VVS-SMS AiO lectures 2004
VVS-SMS AiO lectures 2004
| Organised at the Korteweg-de Vries Institute for Mathematics of the Universiteit van Amsterdam under the auspices of the Section Mathematical Statistics (SMS) of the Dutch Society for Statistics and Operations Research (VVS) |
| 13.30-14.15 | Philip Mokveld | Efficient estimation in the accelerated failure time model under cross sectional sampling |
| 14.15-15.00 | Leila Mohammadi | On threshold-based classification rules |
| 15.00-15.30 | Coffee break | |
| 15.30-16.15 | Rachel Brouwer | Forest-fire models and self-destructive percolation |
| 16.15-17.00 | Misja Nuyens | The maximum queue length in the M/G/1 FB queue |
| 17.00 | Drinks |
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Rachel Brouwer: Forest-fire models and self-destructive percolation There are several ways to study forest fires using percolation techniques. In this talk we mention some possibilities. We will mainly focus on what we call self-destructive percolation: Consider site percolation on an infinite graph in which the sites are occupied with probability p and vacant with probability 1-p, independently of each other. Now suppose that, by some "catastrophe" all sites that are in the infinite occupied cluster become vacant. Finally each site that is vacant after the catastrophe, gets an extra enhancement to become occupied, i.e. each vacant site becomes occupied with probability a, independent of each other. When p is larger than but close to the critical value one might believe (for 'nice' graphs) that only a small probability a is needed to have an infinite occupied cluster in the final configuration. This appears to be the case for the binary tree. However, we strongly conjecture that it is not true for the square lattice. |
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Leila Mohammadi: On threshold-based classification rules This lecture contains some theories in statistical learning problems in the nonparametric setting. Suppose we are given n i.i.d. copies of a random variable (X,Y), where X is an instance and Y is a label, -1 or 1 . We define a classifier h as a function with values -1 and 1 and we assume H denotes a class of classifiers. If X is one dimensional and for some parametric cases of H such as the classifiers with K thresholds, we estimate the parameters by the minimizer of the classification error in the sample and we show that the cube root asymptotic results hold under some conditions (see also Mohammadi and van de Geer (2003)). We obtain the asymptotic distributions of the estimators. If one of the thresholds is at the border of the space of X, then the asymptotic result is different and convergence is quicker. We also consider the case that X is multidimensional and show that similar results hold when the classifiers are 1 on halfspaces. In a simple case, we show that the rate of convergence of the empirical risk minimizer is optimal. We also propose some algorithms to find the empirical risk minimizers in one dimensional case. |
| Philip Mokveld:
Efficient estimation in the accelerated failure time model under
cross sectional sampling Consider estimation of the regression parameter in the accelerated failure time model, when data are obtained by cross sectional sampling. It is possible under regularity of the model to construct an efficient estimator of the unknown Euclidian regression parameter whether the distribution of the covariate vector is known or not. For each of these cases a different techniques is used. |
| Misja Nuyens:
The maximum queue length in the M/G/1 FB queue In this talk we study the maximum queue length M, in terms of the number of customers present, in a busy period in the M/G/1 queue. The distribution of M depends both on the service time distribution and on the service discipline. Assume that the service times have a logconvex density and the service discipline is the so-called Foreground Background (FB) discpline. The FB service discipline gives service to the customer(s) that have received the least amount of service so far. It is shown that under these assumptions the tail of M is bounded by an exponential tail. |