution is commonly used in statistical asymptotic theory to calculate approximations to sample statistic distributions of orders greater than n-1/2 [3].
References
[1] S. Kocherlakota , K. Kocherlakota & N. Balakrishnan (1985) Effects of nonnormality on the spart for the correlation coefficient: bivariate edgeworth series distribution, Journal of Statistical Computation and Simulation, 23:1-2, 41-51, DOI: 10.1080/00949658508810857
[2] Johnson, Kotz, and Balakrishnan, (1994), Continuous Univariate Distributions, Volumes I and II, 2nd. Ed., John Wiley and Sons.
[3] A. Adeyeye. Asymptotic Distribution of Probabilities of Misclassification for Edgeworth Series Distribution (ESD). Engineering Mathematics. 2020; 4(1): 1-9 http://www.sciencepublishinggroup.com/j/engmath doi: 10.11648/j.engmath.20200401.11