Inferential Statistics 369
where σX^2 denotes the common variance of each drawing. This variance can
be minimized with respect to the coefficients ai and we obtain the best lin-
ear unbiased estimator (BLUE) or minimum-variance linear unbiased esti-
mator (MVLUE). We have to be aware, however, that we are not always able
to find such an estimator for each parameter.
An example of a BLUE is given by the sample mean x. We know that
ani= (^1). This not only guarantees that the sample mean is unbiased for the
population mean μ, but it also provides for the smallest variance of all unbi-
ased linear estimators. Therefore, the sample mean is efficient among all
linear estimators. By comparison, the first draw is also unbiased. However,
its variance is n times greater than that of the sample mean.
Confidence Intervals
In making financial decisions, the population parameters characteriz-
ing the respective random variable’s probability distribution needs to be
known. However, in most realistic situations, this information will not
be available. In this section, we deal 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 section. 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 sec-
tion on the construction of confidence intervals. We begin with the presenta-
tion 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 error in the context of confidence intervals, which is related to
the confidence level.
Confidence Level and Confidence interval
In the previous section, we inferred the unknown parameter with a single
estimate. The likelihood of the estimate exactly reproducing the true param-
eter value may be negligible. Instead, by estimating an interval, which we
may denote by Iθ, we use a greater portion of the parameter space and not