Optimal integration of two or three PPS surveys with common sample size n > 1

Abstract

We consider a plan $P$ for the integration of $k$ surveys in the special case of a sample size one for each survey and $n$ independent repetitions of $P$ so as to ensure a sample size $n$ for each survey. We restrict our attention only to plans of this type which we denote by $Pˆn$. A plan is called optimal if it minimizes the expected number of distinct units in the integrated survey. It is shown that when $k=2$ and $P$ is obtained through the Mitra-Pathak algorithm then $P$ is indeed optimal in the above sense. The same is also true for $k=3$ if $\theta_2\leq 1$. We recall that $\theta_2=\sumˆ N_{j=1}P_{(2)j}$, where $P_{ij}$ is the probability of selecting the $j$th population unit as specified by the $i$th survey and $P_{(1)j}\leq P_{(2)j}\leq P_{(3)j}$ are the ordered values of $P_{1j}$, $P_{2j}$ and $P_{3j}$ are arranged in increasing order. When $\theta_2>1$ we identify a plan $P$ which is optimal for $n=1$ and has the following properties: $Pˆn$ is optimal for sufficiently large sample sizes $n$. A sufficient condition is stated under which $Pˆn$ is optimal for all sample sizes $n$. Numerical computation shows that even when $Pˆn$ is not optimal the loss in using $Pˆn$ is numerically insignificant.