Monte Carlo methods are a fundamental tool in many areas of statistics. In this thesis,
we will examine these methods, especially for rare event simulation. We are mainly
interested in the computation of multivariate normal probabilities and in constructing
hitting thresholds in survival analysis models.
Firstly, we develop an algorithm for computing high dimensional normal probabilities.
These kinds of probabilities are a fundamental tool in many statistical
applications. The new algorithm exploits the diagonalisation of the covariance matrix
and uses various variance reduction techniques. Its performance is evaluated via
a simulation study. The new method is designed for computing small exceedance
probabilities.
Secondly, we introduce a new omnibus cumulative sum chart for monitoring in
survival analysis models. By omnibus we mean that it is able to detect any change.
This chart exploits the absolute differences between the Kaplan-Meier estimator and
the in-control distribution over specific time intervals. A simulation study is presented
that evaluates the performance of our proposed chart and compares it to existing
methods.
Thirdly, we apply the method of adaptive multilevel splitting for the estimation of
hitting probabilities and hitting thresholds for the survival analysis cumulative sum
charts. Simulation results are presented evaluating the benefits of adaptive multilevel
splitting.
Finally, we extend the idea of adaptive multilevel splitting by estimating not
just a hitting probability, but the whole distribution function up to a certain point.
A theoretical result is proved that is used to construct confidence bands for the distribution function conditioned on lying in a closed interval.