Bootstrap近似的定性稳健性
Qualitative robustness for bootstrap approximations
摘要：
统计估计量的一个重要性质是定性稳健性，即数据分布的微小变化只会导致估计量分布的微小机会。此外，在实践中，数据的分布通常是未知的，因此可以使用Bootstrap近似来近似估计量的分布。因此，定性稳健性的统计估计下的自助近似是一个理想的属性。目前大多数定性稳健性的理论研究假设独立同分布的随机变量对。然而，在实践中，这一假设并不成立。因此，我们研究非独立同分布的自举近似的定性稳健性。随机变量，例如$α$ -混合和弱相关过程。在i.i.d.通过统计算子的连续性来保证案例定性的鲁棒性，统计算子代表估计量，参见Hampel（1971）和Cuevas和Romo（1993）。我们表明，定性的稳健性的自举近似仍然是确保统计算子是连续的假设下，并在一个额外的假设下的随机过程。特别是，我们需要一个收敛条件的经验措施的基本过程，所谓的Varadarajan属性。
英文摘要：
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 $α$-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.
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