Variance estimator in complex surveys using linear regression with expansion factor as independent variable

In probability sampling, variance estimation of an estimated mean or total requires developing a mathematical expression that depends on the design used to extract a sample. These formulae can be difficult to build and sometimes involve computation of joint inclusion probabilities of selection, which can be hard to obtain. For some sampling designs it is not possible to obtain an unbiased estimator of the variance. These designs include the selection of one element or one large primary sampling unit within some strata, or systematic selection of units or primary sampling units within strata. The problem of variance estimation may also arise from an analytical perspective, while estimating means or totals in unplanned domains, it is possible to arrive at only one unit or cluster within some strata. In this article, we propose a linear regression variance estimator which is very simple to compute and gives a solution to the aforementioned problems. Some examples using different designs are given.