Talk:Edgeworth series

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Mathematics Mid‑priority

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In this article as it originally appeared, the statement of the central limit theorem was wrong. The error was copied from the cited 1958 paper. Here is a simple counterexample: Suppose X₁ = 0 or 1, each with probability 1/2, and X_i = 0 with probability 1. Then the expression that was asserted to approach the normal c.d.f. does not. I have inserted the additional hypothesis that the sum of the variances diverges to infinity. Michael Hardy 02:34, 10 May 2004 (UTC)[reply]

I have changed f(x) to F(x) in the statement about the convergence at the end of the Gram-Charlier A series. I think that is the function which was meant. Lafrontera 17:51, 10 February 2007 (UTC)[reply]

I changed the priority to mid because (1) Edgeworth expansions are taught first or second term in the core of a stats PhD; (2) they are also taught to some masters students; and, (3) they are used in other fields (such as astronomy). Someday soon I'll update the page for the general form of the expansion. --Cumulant 20:55, 21 August 2007 (UTC)[reply]

The notation "F" appears to be used inconsistently. It is introduced in the G-C section as a density (nonstandard usage!), but explicitly defined as the (sample) cumulative distribution F_n in the Edgeworth section. Later, however, it is referred to as a "density", despite being linked with \Phi, the normal cdf. Could an expert sort this out? Gilgalad55 (talk) 19:06, 3 August 2010 (UTC)[reply]

I agree with this point; F is sometimes used as a density and sometimes as a cdf. This is incorrect and extremely confusing — Preceding unsigned comment added by 72.77.85.3 (talk) 21:48, 28 November 2014 (UTC)[reply]

Gram-Charlier[edit]

Who are they? Albmont 17:02, 27 July 2007 (UTC)[reply]

Good question. I'm not sure, but I suspect Gram is the same person whose name appears in the phrase Gram-Schmidt process, and that Charlier is the one whose name appears in Poisson-Charlier polynomials. Michael Hardy 21:21, 27 July 2007 (UTC)[reply]

Gram is the same; suspect (but haven't confirmed) that Charlier is also the same. FWIW, Gram and Charlier were mathematicians living around the end of the nineteenth century. The Gram-Charlier Series was actually found first by Thiele who gets no credit (thus partially fulfilling Stigler's law of eponymy).

Thiele's work was published in 1871 as (Danish) "En mathematisk Formel for Dødeligheden, prøvet paa en af Livsforsikringanstalten af 1871 benyttet Erfaringrække"; an English translation followed in 1872. Gram's article was published in 1883 as (German) "Über die Entwickelung reeler Funktionen in Reihen mittelstder Methode der kleinsten Quadrate". Charlier? Eh. Late to the party from what I can tell -- although if someone can figure out how he enters in, I'd love to hear it.--Cumulant 19:32, 15 August 2007 (UTC)[reply]

What is n?[edit]

I think the definition of n in Edgeworth (after Gram-Charlier) is confusing. It says that the two are the same thing except that Edgeworth is grouped by powers of n, but GC has no n at all. What would it even mean to follow the GC derivation and then group it by n part way through when there is no n. After many reads, my best understanding is that n is the number of these iid variables summed together that we are trying to estimate the cdf and pdf of. So, n would usually be set to 1 I would think?????? — Preceding unsigned comment added by 75.182.7.143 (talk) 22:24, 12 January 2012 (UTC)[reply]

Edgeworth Series[edit]

This section is hard to read. There seems to be an error in the computation of the rth cumulant, and it is not at all clear how to "group by powers of n." This section should be rewritten along the lines of http://sites.stat.psu.edu/~drh20/asymp/fall2004/lectures/edgeworth.pdf — Preceding unsigned comment added by 165.134.212.74 (talk) 21:56, 30 January 2018 (UTC)[reply]