vector with length $p$, ${\bm{{\beta}}}$ is an unknown vector of parameters to be estimated, and $\epsilon$ denotes the random error with mean $\hbox{E}(\epsilon) =0$ and variance $\hbox{Var}(\epsilon)=\sigma^2>0$. Building a fixed-sized confidence set for ${\bm{{\beta}}}$ with a prespecified coverage probability is a classical problem, and some authors have extended the thinking of [@Chow1965] to linear regression models [@albert1966; @gleser1965; @srivastava1971]. Since then, many papers have appeared in the literature that feature various setups and perspectives [@GRAMBSCH1989; @ChangMart1992; @MuthuPoruthotage2013; @Vorobeichikov:2017].

Assume that there is already a very large data set available for analysis, similar to that seen in most ‘big data’ scenarios. When its size is too large—such that using the data set all at once is impractical—the divide-and-conquer method is an economical, hands-on approach that requires fewer changes to existing tools (e.g. hardware and software). If we simultaneously run $M$ independent estimation procedures (e.g. use $M$ independent machines), then we will definitely reduce the computation time; at this point, how to partition the data set appropriately into $M$ partitions becomes an issue. Most practitioners follow the conventional sampling method by treating the available data set as a pseudo-population. Besides those studies that examine how to divide the original data set without 