Talk:Pearson distribution

comment

This article is neither complete nor correct. It needs the definition of Pearson's system of distribution and a discussion of Pearson distributions up through type XII. And I suspect that the functional form of the type IV is wrong. Ϙ 21:07, 7 July 2007 (UTC)

I've scrapped it and replaced it with my working draft. It's still obviously incomplete, but less so than before. I think it would be fair to say that only types I through VII form a coherent system. Types VIII through XII were more of an afterthought. --MarkSweep (call me collect) 00:09, 8 July 2007 (UTC)

That is a vast improvement. The extra types seem to be various special cases that he didn't discuss in the first paper. Type VII is the student-t distribution, X the exponential, XI are power laws, and I don't recognize types VIII, IX and XII. I still don't understand why Pearson thought Eq. 1 was the right way to derive probability distributions Ϙ 00:34, 8 July 2007 (UTC)

There seems to be an error in the derivation of type IV. If b2 > 3 (as indicated by the figure, and from the negative sign of the discriminant), how can m := 1/2*b2 > 0.5? Labecks 12:57, 8 October 2007 (UTC)

Basic equation

Chapter 4 of the Elderton/Johnson book cited in References at the end of the article gives the reasoning behind Pearson's choice of Equation 1. He assumed that most of the useful density curves were smooth, shaped like a witch's hat (unimodal), and had tails having high contact with the horizontal axis (that is, p' = 0 when p = 0). To force the derivative to be zero at both the tail(s) and at the mode, the controlling equation for p(x) had to be of the form p' = p*(a + x)/F(x), where x = -a is the abscissa of the mode, and hence a place where p' should vanish. The denominator F(x) was then expanded as the first three terms of a Taylor series, and lambda was added to allow horizontal shifts to be made in order to write the resulting density functions as conveniently as possible (e.g., to allow placement of the origin at the mean or mode of the distribution). The coefficients of the Taylor expansion are functions of the moments of the distributions; the more terms used, the more moments must be computed. Since higher moments (beyond 4th) are very sensitive to sampling variations, Pearson judged that three terms would provide the necessary variety of possible density functions without introducing too many of the sampling fluctuations associated with higher moments. Fizzbowen (talk) 06:09, 8 December 2009 (UTC)

This seems good info. Possibly better to find something in Pearson's actual writings, but no reason why this should not go in, with the citation mentioned. It might be best to move the existing stuff about this topic and join it with the above in a new section to improve the structure of the article. Melcombe (talk) 10:58, 8 December 2009 (UTC)

follows a?

What is the meaning of the "follows a" language inserted in the this diff, and of the sentence fragments that follow. I started to clean up the typography of things like gamma, but then didn't understand what the editor who made these sections was trying to say. Dicklyon (talk) 06:04, 19 March 2011 (UTC)

Type I

I noticed that the section on Type I refers to roots $a_{1}$ and $a_{2}$ that are of opposite sign. I think that the subsequent comment, $a_{1}<0<a_{2}$, may be incorrect, as it seems that this is only true if $b_{2}>0$. If $b_{2}<0$, which seems to be possible, then $a_{2}<0<a_{1}$. In some sense it shouldn't matter which root is called $a_{1}$ and which is called $a_{2}$, but I wonder if it might impact the way the rest of the section is written. 130.88.24.203 (talk) 23:34, 10 January 2015 (UTC)

Tabulation

I think the readability of the article will be greatly enhanced by summarising all the Types in tabular form. Manoguru (talk) 23:52, 7 July 2016 (UTC)