Variance issue in systematic sampling

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Empirical approximation of error variance

Again and again: there is no design-unbiased variance estimator in systematic sampling. If we are interested in the true error variance, the only way is to very often repeat the systematic sample and calculate the variance of all the estimations produced; that is then an empirical approximation to the parametric error variance which is the closer to the unknown true value the larger the number of repetitions is. Of course, this is not a viable approach for practical implementation, but it is something that can be done in computer simulations.

Using SRS estimators

What is most frequently done for variance estimation in systematic sampling is that the simple random sampling framework of estimators is applied. It is clear and known that these estimators are not unbiased for systematic sampling but they yield consistently over-estimations of the true error variance; this positive bias can be considerable. We call this sort of estimation a “conservative estimation”: we know that the true error is less (in many cases much less) than the estimation that has been calculated. An example is presented further down in chapter ‎6.2.2, where the area estimation by dot grids is presented.

Remember

For error variance estimation we use to apply the simple random sampling estimator

\(s_\bar y^2=\frac{s^2}{n}\)

(essentially, because we do not \(k\) now better …). This, however, is not an unbiased estimator but produces an overestimation of the true error variance.

References

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