# Confidence band

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. Confidence bands are often used as part of the graphical presentation of results in a statistical analysis. Confidence bands are closely related to confidence intervals, which represent the uncertainty in an estimate of a single numerical value.

## 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 $\hat{f}(x)\pm w(x)$ 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 $\hat{f}(x)$ 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 $\hat{f}(x)\pm w(x)$ 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.

Confidence bands for simulated data depicting the proportion of voters supporting a given candidate in election, as a function of the voters' ages. Pointwise 95% confidence bands, and simultaneous 95% confidence bands constructed using the Bonferroni method are shown.

## Confidence bands in regression analysis

Confidence bands commonly arise in regression analysis.[1] In the case of a simple regression involving a single independent variable, results can be presented in the form of a plot showing the estimated regression line along with either point-wise or simultaneous confidence bands. Commonly used methods for constructing simultaneous confidence bands in regression are the Bonferroni and Scheffé methods.

Confidence bands for a simple linear regression analysis using simulated data. Pointwise 95% confidence bands, and simultaneous 95% confidence bands constructed using Scheffé's method are shown.

## Other applications of confidence bands

Confidence bands arise whenever a statistical analysis focuses on estimating a function.

For example, confidence bands can be constructed around estimates of the empirical distribution function. Simple theory allows the construction of point-wise confidence intervals, but it is also possible to construct a simultaneous confidence band for the cumulative distribution function as a whole by inverting the Kolmogorov-Smirnov test, or by using non-parametric likelihood methods .[2]

Confidence bands have also been devised for estimates of density functions, spectral density functions[3] , quantile functions, scatterplot smooths, survival functions, and characteristic functions.

## Prediction bands

Prediction bands are related to prediction intervals in the same way that confidence bands are related to confidence intervals. Prediction bands commonly arise in regression analysis. The goal of a prediction band is to cover with a prescribed probability the values of one or more future observations from the same population from which a given data set was sampled. Just as prediction intervals are wider than confidence intervals, prediction bands will be wider than confidence bands.

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

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

where y* is an observation taken from the data-generating process at the given point x that is independent of the data used to construct the point estimate $\hat{f}(x)$ and the confidence interval w(x). This is a pointwise prediction interval. It would be possible to construct a simultaneous interval for a finite number of independent observations using, for example, the Bonferroni method to widen the interval by an appropriate amount.

## References

1. ^ 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.
2. ^ Owen, A.B. (1995). "Nonparametric likelihood confidence bands for a distribution function". Journal of the American Statistical Association (American Statistical Association) 90 (430): 516–521. doi:10.2307/2291062. JSTOR 2291062.
3. ^ Neumann, M.H.; Paparoditis, E. (2008). "Simultaneous confidence bands in spectral density estimation". Biometrika 95 (2): 381. doi:10.1093/biomet/asn005.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Confidence interval — This article is about the confidence interval. For Confidence distribution, see Confidence Distribution. In statistics, a confidence interval (CI) is a particular kind of interval estimate of a population parameter and is used to indicate the… …   Wikipedia

• Confidence region — In statistics, a confidence region is a multi dimensional generalization of a confidence interval. It is a set of points in an n dimensional space, often represented as an ellipsoid around a point which is an estimated solution to a problem,… …   Wikipedia

• confidence — [[t]kɒ̱nfɪdəns[/t]] ♦♦ confidences 1) N UNCOUNT: usu N in n If you have confidence in someone, you feel that you can trust them. I have every confidence in you... This has contributed to the lack of confidence in the police... His record on… …   English dictionary

• Band on the Run — For the title song from the album, see Band on the Run (song). Band on the Run Studio album by Paul McCartney & …   Wikipedia

• Band, the — ▪ American rock group Introduction       Canadian American band that began as the backing group for both Ronnie Hawkins and Bob Dylan (Dylan, Bob), then branched out on its own in 1968. The Band s pioneering blend of traditional country (country… …   Universalium

• band — I (New American Roget s College Thesaurus) n. strip, stripe, filament; brassard; belt, strap; group, crowd; orchestra, brass or military band. See assemblage, music. II (Roget s IV) n. 1. [A beltlike strip] Syn. ribbon, belt, line, strip, tape,… …   English dictionary for students

• Gentleman & the Far East Band — Gentleman im März 2005 nach einem Konzert in Stuttgart Gentleman (bürgerlich: Tilmann Otto; * 19. April 1975 in Osnabrück) ist ein deutscher Roots Reggae , Conscious Reggae , und Dancehall Musiker. Inhaltsverzeichnis …   Deutsch Wikipedia

• In Strict Confidence — Infobox musical artist Name = In Strict Confidence Img capt = Antje Schulz and Dennis Ostermannrfu c|2007 09 06 Background = group or band Origin = flagicon|Germany Germany Genre = Electro industrial Years active = 1989 ndash;Present Label = Zoth …   Wikipedia

• British Band — Participant in the Black Hawk War This 1854 artist rendering depicts members of the British Band at Stillman s Run …   Wikipedia

• Far East Band — Gentleman Far East Band auf dem Southside Festival Die Far East Band ist eine sechsköpfige, deutsche Band, die den Reggaemusiker Gentleman sowie andere Künstler bei ihren Auftritten begleitet. Hervorgegangen 1997 aus der Leipziger Band Messer… …   Deutsch Wikipedia