- Thesis:
*Master in Mathematics*- Author:
**Sergio Filipe Brenner Miguel**- Title:
*Data-driven Laguerre estimation under multiplicative censoring*- Supervisor:
- Jan JOHANNES
- Abstract:
- In this Master’s thesis we study the nonparametric estimation of the
density
*f*and the survival function*S*of a positive real-valued random variable*X*which is subject to multiplicative censoring. More precisely, rather than observing*X*directly we only have access to the product*Y = XU*where*X*and the*β(1, k)*-distributed random variable*U*are independent. Given an i.i.d. sample of*Y*the proposed nonparametric estimator relies on a projection onto the Laguerre basis, which we introduce und discuss in detail. Under varying regularity conditions and certain smoothness condition on*f*(respectively on*S*) we derive upper bounds for the mean integrated squared error of the Laguerre projection estimators. We show, that in the case of direct observations, that is*k = 0*, the upper bound provides up to a constant also a lower bound for the mean integrated squared error of the estimator, and hence the estimator is optimal in this situation. However, the optimal estimator relies on suitable a choice of a dimension parameter, which depends on characteristics of*f*(respectively*S*) which are not known in practice. Therefore, we introduce and analyse a data-driven choice of the dimension parameter. We show that the data-driven estimator still can attain the optimal bounds. In the end, the theoretical results are illustrated by a Monte-Carlo simulation of several examples. - Reference:
- D. Belomestny, F. Comte and V. Genon-Catalot.
*Nonparametric Laguerre estimation in the multiplicative censoring model*Electronic Journal of Statistics, 10(2):3114–315, 2016.