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Hello math geniuses! Compsci dude here.

I'm trying to model the price distribution of a category of goods, and figure a LogNormalDistribution would provide a good fit. The problem is, I don't know how to do this with the data I have available.

I have several data points available: C(x1) to C(xn). For a given x, C(x) is the cumulative percentage of all items cheaper than x dollars. That is, if C(x) = .5, x is the median.

Given points C(x1) to C(xn), how can I find the best fit Log Normal Distribution? I suppose the question really is: how do I find the best fit log normal cumulative distribution function for a given set of points, and how "good" will the fit be?

Thanks a lot for the help!

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You might try math.stackexchange –  Will Jagy Jun 17 '11 at 3:55
    
I'll give it a shot, thanks. –  Franky Jun 17 '11 at 3:57
    
Is MathOverflow the more advanced version of math.stackexchange? Please excuse my ignorance. –  Franky Jun 17 '11 at 3:58
    
Yes. From the faq, "MathOverflow's primary goal is for users to ask and answer research level math questions, the sorts of questions you come across when you're writing or reading articles or graduate level books." –  Ricky Demer Jun 17 '11 at 4:17
    
Or even try stats.stackexchange.com, which is better for statistical questions –  David Roberts Jun 17 '11 at 6:10
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closed as off topic by Will Jagy, David Roberts, Andres Caicedo, Todd Trimble, Ryan Budney Jun 17 '11 at 22:12

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1 Answer

A simple estimation:

1) $y_i=\log(x_i)$

2) $\hat\mu = \sum_i y_i (F((y_i+y_{i+1})/2)-F((y_i+y_{i-1})/2)$

3) $\hat\sigma= (\sum_i y_i^2 (F((y_i+y_{i+1})/2)-F((y_i+y_{i-1})/2)) -\mu^2$.

Then $x \sim \mathrm{Lognormal}(\hat\mu,\hat\sigma)$

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What function is F() ? –  Franky Jun 20 '11 at 2:34
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