Chapter 3

Fundamentals of Statistics*

The preceding chapter was mainly concerned with the theory of probability, including distribution theory. In practice, researchers have to find methods to choose among distributions and to estimate distribution parameters from real data. The subject of sampling brings us now to the theory of statistics. Whereas probability assumes the distributions are known, statistics attempts to make inferences from actual data.

Here, we sample from the distribution of a population, say the return on a stock market index, to make inferences about the population. Issues of interest are the choices of the best distribution and of the best parameters.

In addition, risk measurement deals with large numbers of random variables. As a result, we also need to characterize the relationships between risk factors. For instance, what is the correlation between U.S. and UK stock indices? This leads to the need to develop decision rules to test hypotheses, for instance whether the volatility for a risk factor remains stable over time, or whether the relationship between these stock indices is significant.

These examples illustrate two important problems in statistical inference: estimation and tests of hypotheses. With estimation, we wish to estimate the value of an unknown parameter from sample data. With tests of hypotheses, we wish to verify a conjecture about the data.

This chapter reviews the fundamental tools of statistics theory for risk managers. Parameter estimation ...