Estimators for the population mean and variance for stratified sampling
The search for unbiased estimators in a suboptimal sample
L.J. Verbeeke (TU Delft - Electrical Engineering, Mathematics and Computer Science)
Alexis Derumigny – Mentor (TU Delft - Statistics)
G. F. Nane – Graduation committee member (TU Delft - Applied Probability)
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Abstract
Dividing a population into subgroups and conducting research on this population including the subgroups comes with a challenge. This stratified sampling relies on information about the share of the subgroups in the population. Sometimes the proportions in the sample are not taken equal to the true proportions of the population. This can be corrected through the use of particular estimators taking these proportions into account.
In this thesis, different estimators for the true population mean and variance are defined and examined in terms of bias and variance in the case of two subgroups. Weighing the measurements according to the true proportions creates unbiased estimators for both the mean and the variance. These unbiased estimators are compared with other, biased, estimators, including naive ones in which the influence of different subgroups is not taken into account. The naive estimators are not only biased, they also have a variance of the same order as the unbiased ones. When the true proportions are not available, one can only take a guess. A guess lying close to the true proportions leads to a smaller bias and therefore a better estimator. This underlines the importance of obtaining sufficient knowledge about the population.