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1. ## Order Statistics

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.

2. Originally Posted by 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.
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.

3. 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.

4. Originally Posted by Jo_M.
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.

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

6. Originally Posted by 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.

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

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

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

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