Confidence Band - Pointwise and Simultaneous Confidence Bands

Pointwise and Simultaneous Confidence Bands

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 with coverage probability 1−α satisfies the following condition separately for each value of x:

{\rm Pr}\Big(\hat{f}(x)-w(x) \le f(x) \le \hat{f}(x)+w(x)\Big) = 1-\alpha,

where is the point estimate of f(x).

The simultaneous coverage probability of a collection of confidence intervals is the probability that all of them cover their corresponding true values simultaneously. In the example above, the simultaneous coverage probability is the probability that the intervals for x=18,19,... all cover their true values (assuming that 18 is the youngest age at which a person can vote). If each interval individually has coverage probability 0.95, the simultaneous coverage probability is generally less than 0.95. A 95% simultaneous confidence band is a collection of confidence intervals for all values x in the domain of f(x) that is constructed to have simultaneous coverage probability 0.95.

In mathematical terms, a simultaneous confidence band with coverage probability 1−α satisfies the following condition:

{\rm Pr}\Big(\hat{f}(x)-w(x) \le f(x) \le \hat{f}(x)+w(x) \;\;\;\;\forall x\Big) = 1-\alpha.

In nearly all cases, a simultaneous confidence band will be wider than a pointwise confidence band with the same coverage probability.

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