- Thesis:
- Bachelor in Mathematics
- Author:
- Clemens Hecht
- Title:
- Optimal adaptive and minimax estimation on R of the derivatives of a density
- Supervisors:
- Sergio Brenner Miguel
- Jan JOHANNES
- Abstract:
- This thesis studies the nonparametric estimation of the d−th derivative of a probability density, with support on R, relying on a sample of i.i.d. observations. We propose a projection estimator based on the Hermite basis on finite dimensional subspaces. The integrated L2-risk is studied and an optimal rate of convergence is given. It is also shown that this rate of convergence is minimax optimal under regularity conditions. Since the optimal rate of convergence depends on an unknown parameter, we provide a fully data driven estimator of the optimal dimension, which minimizes the bias-variance trade-off automatically. Further simulations show that the estimator is not only convincing in theory but also in practice. In the end, we compare our estimator with a more intrinsic approach to the estimator of the d−th derivative.
- References:
- F. Comte, C. Duval, and O. Sacko. Optimal adaptive estimation on R or R+ of the derivatives of a density, HAL Id: hal-02296067, 2019.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |