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Last edited on
Oct 18, 2021 by JJ
.
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.