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There are at least two ways people look at statistical data:

A. For mathematicians, scientists, engineers, economists and such the most familiar distribution parameters would be analytical: mean, variance, and other central moments, maybe characteristic functions.

B. Everybody else would consider median and other percentiles, as well as max & min values for finite distributions.

Therefore a couple of questions:

  1. How would one connect the above? For example, given a finite number of moments, say m1-m4, how one would estimate median and quartiles?

  2. For finite distribution of size N, how one would estimate expected max & min values based on the 1st M moments, M << N?

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1 Answer 1

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Question 1. It is impossible to give any estimate without extra assumptions. Take a random variable which takes only two values: $a>0$ and $b<0$, both with probability $1/2$. The median is zero (see the remark below). All moments can be arbitrarily large or arbitrarily small: to make them large choose $a$ very large, and $b$ moderate.

Remark. If you disagree that the median of this random variable is $0$, modify this distribution by making it continuous with density strictly positive on $[a,b]$, but very small between $a$ and $b$. So that most of the density sits near $a$ and $b$.

With question 2, there are of course a trivial estimates: for example, if the max is $a>0$ then the $m$-th moment is at most $a^m$. Same with min. There is nothing else in general.

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  • $\begingroup$ Thanks, your answer to #1 is very illuminating. I think your remark on the location of median could keep the distribution finite by placing N values at a, N values at b, and 1 value at 0. About answer to #2, I wish there were stronger estimates than that: in applications one sometimes starts with limited info on the distribution, such as moments 0-4, and then has to detect outliers, the values beyond projected based on said moments max & min values. What would be the best method for that? $\endgroup$
    – Michael
    Aug 30, 2013 at 23:15

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