A Unique Unbiased Estimator with an Interesting Property

Suppose an unknown proportion p of the elements of an infinite population have size a > 0, but the remainder all have size b > a.We draw a random sample X of size 3, (x 1, x 2, x 3), from the infinite population.We then draw from X a sample Y of two elements successively without replacement and with probability proportional to size (PPSWOR sampling).Then it turns out that, when only Y is observable, there is a unique unbiased estimator of p, which has the interesting property of being a ridiculous unbiased estimator (RUBE), since it takes values outside of (0, 1).This is also the unique unbiased estimator for conditionally estimating the proportion of a's in X.