# Qualitative robustness for bootstrap approximations

## TL;DR

This paper investigates the qualitative robustness of bootstrap approximations for non-i.i.d. data, extending existing theory to dependent processes like $eta$-mixing, and establishes conditions under which robustness is preserved.

## Contribution

It extends the theory of qualitative robustness of bootstrap methods to dependent data processes, introducing new conditions for robustness beyond i.i.d. assumptions.

## Key findings

- Qualitative robustness holds for certain dependent processes under specific conditions.
- Continuity of the statistical operator is crucial for robustness.
- A convergence condition called the Varadarajan property is necessary for robustness in dependent data.

## Abstract

An important property of statistical estimators is qualitative robustness, that is small changes in the distribution of the data only result in small chances of the distribution of the estimator. Moreover, in practice, the distribution of the data is commonly unknown, therefore bootstrap approximations can be used to approximate the distribution of the estimator. Hence qualitative robustness of the statistical estimator under the bootstrap approximation is a desirable property. Currently most theoretical investigations on qualitative robustness assume independent and identically distributed pairs of random variables. However, in practice this assumption is not fulfilled. Therefore, we examine the qualitative robustness of bootstrap approximations for non-i.i.d. random variables, for example $\alpha$-mixing and weakly dependent processes. In the i.i.d. case qualitative robustness is ensured via the continuity of the statistical operator, representing the estimator, see Hampel (1971) and Cuevas and Romo (1993). We show, that qualitative robustness of the bootstrap approximation is still ensured under the assumption that the statistical operator is continuous and under an additional assumption on the stochastic process. In particular, we require a convergence condition of the empirical measure of the underlying process, the so called Varadarajan property.