# Confidence bands

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A **confidence band** is an object used in a statistical analysis, and most usually occurs as part of a graphical presentation of results. Confidence bands are closely allied in concept to confidence intervals, but the interpretation is subtly different.

One application where confidence bands are used is in regression analysis, in the case of a simple regression involving a single dependent variable. Results can be presented in the form of a plot showing the estimated regression line. A first extension of this is to include lines showing point-wise confidence limits: the lines are drawn through the upper and lower limits obtained treating each possible value of the dependent variable separately. Two cases are commonly dealt with:

- confidence limits for the location of "true" regression line;
- confidence limits for the value of a new observation from the same population as the experimental data.

A second extension of the plot is to include confidence bands: these are lines derived from the observed dataset which define a region such that, across notional repetitions of the experiment, regions devised in the same way have a given probability that the true regression line lies entirely within such a region.

A confidence band typically has a similar shape to the region defined by the limits of the point-wise confidence intervals, but is rather wider.

A second type of application where confidence bands are available are those deriving from the empirical distribution function. Again, simple theory allows the construction of point-wise confidence intervals, but it is also possible to construct a confidence band for the distribution function as a whole. See Setting confidence limits for a distribution function for example, or Owen (1995).

Confidence bands have also been devised for applications to spectral density estimation.

## References

- Liu W., Lin S., Piegorsch W.W.. (2008) "Construction of Exact Simultaneous Confidence Bands for a Simple Linear Regression Model".
*International Statistical Review*, 76 (1), 39–57 doi:10.1111/j.1751-5823.2007.00027.x - Owen A.B. (1995) "Nonparametric likelihood confidence bands for a distribution function".
*Journal of the American Statistical Association*, 90, 516–521. - Neumann, M.H., Paparoditis, E. (2008) "Simultaneous confidence bands in spectral density estimation". Biometrika, 95 (2). doi:10.1093/biomet/asn005

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