Stratified Sampling & Notation

[drhugrobot]

In the ASM manual for the MFE exam, the authors use the notation u sub 4i+1, u sub 4i+2, u sub 4i+3 and u sub 4i+4 to denote which uniform number corresponds to each strata.

Examples then go on to match u sub 1 to the first strata (u sub 4i+1) and u sub 2 to the second strata (u sub 4i+2), so I assume the index i is always zero. I must be missing something here, what am I not understanding?

For example, if we had ten strata, would the notation then be u sub 10i+1, u sub 10i+2 ... u sub 10i+10?

[NoMoreExams]

It's easiest to work through an example to make sense of notation. Have you checked out sample 57?

[Nokio720]

In the ASM manual for the MFE exam, the authors use the notation u sub 4i+1, u sub 4i+2, u sub 4i+3 and u sub 4i+4 to denote which uniform number corresponds to each strata.

Examples then go on to match u sub 1 to the first strata (u sub 4i+1) and u sub 2 to the second strata (u sub 4i+2), so I assume the index i is always zero. I must be missing something here, what am I not understanding?

For example, if we had ten strata, would the notation then be u sub 10i+1, u sub 10i+2 ... u sub 10i+10?

The index is not zero, and you don't need an index of 10 for 10 uniform numbers. Here is a way you may interpret this section

Case 1: Suppose a want to "strata" four uniform numbers a_1,a_2,a_3,a_4 into the intervals (0,.25], (.25,.5], (.5,.75], (.75,1]. Then the keeping in line with the manual,

a_1 will go to [0,.25),
a_2 will go to [.25,.5)
a_3 will go to [.5,.75)
a_4 will go to [.75,1)

Case 2: If you have more than 4 uniform numbers that you want to "strata"
a_1,a_2,..,a_5,..,a_8. The question is once you get to the 5th number where do you place it? The 4th interval, the 1st? The book suggests that you start at the first interval or

a_1, a_5 will go to [0,.25),
a_2, a_6 will go to [.25,.5)
a_3, a_7 will go to [.5,.75)
a_4, a_8 will go to [.75,1)

The book uses the phrase for every i number 4i+1 goes to the 1st strata ....
You should think of it as if a number is of the form 4i+1 it goes to the 1st strata, 41+2 to the 2nd strata . . . 4i+4 goes to the 4th strata. You can see that 5=4(1) +1 is in the 1st strata, 6=4(1)+2 is in the second, and 8=4(1)+4 to the 4th strata. Its merely a means of accounting for a scenario of stratifying more that 4 numbers.

So if by chance you had 24 uniform numbers you know the 21st = 4(5)+1 goes to the 1st strata, ...and so on.

Hope this helps!

[drhugrobot]

''', thanks for the great explanation. I completely understand the notation now. Thank you!