Simulational Algorithms

Many statistical problems are far too complex to be solved explicitly using pure paper-and-pencil mathematics. For example, many industrial systems consist of hundreds of inter-related variables with possibly different probability distributions. In modelling such a complex system mathematically, without the use of heavy computation, many assumptions will be made and many variables assumed constant or negligible in their contribution in order to arrive at a tractable model. However, with high speed computers (getting higher in speed every few months, it seems) readily available to virtually all practitioners and researchers, more and more complex systems may be modelled via statistical simulation in which many variables can be made to vary simultaneously, and the process may be studied with surprisingly high accuracy; we can finally “fit a model to the data” rather than fitting the data to a “friendly” model. As one becomes more familiar with problems in statistics and mathematics, and we see that even innocent-looking problems are sometimes impossible to solve explicitly, the appreciation for simulation and computing power grows rapidly. Many of us have already mused, “It is a wonder that any problems at all were solved without the use of computers.”