## Computing with Confidence

Confidence boxes ("c-boxes") tell you confidence intervals for a parameter at any confidence level you like.  For instance, the confidence box depicted below yields several confidence intervals for the parameter θ.  Although you can't generally compute with confidence intervals, you can compute with confidence boxes, and you can get arbitrary confidence intervals for the results.

Confidence boxes can be computed in a variety of ways directly from random sample data.  There are confidence boxes for both parametric problems where the family of the underlying distribution from which the data were randomly generated is known (including normal, lognormal, exponential, binomial, Poisson, etc.), and nonparametric problems in which the shape of the underlying distribution is unknown.  Confidence boxes account for the uncertainty about a parameter that comes from the inference from observations, including the effect of small sample size, but also the effects of imprecision in the data and demographic uncertainty which arises from trying to characterize a continuous parameter from discrete data observations.

When confidence boxes have the form of probability boxes, they can be propagated through mathematical expressions using the ordinary machinery of probability bounds analysis, and this allows analysts to compute with confidence, both figuratively and literally, because the results also have this confidence interpretation.

Confidence boxes are imprecise generalizations of traditional confidence distributions.  Like Student's t distribution, they encode frequentist confidence intervals for parameters of interest at every confidence level.  They are analogous to Bayesian posterior distributions in that they characterize the inferential uncertainty about distribution parameters estimated from sparse or imprecise sample data, but they have a purely frequentist interpretation that makes them useful in engineering because they offer a guarantee of statistical performance through repeated use.  Unlike traditional confidence intervals which cannot usually be propagated through mathematical calculations, c-boxes can be used in calculations to yield results that also admit the same confidence interpretation.  For instance, they can be used to compute probability boxes for both prediction and tolerance distributions.  They are easy to construct and use in calculations; see the software page for R functions to construct several c-boxes.

Note that c-boxes are completely different from confidence bands such as the Kolmogorov-Smirnov distributional bands which are nonparametric confidence limits at some particular confidence level for the distribution from which sample data were randomly drawn.  C-boxes encode confidence intervals at all possible confidence levels at the same time.