Talk:Order statistic

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I created the page on order statistics. Let us see if we can merge these pages together..

No offence, but for how important this topic is in statistics and how elegant its theory is, the article does an atrocious job. — Miguel 14:07, 2005 May 1 (UTC)

Not done yet, but at least now the article does not just dump a pile of equations on the reader without explanation or context. — Miguel 15:04, 2005 May 1 (UTC)

[edit] Old derivation saved for reference

Let be iid continuously distributed random variables, and X_{(1)}, X_{(2)}, \ldots, X_{(n)} be the corresponding order statistics. Let f(x) be the probability density function and F(x) be the cumulative distribution function of Xi. Then the probability density of the kth statistic can be found as follows.

f_{X_{(k)}}(x)={d \over dx} F_{X_{(k)}}(x) ={d \over dx}P\left(X_{(k)}\leq x\right)={d \over dx}P(\mathrm{at}\ \mathrm{least}\ k\ \mathrm{of}\ \mathrm{the}\ n\ X\mathrm{s}\ \mathrm{are}\leq x)
={d \over dx}P(\geq k\ \mathrm{successes}\ \mathrm{in}\ n\ \mathrm{trials})={d \over dx}\sum_{j=k}^n{n \choose j}P(X_1\leq x)^j(1-P(X_1\leq x))^{n-j}
={d \over dx}\sum_{j=k}^n{n \choose j} F(x)^j (1-F(x))^{n-j}
=\sum_{j=k}^n{n \choose j} \left(jF(x)^{j-1}f(x)(1-F(x))^{n-j} +F(x)^j (n-j)(1-F(x))^{n-j-1}(-f(x))\right)
=\sum_{j=k}^n\left(n{n-1 \choose j-1}F(x)^{j-1}(1-F(x))^{n-j} - n{n-1 \choose j} F(x)^j(1-F(x))^{n-j-1} \right)f(x)
=nf(x)\left(\sum_{j=k-1}^{n-1} {n-1 \choose j} F(x)^j (1-F(x))^{(n-1)-j} - \sum_{j=k}^n {n-1 \choose j} F(x)^j (1-F(x))^{(n-1)-j}\right)

and the sum above telescopes, so that all terms cancel except the first and the last:

=nf(x)\left({n-1 \choose k-1} F(x)^{k-1} (1-F(x))^{(n-1)-(k-1)} - \underbrace{{n-1 \choose n} F(x)^n (1-F(x))^{(n-1)-n}}\right)

and the term over the underbrace is zero, so:

=nf(x){n-1 \choose k-1} F(x)^{k-1} (1-F(x))^{(n-1)-(k-1)}
={n! \over (k-1)!(n-k)!} F(x)^{k-1} (1-F(x))^{n-k} f(x).


COMMENT BY an actuary (knows prob + stat, not academic) The section "Distribution of each order statistic of an absolutely continuous distribution" would be clearer if it went like this:

1. Explain that you will derive the CDF and then take its derivative to get the pdf. 2. Derive the CDF. 3. Take the derivative.

The section Probability Distributions of Order Statistics should (optimally) reference another article for why F(X) ~ uniform. It is not obvious to newbies.

The interpolatory comments (such as the one about time series) should be distinguished somehow (i.e., with parentheses) so that the reader knows that they are not central to the argument of the article.

---End last guy's comment---

I disagree with your first point. It's easy to get an expression for the CDF, so it's the obvious starting point for a proof. But by far the easiest way to get the sum for the CDF into closed form is by differentiating it, fiddling with it until it's in closed form, then integrating it - which gives you the PDF along the way. It makes for a slightly confusing proof, but the alternatives - either trying to find an expression for the PDF from first principles, or trying to deal with that sum without differentiating it - seem deeply unpleasant. (Plus, at least according to Mathematica, the CDF's closed form seems to involve hypergeometric functions - making it a lot more complicated than the PDF.)

86.3.124.147 (talk) 22:53, 2 April 2008 (UTC)

[edit] Attention needed

I have placed an "expert needed" tag on this article partly because of the empty sections but mainly because of the relation between the parts that derive the distributions of the order statistics. The first part of what is there might be considered a direct approach and is probably OK for that. But a more advanced approach would be to start from the uniform distributiuon case, and to derive the more general case from this, which involves less complicated formulae. I am slightly unhappy about the 'du' approach taken for the uniform case. I do think the "uniform" part needs to be finished off by giving an expicit statement for the distribition function, possibly using an incomplete beta function, but certainly as an integral ... from which the density for the general case could then be derived. Melcombe (talk) 08:51, 11 April 2008 (UTC)