# Confidence Bands (confidence + bands)

Distribution by Scientific Domains
Distribution within Mathematics and Statistics

Kinds of Confidence Bands

 simultaneous confidence bands

## Selected Abstracts

### Construction of Exact Simultaneous Confidence Bands for a Simple Linear Regression Model

INTERNATIONAL STATISTICAL REVIEW, Issue 1 2008
Wei Liu
Summary A simultaneous confidence band provides a variety of inferences on the unknown components of a regression model. There are several recent papers using confidence bands for various inferential purposes; see for example, Sun et al. (1999), Spurrier (1999), Al-Saidy et al. (2003), Liu et al. (2004), Bhargava & Spurrier (2004), Piegorsch et al. (2005) and Liu et al. (2007). Construction of simultaneous confidence bands for a simple linear regression model has a rich history, going back to the work of Working & Hotelling (1929). The purpose of this article is to consolidate the disparate modern literature on simultaneous confidence bands in linear regression, and to provide expressions for the construction of exact 1 ,, level simultaneous confidence bands for a simple linear regression model of either one-sided or two-sided form. We center attention on the three most recognized shapes: hyperbolic, two-segment, and three-segment (which is also referred to as a trapezoidal shape and includes a constant-width band as a special case). Some of these expressions have already appeared in the statistics literature, and some are newly derived in this article. The derivations typically involve a standard bivariate t random vector and its polar coordinate transformation. Résumé Un intervalle de confiance simultanée fournit une variété d'inférences sur les composantes inconnues d'un modéle de régression. Plusieurs articles récents utilisent des intervalles de confiance dans des buts variés; voir par exemple Sun, Raz et Faraway (1999), Spurrier (1999), Al-Saidy et al. (2003), Liu, Jamshidian et Zhang (2004), Bhargava et Spurrier (2004), Piegorsch et al. (2005), Liu et al. (2007). La construction d'intervalles de confiance simultanés pour un simple modéle de régression linéaire a une histoire riche, qui remonte aux travaux de Working et hotelling (1929). L'objet de cet article est de consolider la littérature moderne disparate sur les intervalles de confiance simultanés dans la régression linéaire, de fournir des expressions pour la construction d'intervalles de confiance simultanés de niveau exact 1 ,, pour un modéle de régression linéaire simple ou pour des formes unilatérales ou bilatérales. Nous concentrons notre attention sur les trois formes les plus reconnues: hyperbolique, à deux segments et à trois segments (qui est aussi appelée forme trapézoïdale et inclut un intervalle de largeur constante comme cas spécial). Certaines de ces expressions sont déjà apparues dans la littérature statistique, d'autres sont nouvellement introduites dans cet article. Les dérivations comprennent typiquement un vecteur aléatoire standard bivarié t et sa transformation en coordonnées polaires. [source]

### Confidence Bands for Low-Dose Risk Estimation with Quantal Response Data

BIOMETRICS, Issue 4 2003
Obaid M. Al-Saidy
Summary. We study the use of simultaneous confidence bands for low-dose risk estimation with quantal response data, and derive methods for estimating simultaneous upper confidence limits on predicted extra risk under a multistage model. By inverting the upper bands on extra risk, we obtain simultaneous lower bounds on the benchmark dose (BMD). Monte Carlo evaluations explore characteristics of the simultaneous limits under this setting, and a suite of actual data sets are used to compare existing methods for placing lower limits on the BMD. [source]

### Construction of Exact Simultaneous Confidence Bands for a Simple Linear Regression Model

INTERNATIONAL STATISTICAL REVIEW, Issue 1 2008
Wei Liu
Summary A simultaneous confidence band provides a variety of inferences on the unknown components of a regression model. There are several recent papers using confidence bands for various inferential purposes; see for example, Sun et al. (1999), Spurrier (1999), Al-Saidy et al. (2003), Liu et al. (2004), Bhargava & Spurrier (2004), Piegorsch et al. (2005) and Liu et al. (2007). Construction of simultaneous confidence bands for a simple linear regression model has a rich history, going back to the work of Working & Hotelling (1929). The purpose of this article is to consolidate the disparate modern literature on simultaneous confidence bands in linear regression, and to provide expressions for the construction of exact 1 ,, level simultaneous confidence bands for a simple linear regression model of either one-sided or two-sided form. We center attention on the three most recognized shapes: hyperbolic, two-segment, and three-segment (which is also referred to as a trapezoidal shape and includes a constant-width band as a special case). Some of these expressions have already appeared in the statistics literature, and some are newly derived in this article. The derivations typically involve a standard bivariate t random vector and its polar coordinate transformation. Résumé Un intervalle de confiance simultanée fournit une variété d'inférences sur les composantes inconnues d'un modéle de régression. Plusieurs articles récents utilisent des intervalles de confiance dans des buts variés; voir par exemple Sun, Raz et Faraway (1999), Spurrier (1999), Al-Saidy et al. (2003), Liu, Jamshidian et Zhang (2004), Bhargava et Spurrier (2004), Piegorsch et al. (2005), Liu et al. (2007). La construction d'intervalles de confiance simultanés pour un simple modéle de régression linéaire a une histoire riche, qui remonte aux travaux de Working et hotelling (1929). L'objet de cet article est de consolider la littérature moderne disparate sur les intervalles de confiance simultanés dans la régression linéaire, de fournir des expressions pour la construction d'intervalles de confiance simultanés de niveau exact 1 ,, pour un modéle de régression linéaire simple ou pour des formes unilatérales ou bilatérales. Nous concentrons notre attention sur les trois formes les plus reconnues: hyperbolique, à deux segments et à trois segments (qui est aussi appelée forme trapézoïdale et inclut un intervalle de largeur constante comme cas spécial). Certaines de ces expressions sont déjà apparues dans la littérature statistique, d'autres sont nouvellement introduites dans cet article. Les dérivations comprennent typiquement un vecteur aléatoire standard bivarié t et sa transformation en coordonnées polaires. [source]

### Small-sample confidence intervals for multivariate impulse response functions at long horizons

JOURNAL OF APPLIED ECONOMETRICS, Issue 8 2006
Elena Pesavento
Existing methods for constructing confidence bands for multivariate impulse response functions may have poor coverage at long lead times when variables are highly persistent. The goal of this paper is to propose a simple method that is not pointwise and that is robust to the presence of highly persistent processes. We use approximations based on local-to-unity asymptotic theory, and allow the horizon to be a fixed fraction of the sample size. We show that our method has better coverage properties at long horizons than existing methods, and may provide different economic conclusions in empirical applications. We also propose a modification of this method which has good coverage properties at both short and long horizons. Copyright © 2006 John Wiley & Sons, Ltd. [source]

### Non-parametric confidence bands in deconvolution density estimation

JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES B (STATISTICAL METHODOLOGY), Issue 3 2007
Nicolai Bissantz
Summary., Uniform confidence bands for densities f via non-parametric kernel estimates were first constructed by Bickel and Rosenblatt. In this paper this is extended to confidence bands in the deconvolution problem g=f*, for an ordinary smooth error density ,. Under certain regularity conditions, we obtain asymptotic uniform confidence bands based on the asymptotic distribution of the maximal deviation (L, -distance) between a deconvolution kernel estimator and f. Further consistency of the simple non-parametric bootstrap is proved. For our theoretical developments the bias is simply corrected by choosing an undersmoothing bandwidth. For practical purposes we propose a new data-driven bandwidth selector that is based on heuristic arguments, which aims at minimizing the L, -distance between and f. Although not constructed explicitly to undersmooth the estimator, a simulation study reveals that the bandwidth selector suggested performs well in finite samples, in terms of both area and coverage probability of the resulting confidence bands. Finally the methodology is applied to measurements of the metallicity of local F and G dwarf stars. Our results confirm the ,G dwarf problem', i.e. the lack of metal poor G dwarfs relative to predictions from ,closed box models' of stellar formation. [source]

### Flexible modelling of neuron firing rates across different experimental conditions: an application to neural activity in the prefrontal cortex during a discrimination task

JOURNAL OF THE ROYAL STATISTICAL SOCIETY: SERIES C (APPLIED STATISTICS), Issue 4 2006
Summary., In many electrophysiological experiments the main objectives include estimation of the firing rate of a single neuron, as well as a comparison of its temporal evolution across different experimental conditions. To accomplish these two goals, we propose a flexible approach based on the logistic generalized additive model including condition-by-time interactions. If an interaction of this type is detected in the model, we then establish that the use of the temporal odds ratio curves is very useful in discriminating between the conditions under which the firing probability is higher. Bootstrap techniques are used for testing for interactions and constructing pointwise confidence bands for the true odds ratio curves. Finally, we apply the new methodology to assessing relationships between neural response and decision-making in movement-selective neurons in the prefrontal cortex of behaving monkeys. [source]

### Empirical likelihood confidence regions for comparison distributions and roc curves

THE CANADIAN JOURNAL OF STATISTICS, Issue 2 2003
Gerda Claeskens
Abstract Abstract: The authors derive empirical likelihood confidence regions for the comparison distribution of two populations whose distributions are to be tested for equality using random samples. Another application they consider is to ROC curves, which are used to compare measurements of a diagnostic test from two populations. The authors investigate the smoothed empirical likelihood method for estimation in this context, and empirical likelihood based confidence intervals are obtained by means of the Wilks theorem. A bootstrap approach allows for the construction of confidence bands. The method is illustrated with data analysis and a simulation study. Résumé: Les auteurs déduisent de la vraisemblance empirique des régions de confiance pour la distribution comparée de deux populations dont on veut tester l'égalité en loi au moyen d'échantillons aléatoires. Une autre application qu'ils considèrent concerne les courbes ROC, qui permettent de comparer les résultats d'un test diagnostique effectué auprès de deux populations. L'estimation proposée par les auteurs dans ce contexte s'appuie sur une méthode de lissage de la vraisemblance empirique qui conduit, gr,ce au théorème de Wilks, aux intervalles de confiance recherchés. Une approche bootstrap permet en outre de construire des bandes de confiance. La méthode est illustrée au moyen de simulations et d'un jeu de données. [source]

### A NEW CONFIDENCE BAND FOR CONTINUOUS CUMULATIVE DISTRIBUTION FUNCTIONS

AUSTRALIAN & NEW ZEALAND JOURNAL OF STATISTICS, Issue 3 2009
Xingzhong Xu
Summary We consider confidence bands for continuous distribution functions. Following a review of the literature we find that previously considered confidence bands, which have exact coverage, are all step-functions jumping only at the sample points. We find that the step-function bands can be constructed through rectangular tolerance regions for an ordered sample from the uniform distribution R(0, 1). We then construct a set of new bands. Two criteria for assessing confidence bands are presented. One is the power criterion, and the other is the average-width criterion that we propose. Numerical comparisons between our new bands and the old bands are carried out, and show that our new bands perform much better than the old ones. [source]

### S41.5: Comparison of simultaneous and pointwise confidence bands for Kaplan-Meier estimators

BIOMETRICAL JOURNAL, Issue S1 2004
Ralf Strobl

### A Global Sensitivity Test for Evaluating Statistical Hypotheses with Nonidentifiable Models

BIOMETRICS, Issue 2 2010
D. Todem
Summary We consider the problem of evaluating a statistical hypothesis when some model characteristics are nonidentifiable from observed data. Such a scenario is common in meta-analysis for assessing publication bias and in longitudinal studies for evaluating a covariate effect when dropouts are likely to be nonignorable. One possible approach to this problem is to fix a minimal set of sensitivity parameters conditional upon which hypothesized parameters are identifiable. Here, we extend this idea and show how to evaluate the hypothesis of interest using an infimum statistic over the whole support of the sensitivity parameter. We characterize the limiting distribution of the statistic as a process in the sensitivity parameter, which involves a careful theoretical analysis of its behavior under model misspecification. In practice, we suggest a nonparametric bootstrap procedure to implement this infimum test as well as to construct confidence bands for simultaneous pointwise tests across all values of the sensitivity parameter, adjusting for multiple testing. The methodology's practical utility is illustrated in an analysis of a longitudinal psychiatric study. [source]

### Functional Generalized Linear Models with Images as Predictors

BIOMETRICS, Issue 1 2010
Philip T. Reiss
Summary Functional principal component regression (FPCR) is a promising new method for regressing scalar outcomes on functional predictors. In this article, we present a theoretical justification for the use of principal components in functional regression. FPCR is then extended in two directions: from linear to the generalized linear modeling, and from univariate signal predictors to high-resolution image predictors. We show how to implement the method efficiently by adapting generalized additive model technology to the functional regression context. A technique is proposed for estimating simultaneous confidence bands for the coefficient function; in the neuroimaging setting, this yields a novel means to identify brain regions that are associated with a clinical outcome. A new application of likelihood ratio testing is described for assessing the null hypothesis of a constant coefficient function. The performance of the methodology is illustrated via simulations and real data analyses with positron emission tomography images as predictors. [source]

### Confidence Bands for Low-Dose Risk Estimation with Quantal Response Data

BIOMETRICS, Issue 4 2003
Obaid M. Al-Saidy
Summary. We study the use of simultaneous confidence bands for low-dose risk estimation with quantal response data, and derive methods for estimating simultaneous upper confidence limits on predicted extra risk under a multistage model. By inverting the upper bands on extra risk, we obtain simultaneous lower bounds on the benchmark dose (BMD). Monte Carlo evaluations explore characteristics of the simultaneous limits under this setting, and a suite of actual data sets are used to compare existing methods for placing lower limits on the BMD. [source]