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 Post subject: Statistical distributions
PostPosted: Thu Jul 14, 2011 10:51 am 
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I need some help with a statistics/calculus problem please...

I have the following four-parameter inverse cumulative distribution function:

Q = A*ln(p) + B*ln(1-p) + C*(p-0.5) + D

p is the cumulative probability or percentile, and A, B, C and D are four independent parameters of the function.

Using my limited/forgotten abilities in mathematics, I managed to calculate the mean of this distribution as follows:

Mean = D - (A + B)

However, I need similar equations for the other 3 moments (standard deviation, skewness and kurtosis) based on the same four parameters (A, B, C and D), and also the corresponding PDF and CDF functions derived from the inverse function above, but this is where I got completely stuck and need some help...

Can anybody please help me with this?

EDIT: since the original problem has been solved (or rather proven that it can't be solved!), I thought that maybe this thread could continue as a general discussion on statistical distributions (in case anyone is interested), and have changed the thread title accordingly.

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If you want something you’ve never had, you’ve got to do something you’ve never done - Thomas Jefferson


Last edited by KelvinS on Fri Jul 22, 2011 6:05 am, edited 4 times in total.

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 Post subject: Re: Any statistics/calculus experts here?
PostPosted: Sun Jul 17, 2011 1:40 pm 
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I'm still struggling with this, even just to rearrange the equation to express p as a function of Q, with constants A, B, C and D. This should be basic algebra, but I can't figure it out...

Q = A*ln(p) + B*ln(1-p) + C*(p-0.5) + D

Can anybody help?

_________________
If you want something you’ve never had, you’ve got to do something you’ve never done - Thomas Jefferson


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 Post subject: Re: Any statistics/calculus experts here?
PostPosted: Wed Jul 20, 2011 3:32 pm 
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Kelvin Stott wrote:
This should be basic algebra


I would have to call that advanced algebra, since Q is a transcendental function of p. I'm not particularly confident that there *is* a closed form inverse for this function.


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 Post subject: Re: Any statistics/calculus experts here?
PostPosted: Wed Jul 20, 2011 4:41 pm 
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Location: The land of dreams, coincedentally located in Alberta
HeavyTanHat wrote:
Kelvin Stott wrote:
This should be basic algebra


I would have to call that advanced algebra, since Q is a transcendental function of p. I'm not particularly confident that there *is* a closed form inverse for this function.

For me, basic algebra is solving for x. If you have to press any keys on your calculator with more than 1 letter on it then it becomes advanced algebra :oops:

Well, I've only got to grade 9 math... should I be afraid of high school algebra?

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 Post subject: Re: Any statistics/calculus experts here?
PostPosted: Fri Jul 22, 2011 4:03 am 
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This site that uses Mathematica did not find a way to re-express the equation to isolate p.
http://www.quickmath.com

The bits p and 1-p in the equation remind me of binomial distributions. Does your source give any name to what your distribution is, or what situations it is supposed to apply to?


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 Post subject: Re: Any statistics/calculus experts here?
PostPosted: Fri Jul 22, 2011 4:53 am 
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nicknicknick wrote:
This site that uses Mathematica did not find a way to re-express the equation to isolate p.
http://www.quickmath.com
Thanks. In fact TomZ has already confirmed via PM that this equation cannot be inverted to give a closed expression of p. At first I was surprised, but now realise that this is the case for many equations.
nicknicknick wrote:
The bits p and 1-p in the equation remind me of binomial distributions. Does your source give any name to what your distribution is, or what situations it is supposed to apply to?
Yes, this is indeed some form of probability distribution function (actually it's technically a quantile function, expressing x in terms of p, rather than the other way round), but I developed this four parameter function myself to provide more flexibility than all of the established distributions. Thus with this function you can easily control the mean, standard deviation, skew and kurtosis, without having to select a completely different distribution function. Pretty neat, eh? :D

_________________
If you want something you’ve never had, you’ve got to do something you’ve never done - Thomas Jefferson


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 Post subject: Re: Any statistics/calculus experts here?
PostPosted: Fri Jul 22, 2011 5:19 am 
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Oh, quantiles, I see. Yes, neat -- I only knew about mean, sd, etc in terms of moments recently through Wikipedia, but not in school for some reason. One thing I figured out about quantiles a couple of years ago that I didn't know when I was in school is that there isn't just one official defined method of calculating them, and the many statistical software packages out there can give different answers using the same data because they are operating under different methods. It bugged me so much when I realized that. For instance, it may be insisted that the 50th percentile must equal the median, but the standard way to find the median of an even number of numbers might give an answer that is different depending on the percentile method that is chosen.

Ha -- I found my notes that I write to myself because I forget details: Quantiles can be defined so that only ranks of the actual members of the data are permitted. Altermatively, quantiles can be defined so that fractional ranks are also permitted, so that a quantile may refer to the ranking of an observation that doesn't actually exist in the data. Such an unseen "observation" can be derived from a calculation involving averaging or interpolating between seen observations.
Excel's PERCENTILE() function: "If k is not a multiple of 1/(n - 1), PERCENTILE interpolates to determine the value at the kth percentile."

I decided that the problem was that quantiles make best sense for theoretical infinite sets of numbers, and there are problems fitting the idea to finite sets of numbers. For instance, for a set of only three numbers, it seems silly to bother trying to define a 17th percentile distinguishable from an 18th percentile.


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 Post subject: Re: Any statistics/calculus experts here?
PostPosted: Fri Jul 22, 2011 5:35 am 
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Yes, you can get differences in the numbers, methods and definitions when measuring small population samples. Even standard deviation has two alternative definitinos and formulae for limited sample sizes, with denominator N or (N-1). But this is not an issue if you're looking at very large samples, or an overall distribution function that describes an infinite population size.

If you're interested in quantiles then I thoroughly recommend reading Statistical Modelling with Quantile Functions, by Warren Gilchrist. This book explains how the traditional approach of looking at statistical distributions in terms of their cumulative distribution function (CDF) and/or probability distribution function (PDF) has been extremely limiting, while quantile functions (inverse CDFs and PDFs) are much more useful, intuitive and also mathematically flexible for statistical modelling.

EDIT: since the original problem has been solved (or rather proven that it can't be solved!), I thought that maybe this thread could continue as a general discussion on statistical distributions (in case anyone is interested), and have changed the thread title accordingly.

_________________
If you want something you’ve never had, you’ve got to do something you’ve never done - Thomas Jefferson


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