Random and fixed replacement sampling plans (Q2640265)

Consider samples of size \(n\geq 2\) from a finite population consisting of N elements where \(N\geq n\). Let the row vector \({\underset{\tilde{}} \pi}=(\pi_ 1,...,\pi_{n-1})\) belong to \(\{0,1\}^{n-1}\). A fixed replacement sampling plan is given as follows. At stage i, \(i=1,2,...,n- 1\), an element is selected at random from the available population elements, the chosen i-th sample element will be replaced into the population before sampling the \((i+1)th\) element if \(\pi_ i=1\), and this will not be done if \(\pi_ i=0\). Consequently, \({\underset{\tilde{}} \pi}=(1,...,1)\) defines sampling with replacement and \({\underset{\tilde{}} \pi}=(0,...,0)\) sampling without replacement. If \({\underset{\tilde{}} \pi}\in [0,1]^{n-1}\) then a random replacement sampling plan is defined as follows. When the i-th sample element has been chosen this element will be replaced with probability \(\pi_ i\) into the population before sampling the next element. A rather weak order relation on the set of fixed replacement sampling plans is introduced and a class of functions on the set of possible samples is defined. It is shown that the expectation of a function in this class is monotone in the ordering of the underlying fixed replacement sampling plans. A similar result with a stronger order relation for random replacement sampling plans is proved for the same class of functions on the possible samples.