Confidence band

A confidence band is used in statistical analysis to represent the uncertainty in an estimate of a curve or function based on limited or noisy data. Similarly, a prediction band is used to represent the uncertainty about the value of a new data-point on the curve, but subject to noise. Confidence and prediction bands are often used as part of the graphical presentation of results of a regression analysis.

Confidence bands are closely related to confidence intervals, which represent the uncertainty in an estimate of a single numerical value. "As confidence intervals, by construction, only refer to a single point, they are narrower (at this point) than a confidence band which is supposed to hold simultaneously at many points."

Suppose our aim is to estimate a function f(x). For example, f(x) might be the proportion of people of a particular age x who support a given candidate in an election. If x is measured at the precision of a single year, we can construct a separate 95% confidence interval for each age. Each of these confidence intervals covers the corresponding true value f(x) with probability 0.95. Taken together, these confidence intervals constitute a 95% pointwise confidence band for f(x).

In mathematical terms, a pointwise confidence band ${\displaystyle {\hat {f}}(x)\pm w(x)}$ with coverage probability 1−α satisfies the following condition separately for each value of x:

where ${\displaystyle {\hat {f}}(x)}$ is the point estimate of f(x).

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