- Thesis:
- Master in Mathematics
- Author:
- Emma Dingel
- Title:
- Bernstein-type exponential inequalities in survey sampling
- Supervisors:
- Jan JOHANNES
- Abstract:
- In this work we consider the problem of estimating the population total from a random survey sample chosen according to a rejective sampling design. The estimator we utilise is the Horvitz-Thompson estimator, which sums up the normalised values of the observed random sample, making it an unbiased estimator. We establish Bernstein-type exponential tail bounds for this Horvitz-Thompson estimator, giving insight on the probability of its deviation from the population total by a certain amount. Since the sampled points are dependent on each other according to a rejective sampling, the classical Bernstein inequality in the independently distributed setting does not straightforwardly apply here. To overcome this problem, we characterise the rejective sampling as a conditional Poisson sampling and use this structure together with an Esscher transformation to derive the desired Bernstein-type exponential tail bounds. After having established these main results, we show how they can be used to derive similar tail bounds for the Horvitz-Thompson estimator with respect to more general sampling designs that are sufficiently close to the rejective sampling in the total variation distance. Lastly, we provide empirical versions of our results.
- Reference:
- P. Bertail and S. Clémençon. Bernstein-type exponential inequalities in survey sampling: Conditional Poisson sampling schemes, Bernoulli 25:4B,3527–3554, 2019.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |