Pointwise Bands Versus Whole-Curve Uncertainty

Why the Coverage Target Matters

An uncertainty band can look like one continuous ribbon around a fitted curve, but its visual appearance does not determine what its coverage statement means. A band made by calculating a 95% interval separately at each predictor value answers a point-by-point question. A band designed for the whole curve answers a stronger question: whether the entire true mean curve is covered at once.

The distinction matters when interpreting a plot. If a pointwise band is labelled “95%,” that does not mean there is a 95% probability that the whole curve lies inside it. To make a whole-curve statement, the calculation must account for uncertainty across the collection of predictor values, not only at one value at a time.

Model and Assumptions

For the formulas below, consider a regression curve that may be nonlinear in the predictor but is linear in its \(p\) coefficients. Write the mean response at predictor value \(x\) as \(m(x)=\mathbf{b}(x)^\mathsf{T}\boldsymbol{\beta}\), where \(\mathbf{b}(x)\) is a specified vector of \(p\) basis terms and \(\boldsymbol{\beta}\) is the coefficient vector. For example, a quadratic curve uses \(\mathbf{b}(x)=(1,x,x^2)^\mathsf{T}\). The observed response is the mean plus an error.

The pointwise t intervals and the simultaneous band derived here rely on the same model conditions: the chosen mean form is correct, the design matrix has full column rank, and the errors are independent with mean zero and common variance \(\sigma^2\). For the exact t and F results, assume also that the errors are normally distributed. In matrix terms, \(\boldsymbol{\varepsilon}\sim N(\mathbf{0},\sigma^2\mathbf{I})\). These conditions describe uncertainty in the fitted mean curve under a fixed model; they do not include uncertainty from choosing the curve form.

Let \(\mathbf{X}\) be the \(n\)-by-\(p\) design matrix whose \(i\)th row is \(\mathbf{b}(x_i)^\mathsf{T}\). The least-squares coefficient estimate is \(\hat{\boldsymbol{\beta}}\), and the estimated mean at \(x\) is \(\hat{m}(x)=\mathbf{b}(x)^\mathsf{T}\hat{\boldsymbol{\beta}}\). The estimated standard error of that mean is

Here \(s\) estimates the error standard deviation using the residual sum of squares divided by \(df=n-p\). This standard error is for estimating the mean curve at \(x\), not for predicting the response of a new individual observation.

Pointwise Confidence Intervals

At one predictor value \(x_0\), the estimated mean \(\hat{m}(x_0)\) is compared with its standard error. Under the stated assumptions, the standardized estimation error follows a t distribution with \(n-p\) degrees of freedom. Thus a two-sided pointwise \(100(1-\alpha)\%\) confidence interval is

The coverage claim is for the chosen fixed \(x_0\): over repeated samples under the model, this procedure covers \(m(x_0)\) in \(100(1-\alpha)\%\) of samples. If a different predictor value is selected, its interval also has that pointwise coverage. Those separate statements do not ensure that all the intervals cover their respective means in the same sample.

Simultaneous Bands for a Whole Curve

A standard simultaneous band for a regression curve linear in its coefficients is the Working–Hotelling band. It uses a larger critical value than the pointwise t interval because its target is simultaneous coverage across predictor values. For a two-sided band with confidence level \(1-\alpha\), the multiplier is \(\sqrt{pF_{p,n-p;1-\alpha}}\), where \(F_{p,n-p;1-\alpha}\) is the \(1-\alpha\) quantile of an F distribution with \(p\) and \(n-p\) degrees of freedom.

Under the stated normal linear model, this band covers the mean curve at every \(x\) in the domain simultaneously with probability at least \(1-\alpha\) over repeated samples. The multiplier comes from a confidence region for the entire coefficient vector, rather than from a separate test at each \(x\).

Why the Simultaneous Band Works

The justification begins with the joint uncertainty in \(\hat{\boldsymbol{\beta}}\). Under the model assumptions, the following ellipsoid is a \(100(1-\alpha)\%\) confidence region for the coefficient vector:

If the true coefficient vector lies in this ellipsoid, then the Cauchy–Schwarz inequality gives, for any predictor value \(x\),

The expression on the left is \(|\hat{m}(x)-m(x)|\). Substituting the ellipsoid’s upper bound into the second factor yields

This inequality holds for every \(x\) whenever the coefficient vector is inside the ellipsoid. Since the ellipsoid has the stated confidence level, the resulting band has at least that simultaneous coverage. In contrast, a pointwise interval controls the error for one selected \(\mathbf{b}(x)\) at a time, not for all such vectors together.

Simultaneous Coverage at a Finite Set of Values

Sometimes the question concerns a few specified predictor values rather than every point on a continuous curve. One simple option is the Bonferroni method. For \(q\) specified values, make each two-sided interval with confidence level \(1-\alpha/q\). The probability that at least one interval misses is no greater than the sum of the individual miss probabilities, which is at most \(q(\alpha/q)=\alpha\). This guarantees at least \(1-\alpha\) joint coverage for those \(q\) means, without requiring independence among the intervals. It does not guarantee coverage between the selected values.

Choosing and Explaining the Coverage Statement

The right interval or band depends on the question being asked. A pointwise interval is appropriate when the target is the mean at a particular value chosen in advance. A simultaneous band is appropriate when the goal is to make a confidence statement about the shape or level of the curve across a domain. A finite family of Bonferroni intervals can address a short list of specified values, but should not be described as covering every point between them.

The Working–Hotelling formula applies when the mean curve is linear in its coefficients and the stated model conditions hold. A curve with nonlinear-looking terms, such as \(x^2\), can still meet this requirement because its coefficients enter linearly. A model that is nonlinear in its parameters does not automatically have this exact band. Also, neither band accounts for uncertainty introduced by selecting a curve form after examining the same data. The confidence claim is conditional on the specified model and its assumptions.

Common Mistake