Order Statistics

[Jo_M.]

Does anyone know of a good website to learn about the theory involved with order statistics.

Also, what are the things about order stats that I should know by heart for the exam (pdf, cdf, etc etc)

[The_Czar52]

Does anyone know of a good website to learn about the theory involved with order statistics.

Also, what are the things about order stats that I should know by heart for the exam (pdf, cdf, etc etc)

The important order statistics (for the purpose of the exam) are the min and max. From what I hear, questions about order statistics don't appear on the exam with high frequency but nonetheless it is something you should know should it come up. I haven't had a textbook that really covered this in depth. I purchased the ASM manual and it covered it. Basically just practice problems that involve order statistics and thoroughly look over the solutions until you understand it. Wikipedia (below) gives a descent explanation and the "SOA 123" (2nd link below) has a lot of practice questions you could do after you have a basic grasp of the information.

max(X_1, X_2,...,X_n) =P(X_1<=x, X_2<=x,...,X_n<=x)
=P(X_1<=x)*P(X_2<=x)*...*P(X_n<=x)
=F_x1(x)*F_x2(x)*...*F_xn(x)

min(X_1, X_2,...,X_n) =P(X_1>=x, X_2>=x,...,X_n>=x)
=P(X_1>=x)*P(X_2>=x)*...*P(X_n>=x)
=(1-P(X_1<=x))*(1-P(X_2<=x))*...*(1-P(X_n<=x))
=S_x1(x)*S_x2(x)*...*S_xn(x)

(where S_xn(x) = 1-F_xn(x) is the survival function)

These are the definitions for max/min. You need to memorize these either by using notecards or by doing a ton of practice problems.

Hope this helps

[url]http://en.wikipedia.org/wiki/Order_statistics[/url]
[url]http://www.casact.org/admissions/studytools/exam1/P-09-05QS.pdf[/url]

[Jo_M.]

Thanks for the definitions. I will look things up. Does anyone know which of Dr. O or Broverman's sample questions are about order stats?

[JGET]

I suggest taking a look at Ross's book " A first course in probability." It doesn't have an extensive chapter on Order Statistics, but its dense and has good exercises.

JGET

[krzysio]

Thanks for the definitions. I will look things up. Does anyone know which of Dr. O or Broverman's sample questions are about order stats?

The ASM Manual (i.e., the one written by Dr. O., i.e., me) has extensive coverage of order statistics.
Yours,
Krzys'

[sineintegral]

Hi,
I have a question about order statistics and I hope this is a good as place as any to ask this:

Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample from the pdf e^(-x), x > 0. What is the P(Y4 > 2)?

My solution:
P(Y4 > 2) = 1 - P(Y4 <= 2) = 1 - [F(2)] ^4 = 1 - [1- e^(-2)] ^4.

Manual's solution:
P(Y4 > 2) = P(X > 2) ^4 = [1 - P(X <=2)]^4 =[1-(1 -e^(-2))]^4 = e^(-8).


Which is correct? :confused-:

[krzysio]

Hi,
I have a question about order statistics and I hope this is a good as place as any to ask this:

Let Y1 < Y2 < Y3 < Y4 be the order statistics of a random sample from the pdf e^(-x), x > 0. What is the P(Y4 > 2)?

My solution:
P(Y4 > 2) = 1 - P(Y4 <= 2) = 1 - [F(2)] ^4 = 1 - [1- e^(-2)] ^4.

Manual's solution:
P(Y4 > 2) = P(X > 2) ^4 = [1 - P(X <=2)]^4 =[1-(1 -e^(-2))]^4 = e^(-8).


Which is correct? :confused-:

I thought it was yours... but I misread it. Manual. Sorry.

Yours,
Krzys'

[cactus smash]

Isn't e^-8 the probability that the minimum is greater than 2?

[sineintegral]

Thanks for the check. As for the question whether the manual is saying what I claim it to be saying, anybody who has access to the manual can check page 34 of the practice exams of Averbach & Mehta.

[ctperng]

Thanks for the check. As for the question whether the manual is saying what I claim it to be saying, anybody who has access to the manual can check page 34 of the practice exams of Averbach & Mehta.

Another way to solve is to use the order statistic density function.

For the highest order, Y4, it has a density function
4*F(x)^(4-1)*f(x) = 4*(1-e^(-x))^3 *e^(-x),

therefore P(Y4 > 2) = int(2 to infinity) of 4*(1-e^(-x))^3 *e^(-x) dx

ctperng

Please check if there are further typos.