- Thesis:
- Master in Mathematics
- Author:
- Charlotte Maschmann
- Title:
- Interarrival density estimation for renewal processes
- Supervisors:
- Jan JOHANNES
- Abstract:
- In this thesis, we deal with the nonparametric estimation of the probability density of the interarrival times of a renewal process R. Our approach utilizes projection estimators, which we detail across three distinct cases and corresponding observation variables. In the first case, we operate under the assumption of continuously observing the renewal process R, allowing us to utilize the true interarrival times to define an estimator. In the other cases, we consider a discrete setting in which information on R is only given at a sampling rate. Based on the incomplete information we construct estimators for the interarrival times, which build the foundation for the subsequent estimators. The discrete observation introduces supplementary variables into the equation, which need to be handled to obtain accurate projection estimators. A straightforward approach involves bounding the error term arising from the additional variables, which is what we do for the second estimator. We obtain a projection estimator mirroring the first one’s behaviour as the samplinge rate approaches 0. In the third case, we consider a deconvolution framework taking into account the density of the new variables emerging from the discrete model. Under additional assumptions, we derive upper bounds for the mean integrated squared error of each estimator, allowing us to prove rates of convergence on Sobolev-Laguerre spaces comparable to minimax optimal rates. In all cases, a data-driven procedure for selecting the dimension parameter of the projection estimator, reaching an automatic bias-variance compromise, is presented. Up to a logarithmic factor, the rates attainable through this approach correspond to the rates for the optimal dimension. We demonstrate the outcome of the theoretical part with a comprehensive simulation study.
- Reference:
- F. Comte and C. Duval. Statistical inference for renewal processes, Scandinavian Journal of Statistics, 45(1):164-193, 2018.
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |