Some order relations between selection and inclusion probabilities for PPSWOR sampling scheme (Q1185342)

A sample of size \(n\) is selected from a finite population with \(N\) units using a probability proportional to size without replacement (PPSWOR) sampling design based on \(P = (p_ 1,\dots,p_ N)\), \(\Sigma p_ i = 1\). Let \(\pi_ i\) be the probability of including the \(i\)th unit in the sample. In general, for any subset \(s\) of the \(N\) units let \(\pi(s)\) denote the probability of including \(s\) in the sample. This paper gives some order relations between the \(p_ i\), \(\pi_ i\), and \(\pi(s)\). For example, if for any \(i\) and \(j\), \(p_ i \geq p_ j\), then \(\pi_ i\geq\pi_ j\) and conversely. Some interesting bounds on the inclusion probabilities in terms of the selection probabilities are also established.