Portfolio managers and risk managers must monitor the behavior of certain random variables such as the daily returns of the stocks contained in the portfolio or the default rate of bonds comprising the bond portfolio under management. In both cases, we need to know the population parameters characterizing the respective random variable's probability distribution. However, in most realistic situations, this information will not be available.
In the previous chapter, we dealt with this problem by estimating the unknown parameter with a point estimator to obtain a single number from the information provided by a sample. It will be highly unlikely, however, that this estimate—obtained from a finite sample—will be exactly equal to the population parameter value even if the estimator is consistent—a notion introduced in the previous chapter. The reason is that estimates most likely vary from sample to sample. However, for any realization, we do not know by how much the estimate will be off.
To overcome this uncertainty, one might think of computing an interval or, depending on the dimensionality of the parameter, an area that contains the true parameter with high probability. That is, we concentrate in this chapter on the construction of confidence intervals.
We begin with the presentation of the confidence level. This will be essential in order to understand the confidence interval that will be introduced subsequently. We then present the probability of ...