Let us consider a case where a ranked set sampling scheme from $n^2$ subjects has been adopted. There are a total of $n$ RSS blocks. So each RSS block contains $n$ subjects. We will follow up them. Some of these subjects will fulfill a certain criterion and get a treatment and some will not be able to fulfill the criterion and will not get the treatment. Again the person who will be able to complete the criterion first in the 1st RSS block, who will be able to fulfill the criterion second in the 2nd RSS block $\ldots$ and last in the last RSS block will receive a "special treatment". So, if all fulfilled the criterion then we would expect $n$ subjects to receive the "special treatment". But we have observed that out of $n^2$ subjects $M$ patients do not fulfill the criterion hence receive no treatment.
The response times of the other $n^2-M$ subjects are known to us. We also know the lifetimes for $M$ non-responders. And it can happen that a subject dies before responding to the criterion. Now what should be the probability that the $i$-th subject gets the special treatment?
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**The way I tried:**
**1)** First of all the probability of getting the **special treatment** for those who do not fulfill the criterion is $0$.
**2)** If we order the response times of the $n^2-M$ patients, then say the response times are $x_1<x_2<\ldots<x_i<\ldots<x_{n^2-M}$. So for $i$ to be the $k$-th smallest in a RSS block $(k-1)$ $x$'s should come before $x_i$ from the $x$'s. Let, among the $M$ non-responders, $M_i$ were observed to die before $x_i$ remitted. There can also be lowest $0$ to highest $m=min\{(n-k),M_i\}$ elements from these $M_i$ non-responders before $x_i$.
After $x_i$, there will be $(n-k-m)$ elements from the rest of $(n^2-i-M_i)$ elements.
I want to find out the probability that the $i$-th subject gets the **special treatment**.
Can anyone help please?
The response times of the other $n^2-M$ subjects are known to us. We also know the lifetimes for $M$ non-responders. And it can happen that a subject dies before responding to the criterion. Now what should be the probability that the $i$-th subject gets the special treatment?
----------
**The way I tried:**
**1)** First of all the probability of getting the **special treatment** for those who do not fulfill the criterion is $0$.
**2)** If we order the response times of the $n^2-M$ patients, then say the response times are $x_1<x_2<\ldots<x_i<\ldots<x_{n^2-M}$. So for $i$ to be the $k$-th smallest in a RSS block $(k-1)$ $x$'s should come before $x_i$ from the $x$'s. Let, among the $M$ non-responders, $M_i$ were observed to die before $x_i$ remitted. There can also be lowest $0$ to highest $m=min\{(n-k),M_i\}$ elements from these $M_i$ non-responders before $x_i$.
After $x_i$, there will be $(n-k-m)$ elements from the rest of $(n^2-i-M_i)$ elements.
I want to find out the probability that the $i$-th subject gets the **special treatment**.
Can anyone help please?