# What is the reason behind getting unbiased and efficient estimator by ranked set sampling?

In ranked set sampling (RSS), we select $n$ random sets, each of size $n$. Then we choose the largest unit from the 1st set, 2nd largest from the 2nd set, and thus $n$th largest from the $n$th set for the actual measurement.

$\bullet$ What is the intuition that a sample thus obtained will give an unbiased estimate of the population mean?

$\bullet$ What is the intuition that a sample thus obtained will yield more efficient estimator than an estimator from random sampling?

I know results of simulation show that RSS gives unbiased and efficient estimation. But without performing simulation, there must be an underlying theme that RSS gives unbiased and efficient estimation for such reasons. What are those reasons?

1. Unbiasedness. This means all units are just as likely to be selected. To see this, first note that all units are just as likely as one another to be selected in one of the $n$ sets. Next note that given a particular collection of $n$ sets, they are equally likely to be in any order. Given an element in a set, it has some rank in that set, say $k$. Then this element will be selected in the sample if the set is the $k$th set considered. It's just as likely to be $k$th as any other position, so the given element is just as likely to be selected as any others.