Abstract
In this paper we present new theoretical results on optimal estimation of certain random quantities based on high frequency observations of a Lévy process. More specifically, we investigate the asymptotic theory for the conditional mean and conditional median estimators of the supremum/infimum of a linear Brownian motion and a strictly stable Lévy process. Another contribution of our article is the conditional mean estimation of the local time and the occupation time of a linear Brownian motion. We demonstrate that the new estimators are considerably more efficient compared to the classical estimators studied in e.g. [6, 14, 29, 30, 38]. Furthermore, we discuss pre-estimation of the parameters of the underly-ing models, which is required for practical implementation of the proposed statistics.
| Original language | English |
|---|---|
| Journal | Electronic Journal of Statistics |
| Volume | 16 |
| Issue | 1 |
| Pages (from-to) | 892-934 |
| Number of pages | 43 |
| ISSN | 1935-7524 |
| DOIs | |
| Publication status | Published - Jul 2022 |
Keywords
- Conditioning to stay positive
- local time
- Lévy processes
- occupation time
- optimal estimation
- self-similarity
- supremum
- weak limit theorems