Thread: Help understanding order statistics formula

This may be a stupid question, but this formula is just not clicking with me.



I understand the rationale behind the factorial, but I do not understand the notation after the factorial. What does it represent? Is it... the individual density functions multiplied by each other? Can someone provide a few examples of how this formula would look in action using real #s? It seems like in the examples I am looking at they always each 1?

Again I apologize for the stupid question but I need a pointer or two here!

Hey MN1987.

There are a few different order statistics formulas that include one for a maximum, one for a minimum and one for a distribution that is so many places from the minimum or the maximum.

What does that one refer to? The one's I've seen for the general cases and the extreme cases are not the same.

I think the formula is one for the joint PDF of the particular sequence of that vector on the LHS. I copy/pasted it from this part on the wiki page:

Order statistic - Wikipedia, the free encyclopedia

What you wrote is not what is in the Wikipedia formula.

Anyway, the formula is based on having the maximum where all values are less than it (i.e for a maximum, you have every realization less than this value and the probability of this will be P(X1 < a)*P(X2 < a)*...*P(Xn < a) if you are looking at all observations being less than a specified maximum).

The minimum uses the reverse argument and the general one looks at a value where so many are less and so many are more where you have say X1 < a, X2 < a but X3 >= a, X4 >= a and so on, and note that in an I.I.D random sample, that P(X and Y) = P(X)P(Y) if X,Y independent.