Order Statistics

[jansder]

I was just wondeirng if anyone could give me some helpful tips as what order statistics can be used for. I know it is the range from smallest to largest. Is there anything that says Order Statistic X is the mean or anything like that? Any advice would be greatly appreciated.

[Jo_M.]

I was just wondeirng if anyone could give me some helpful tips as what order statistics can be used for. I know it is the range from smallest to largest. Is there anything that says Order Statistic X is the mean or anything like that? Any advice would be greatly appreciated.

Basically, you have a sample of X_i's (X_1, X_2,...), and each of these random variables have the same probability distribution (exponential, for example). Then, you choose a certain number n of these X_i's, look at their value, and place them in increasing order. After putting them in order, they become an ordered sample (from Y_1 to Y_n), from which you can derive many useful information.

One thing you can get is the cdf of the largest ordered stat (labelled Y_n), then get its pdf using derivation:

F_yn(yn) = [F_x(yn)]^n where F_x is the cdf of any X_i

Another is the cdf of the smallest ordered stat (labelled Y_1):

F_y1(y1) = [1-F_x(y1)]^n

You could also find the joint pdf of the whole ordered sample:

f_yi(y1,...yn) = n! *f_x(y1) * f_x(y2)*...*f_x(yn)

You could find also find the joint pdf of 2 ordered stats (y_j, y_k):

f_yj,yk (yj,yk) = n!/((j-1)!*(k-j-1)!*(n-k)!) *[Fx(yj)]^(j-1) *[Fx(yk)-Fx(yj)]^(k-j-1) *[1-Fx(yk)]^(n-k) * fx(yj) * fx(yk)

Yes, this last formula is huge. You probably don't understand a thing, but write it out, and practice a bit with it an it'll blend into your brain.

[jansder]

Can you give me an example of a problem that uses order statistics? Preferablly one that doesn't say find the Order Statistics. One that implies that you use order statistics.

[jthias]

Basically, you have a sample of X_i's (X_1, X_2,...), and each of these random variables have the same probability distribution (exponential, for example). Then, you choose a certain number n of these X_i's, look at their value, and place them in increasing order. After putting them in order, they become an ordered sample (from Y_1 to Y_n), from which you can derive many useful information.

One thing you can get is the cdf of the largest ordered stat (labelled Y_n), then get its pdf using derivation:

F_yn(yn) = [F_x(yn)]^n where F_x is the cdf of any X_i

Another is the cdf of the smallest ordered stat (labelled Y_1):

F_y1(y1) = [1-F_x(y1)]^n

You could also find the joint pdf of the whole ordered sample:

f_yi(y1,...yn) = n! *f_x(y1) * f_x(y2)*...*f_x(yn)

You could find also find the joint pdf of 2 ordered stats (y_j, y_k):

f_yj,yk (yj,yk) = n!/((j-1)!*(k-j-1)!*(n-k)!) *[Fx(yj)]^(j-1) *[Fx(yk)-Fx(yj)]^(k-j-1) *[1-Fx(yk)]^(n-k) * fx(yj) * fx(yk)

Yes, this last formula is huge. You probably don't understand a thing, but write it out, and practice a bit with it an it'll blend into your brain.

A shorter formula for the last formula is to set U = Fx(x) = F(x), V = Fx(y) = F(y) . Yi and Yj are the ith and jth order statistics respectively of a random sample of size n taken from X.

Since U and V are cumulative distribution functions of a continuous random variable X, it follows that each of them is uniformly distributed on [0,1], so

f_yi,yj (x,y) = n!/((i-1)!*(j - i -1)!*(n - j)!) *[F(x)]^(i-1) *[F(y)-F(x)]^(j - i -1) *[S(y)]^(n-j) * f(x) * f(y)

is equivalent to the following shorter formula

f_u,v (u,v) = n!/((i-1)!*(j - i -1)!*(n - j)!) *u^(i-1) *[v - u]^(j - i - 1) *[1 - v]^(n-j)

Thanks go out to ctperng for helping me on this one a while back when I had trouble making the connection between the longer and shorter versions of the formula because of the cdf being uniformly distributed. Also thanks to Dr. O for motivating me to show that cdf of a continous rv is always uniformly distributed on [0,1].

hehe...haven't done this in a while..got a little lost with all these subscripts

Don't spend too much time on this, as using complicated formulas like this one or having to apply the fact that the cdf of a continous rv is uniformly distributed on [0,1] to solve an exam problem is very unlikely.

[JDav]

Can you give me an example of a problem that uses order statistics? Preferablly one that doesn't say find the Order Statistics. One that implies that you use order statistics.

X and Y are both exponential r.v.'s with mean 2. Find E[max{X,Y}].

[Jo_M.]

Can you give me an example of a problem that uses order statistics? Preferablly one that doesn't say find the Order Statistics. One that implies that you use order statistics.

Here's a link to a thread I started a while ago: [URL="http://actuary.com/actuarial-discussion-forum/showthread.php?t=5341"]http://actuary.com/actuarial-discussion-forum/showthread.php?t=5341[/URL]

This is the thread that motivated me to understand order stats.

Look at reply #20, I posted a good problem involving order stats.

[jansder]

I will take a look at this. What kinds of questions would involve a bivariate distribution besides one that says it is one?