Minimum-area confidence sets for a normal distribution

When making inference on a normal distribution, one often seeks either a joint confidence region for the two parameters or a confidence band for the cumulative distribution function. A number of methods for constructing such confidence sets are available, but none of these methods guarantees a minimum-area confidence set. In this paper, we derive both a minimum-area joint confidence region for the two parameters and a minimum-area confidence band for the cumulative distribution function. The minimum-area joint confidence region is asymptotically equivalent to other confidence regions in the literature, but the minimum-area confidence band improves on existing confidence bands even asymptotically.     

The graphical advantages of finite interval confidence band procedures

When presented as graphical illustrations, regression surface confidence bands for linear statistical models quickly convey detailed information about analysis results. A taut confidence band is a compact set of curves which are estimation candidates for the unobservable, fixed regression curve. The bounds of the band are usually plotted with the estimated regression curve and may be overlaid by a scatter-plot of the data to provide an integrated visual impression. Finite-interval confidence bands offer the advantages of clearer interpretation and improved efficiency and avoid visual ambiguities inherent to infinite-interval bands. The definitive characteristic of a finite-interval confidence band is that it is only necessary to plot it over a finite interval in order to visually communicate all its information. In contrast, visual representations of infinite-interval bands are not fully informative and can be misleading. When an infinite-interval band is plotted, and therefore truncated, substantial information given by its asymptotic behavior is lost. Many curves that are clearly within the plotted portion of the infinite interval confidence band eventually cross a boundary. In practice, a finite-interval band can always be easily obtained from any infinite-interval band. This article focuses on interpretational considerations of symmetric confidence bands as graphical devices.     

Simultaneous Inference For The Mean Function Based on Dense Functional Data

A polynomial spline estimator is proposed for the mean function of dense functional data together with a simultaneous confidence band which is asymptotically correct. In addition, the spline estimator and its accompanying confidence band enjoy oracle efficiency in the sense that they are asymptotically the same as if all random trajectories are observed entirely and without errors. The confidence band is also extended to the difference of mean functions of two populations of functional data. Simulation experiments provide strong evidence that corroborates the asymptotic theory while computing is efficient. The confidence band procedure is illustrated by analyzing the near infrared spectroscopy data.     

Confidence band for percentile line in multiple linear regression

We deal with the problem of estimating constructing a confidence band for the 100γth percentile line in the multiple linear regression model with independent identically normally distributed errors. A method for computing the exact Scheffé type confidence band over a limited space of the particular covariates region is suggested. A confidence band depends on an estimator of the percentile line. The confidence bands based on the different estimators of the percentile line are compared with respect to the average bandwidth.     

Optimal simultaneous confidence bands in simple linear regression

A simultaneous confidence band provides useful information on the plausible range of an unknown regression model. For simple linear regression models, the most frequently quoted bands in the statistical literature include the hyperbolic band and the three-segment bands. One interesting question is whether one can construct confidence bands better than the hyperbolic and three-segment bands. The optimality criteria for confidence bands include the average width criterion considered by Gafarian (1964) and Naiman (1984) among others, and the minimum area confidence set (MACS) criterion of Liu and Hayter (2007). In this paper, two families of exact 1−α$1-\alpha$ confidence bands, the inner-hyperbolic bands and the outer-hyperbolic bands, which include the hyperbolic and three-segment bands as special cases, are introduced in simple linear regression. Under the MACS criterion, the best confidence band within each family is found by numerical search and compared with the hyperbolic band, the best three-segment band and with each other. The methodologies are illustrated with a numerical example and the Matlab programs used are available upon request.     

Simultaneous confidence band for the difference of segmented linear models

Consider comparing between two treatments a response variable, whose expectation depends on the value of a continuous covariate in some nonlinear fashion. We fit separate segmented linear models to each treatment to approximate the nonlinear relationship. For this setting, we provide a simultaneous confidence band for the difference between treatments of the expected value functions. The treatments are said to differ significantly on intervals of the covariate where the simultaneous confidence band does not contain zero. We consider segmented linear models where the locations of the changepoints are both known and unknown. The band is obtained from asymptotic results.     

Simultaneous confidence bands for nonlinear regression models

The maximization and minimization procedure for constructing confidence bands about general regression models is explained. Then, using an existing confidence region about the parameters of a nonlinear regression model and the maximization and minimization procedure, a generally conservative simultaneous confidence band is constructed about the model. Two examples are given, and some problems with the procedure are discussed     

Confidence bands for the Michaelis-Menten kinetic model

The maximization and minimization procedure for constructing confidence bands for the general nonlinear regression model is explained. Then, using the maximization and minimization procedure, a conservative confidence band for the Michaelis-Menten kinetic model used in enzyme kinetics is constructed.     

Constant width simultaneous confidence bands in multiple linear regression with predictor variables constrained in intervals

In the last fifty years, a great deal of research effort has been made on the construction of simultaneous confidence bands for a linear regression function. Two most frequently quoted confidence bands in the statistics literature are the Scheffé type and constant width bands over a given rectangular region of the predictor variables. For the constant width bands, a method is given by Gafarian [Gafarian, A.V., 1964, Confidence bands in straight line regression. Journal of the American Statistical Association, 59, 182–213.] for the calculation of critical constants only for the special case of one predictor variable. In this article, a method is proposed to construct constant width bands when there are any number of predictor variables. A new criterion for assessing a confidence band is also proposed; it is the probability that a confidence band excludes a false regression function and can be viewed as the power function of a test associated, naturally, with a confidence band. Under this criterion, a numerical comparison between the Scheffé type and constant width bands is then carried out. It emerges from this comparison that the constant width bands can be better than the Scheffé type bands for certain designs.     

Simultaneous Confidence Tubes in Multivariate Linear Regression

Simultaneous confidence bands have been shown in the statistical literature as powerful inferential tools in univariate linear regression. While the methodology of simultaneous confidence bands for univariate linear regression has been extensively researched and well developed, no published work seems available for multivariate linear regression. This paper fills this gap by studying one particular simultaneous confidence band for multivariate linear regression. Because of the shape of the band, the word ‘tube’ is more pertinent and so will be used to replace the word ‘band’. It is shown that the construction of the tube is related to the distribution of the largest eigenvalue. A simulation‐based method is proposed to compute the 1 ? α quantile of this eigenvalue. With the computation power of modern computers, the simultaneous confidence tube can be computed fast and accurately. A real‐data example is used to illustrate the method, and many potential research problems have been pointed out.     

Simultaneous bootstrap confidence bands in regression

We suggest pivotal methods for constructing simultaneous bootstrap confidence bands in regression. Most attention is given to the problem of simple linear regression, but our techniques admit trivial extension to other cases, including polynomial regression. The advantages of our bootstrap approach are twofold. Firstly, the bootstrap allows a very general distribution for the errors, and secondly, it admits a wide variety of shapes for the confidence band. In our technique the shape of each envelope of the band is determined by a general template, chosen by the experimenter, and bootstrap methods are used to select the scale of the template.     

Confidence bands for the bias of linear estimators of a function of one variable

The choice of a parametric family of curves used in fitting a response in dependence of an exploratory variable is always debatable, since the fitted curve is merely an approximation of the true response curve. The present paper discusses the computation of (simultaneous) confidence bands for the bias of a fitted curve. Our development is based on integral expressions for the bias using derivatives of the unknown response curve. Such formulas can be used to convert confidence bands on the derivative(s) into confidence band(s) for the bias.     

Confidence Bands for the Difference of Two Survival Curves Under Proportional Hazards Model

A common approach to testing for differences between the survival rates of two therapies is to use a proportional hazards regression model which allows for an adjustment of the two survival functions for any imbalance in prognostic factors in the comparison. When the relative risk of one treatment to the other is not constant over time the question of which therapy has a survival advantage is difficult to determine from the Cox model. An alternative approach to this problem is to plot the difference between the two predicted survival functions with a confidence band that provides information about when these two treatments differ. Such a band will depend on the covariate values of a given patient. In this paper we show how to construct a confidence band for the difference of two survival functions based on the proportional hazards model. A simulation approach is used to generate the bands. This approach is used to compare the survival probabilities of chemotherapy and allogeneic bone marrow transplants for chronic leukemia.     

Simultaneous Confidence Bands for Linear Regression with Covariates Constrained in Intervals

Abstract. The focus of this article is on simultaneous confidence bands over a rectangular covariate region for a linear regression model with k>1 covariates, for which only conservative or approximate confidence bands are available in the statistical literature stretching back to Working & Hotelling (J. Amer. Statist. Assoc. 24 , 1929; 73–85). Formulas of simultaneous confidence levels of the hyperbolic and constant width bands are provided. These involve only a k‐dimensional integral; it is unlikely that the simultaneous confidence levels can be expressed as an integral of less than k‐dimension. These formulas allow the construction for the first time of exact hyperbolic and constant width confidence bands for at least a small k(>1) by using numerical quadrature. Comparison between the hyperbolic and constant width bands is then addressed under both the average width and minimum volume confidence set criteria. It is observed that the constant width band can be drastically less efficient than the hyperbolic band when k>1. Finally it is pointed out how the methods given in this article can be applied to more general regression models such as fixed‐effect or random‐effect generalized linear regression models.     

A Simultaneous Confidence Band for Dense Longitudinal Regression

We present a method of using local linear smoothing to construct simultaneous confidence bands for the mean function of densely spaced functional data. Our approach works well under mild conditions. In addition, the local linear estimator and its accompanying confidence band enjoy semiparametric efficiency in the sense that they are asymptotically equivalent to the counterparts obtained from the random trajectories entirely observed without errors. We illustrate the performance of the proposed confidence band through a simulation study. Furthermore, an application in food science is presented.     

Percentage points of a weighted Kolmogorov-Smirnov statistic

Let F and G be the cumulative distribution functions corresponding to two independent random variables. Define the shift function, Δ(x), by F(x)=G(x+Δ(x)). Doksum and Sievers (1976) compared two confidence bands for Δ(x). The confidence band they found to be best requires the percentage points of a weighted form of the Kolmogorov-Smirnov statistic. The goal in this paper is to supply a table of some of the exact percentage points.     

Generalised K-S confidence regions: some exact results

One-sided confidence regions for continuous cumulative distribution function are constructed using empirical cumulative distribution functions and the generalized Kolmogorov-Smirnov distance. The band width of such regions becomes narrower in the right or left tail of the distribution. Significance levels necessary for implementation are given. Some other K-S type distances useful in constructing a confidence region with nonconstant width are also included.     

Confidence bands for the difference of two survival functions under the additive risk model

In many clinical studies, a commonly encountered problem is to compare the survival probabilities of two treatments for a given patient with a certain set of covariates, and there is often a need to make adjustments for other covariates that may affect outcomes. One approach is to plot the difference between the two subject-specific predicted survival estimates with a simultaneous confidence band. Such a band will provide useful information about when these two treatments differ and which treatment has a better survival probability. In this paper, we show how to construct such a band based on the additive risk model and we use the martingale central limit theorem to derive its asymptotic distribution. The proposed method is evaluated from a simulation study and is illustrated with two real examples.     

Exact Simultaneous Confidence Bands for Quadratic and Cubic Polynomial Regression with Applications in Dose Response Study

Exact simultaneous confidence bands (SCBs) for a polynomial regression model are available only in some special situations. In this paper, simultaneous confidence levels for both hyperbolic and constant width bands for a polynomial function over a given interval are expressed as multidimensional integrals. The dimension of these integrals is equal to the degree of the polynomial. Hence the values can be calculated quickly and accurately via numerical quadrature provided that the degree of the polynomial is small (e.g. 2 or 3). This allows the construction of exact SCBs for quadratic and cubic regression functions over any given interval and for any given design matrix. Quadratic and cubic regressions are frequently used to characterise dose response relationships in addition to many other applications. Comparison between the hyperbolic and constant width bands under both the average width and minimum volume confidence set criteria shows that the constant width band can be much less efficient than the hyperbolic band. For hyperbolic bands, comparison between the exact critical constant and conservative or approximate critical constants indicates that the exact critical constant can be substantially smaller than the conservative or approximate critical constants. Numerical examples from a dose response study are used to illustrate the methods.     

A ray method of confidence band construction for multiple linear regression models

This paper addresses the problem of confidence band construction for a standard multiple linear regression model. A “ray” method of construction is developed which generalizes the method of Graybill and Bowden [1967. Linear segment confidence bands for simple linear regression models. J. Amer. Statist. Assoc. 62, 403–408] for a simple linear regression model to a multiple linear regression model. By choosing suitable directions for the rays this method requires only critical points from t-distributions so that the confidence bands are easy to construct. Both one-sided and two-sided confidence bands can be constructed using this method. An illustration of the new method is provided.