Simultaneous Confidence Bands Research Articles

Simultaneous confidence bands for multiple comparisons of several percentile lines

Simultaneous confidence bands for multiple comparisons of several percentile lines

Oracle-efficient estimation and trend inference in non-stationary time series with trend and heteroscedastic ARMA error

The non-stationary time series often contain an unknown trend and unobserved error terms. The error terms in the proposed model consist of a smooth variance function and the latent stationary ARMA series, which allows heteroscedasticity at different time points. The theoretically justified two-step B-spline estimation method is proposed for the trend and variance function in the model, and then residuals are obtained by removing the trend and variance function estimators from the data. The maximum likelihood estimator (MLE) for the latent ARMA error coefficients based on the residuals is shown to be oracally efficient in the sense that it has the same asymptotic distribution as the infeasible MLE if the trend and variance function were known. In addition to the oracle efficiency, a kernel estimator is obtained for the trend function and shown to converge to the Gumbel distribution. It yields an asymptotically correct simultaneous confidence band (SCB) for the trend function, which can be used to test the specific form of trend. A simulation-based procedure is proposed to implement the SCB, and simulation and real data analysis illustrate the finite sample performance.

Confidence sets for a level set in linear regression.

Regression modeling is the workhorse of statistics and there is a vast literature on estimation of the regression function. It has been realized in recent years that in regression analysis the ultimate aim may be the estimation of a level set of the regression function, ie, the set of covariate values for which the regression function exceeds a predefined level, instead of the estimation of the regression function itself. The published work on estimation of the level set has thus far focused mainly on nonparametric regression, especially on point estimation. In this article, the construction of confidence sets for the level set of linear regression is considered. In particular, level upper, lower and two-sided confidence sets are constructed for the normal-error linear regression. It is shown that these confidence sets can be easily constructed from the corresponding level simultaneous confidence bands. It is also pointed out that the construction method is readily applicable to other parametric regression models where the mean response depends on a linear predictor through a monotonic link function, which include generalized linear models, linear mixed models and generalized linear mixed models. Therefore, the method proposed in this article is widely applicable. Simulation studies with both linear and generalized linear models are conducted to assess the method and real examples are used to illustrate the method.

Oracle-efficient estimation for the mean function of missing covariate data based on noparametrically estimated selection probabilities

A weighted local linear estimator for the mean function is constructed based on nonparametrically estimated selection probabilities when covariates are missing at random. The weighted estimator is shown to be oracle-efficient, meaning it is uniformly as efficient as the infeasible estimator that uses the true selection probabilities assumed to be known as a prior. By applying oracle efficiency and the extreme value distribution of the weighted local linear estimator, asymptotically accurate simultaneous confidence bands for the mean function are derived. This approach avoids the risk of model misspecification for the missingness mechanism. Simulation studies are conducted to examine the finite sample performance, which supports the asymptotic results. The proposed methods are illustrated through the analysis of two real data sets.

Simultaneous inference and uniform test for eigensystems of functional data

The asymptotically correct confidence interval (CI) and simultaneous confidence band (SCB) of any individual eigenvalue and eigenfunction are constructed under dense functional data through B-spline smoothing. Besides, uniform inference procedures for eigensystems with a diverging number of components are novelly developed. The proposed estimators for functional eigensystems employ “oracle” efficiency up to order n, which means they are asymptotically indistinguishable from the estimators conducted by completely observed trajectories, and enjoy computational efficiency with much more convenient spectrum decomposition forms. Furthermore, an extension to two-sample problems is also investigated. Numerical simulation results strongly corroborate the asymptotic theory. Real data analysis for ElectroEncephalogram (EEG) data illustrates applicability of the developed methods.

Causal inference with a functional outcome

Abstract This article presents methods to study the causal effect of a binary treatment on a functional outcome with observational data. We define a Functional Average Treatment Effect (FATE) and develop an outcome regression estimator. We show how to obtain valid inference on the FATE using simultaneous confidence bands, which cover the FATE with a given probability over the entire domain. Simulation experiments illustrate how the simultaneous confidence bands take the multiple comparison problem into account. Finally, we use the methods to infer the effect of early adult location on subsequent income development for one Swedish birth cohort.

Simultaneous tests and confidence bands for Stata estimation commands

Stata estimation commands that implement frequentist methods produce an output table that contains multiple tests and multiple confidence intervals. Presumably, the multiple tests and multiple confidence are designed to help determine which parameters are responsible for a possible rejection of the overall null hypothesis of no effect. When taken by itself, each test and each confidence interval provides valid inference about the null hypothesis of no effect for each parameter at the specified error rate. However, simultaneously using two or more of these tests or confidence intervals provides inference at an error rate that exceeds the one specified. In this article, I discuss the sotable command, which provides p-values that are adjusted for the multiple tests and a confidence band that can be used to simultaneously test multiple parameters for no effect after almost all frequentist estimation commands. I also provide an introduction to the literature on simultaneous inference.

Comparing time varying regression quantiles under shift invariance

This article investigates whether time-varying quantile regression curves are the same up to the horizontal shift or not. The errors and the covariates involved in the regression model are allowed to be locally stationary. We formalize this issue in a corresponding non-parametric hypothesis testing problem, and develop an integrated-squared-norm based test (SIT) as well as a simultaneous confidence band (SCB) approach. The asymptotic properties of SIT and SCB under null and local alternatives are derived. Moreover, the asymptotic properties of these tests are also studied when the compared data sets are dependent. We then propose valid wild bootstrap algorithms to implement SIT and SCB. Furthermore, the usefulness of the proposed methodology is illustrated via analysing simulated and real data related to COVID-19 outbreak.

Statistical inference for high-dimensional panel functional time series

AbstractIn this paper, we develop statistical inference tools for high-dimensional functional time series. We introduce a new concept of physical dependent processes in the space of square integrable functions, which adopts the idea of basis decomposition of functional data in these spaces, and derive Gaussian and multiplier bootstrap approximations for sums of high-dimensional functional time series. These results have numerous important statistical consequences. Exemplarily, we consider the development of joint simultaneous confidence bands for the mean functions and the construction of tests for the hypotheses that the mean functions in the panel dimension are parallel. The results are illustrated by means of a small simulation study and in the analysis of Canadian temperature data.

Simultaneous Confidence Band Approach for Comparison of COVID-19 Case Counts.

The outbreak of the novel coronavirus (COVID-19) was declared to be a global emergency in January of 2020, and everyday life throughout the world was disrupted. Among many questions about COVID-19 that remain unanswered, it is of interest for society to identify whether there is any significant difference in daily case counts between males and females. The daily case count sequences are correlated due to the nature of a contagious disease, and contain a nonlinear trend owing to several unexpected events, such as vaccinations and the appearance of the delta variant. It is possible that these unexpected events have changed the dynamical system that generates data. The classic t-test is not appropriate to analyze such correlated data with a nonconstant trend. This study applies a simultaneous confidence band approach in an attempt to overcome these difficulties; that is, a simultaneous confidence band for the trend of an autoregressive moving-average time series is constructed using B-spline estimation. The proposed method is applied to the daily case count data of seniors of both genders (at least 60 years old) in the State of Ohio from April 1, 2020 to March 31, 2022, and the result shows that there is a significant difference at the 95% confidence level between the two gender case counts adjusted for the population sizes.

Extended biasplot command to assess bias, precision, and agreement in method comparison studies

Recently, a new statistical methodology to assess the bias and precision of a new measurement method, which circumvents the deficiencies of the Bland and Altman (1986, Lancet 327: 307–310) limits of agreement method, was developed by Taffé (2018, Statistical Methods in Medical Research 27: 1650–1660). Later, the methodology was extended to assess the agreement. In addition, to allow for inferences, simultaneous confidence bands around the bias, precision, and agreement lines were developed (Taffé, 2020, Statistical Methods in Medical Research 29: 778–796). The goal of this article is to introduce the extended biasplot command, which implements these latest developments, and to illustrate its use by applying it to simulated data included with the command. Note that the Taffé method assumes that there are several measurements by one of the two measurement methods and possibly as few as one measurement by the other for each individual. The repeated measurements need not come from the reference standard but from any of the two measurement methods. This is a great advantage because it may sometimes be more feasible to gather repeated measurements either with the reference standard or the new measurement method.

Improving Kernel-Based Nonasymptotic Simultaneous Confidence Bands

The paper studies the problem of constructing nonparametric simultaneous confidence bands with nonasymptotic and distribition-free guarantees. The target function is assumed to be band-limited and the approach is based on the theory of Paley-Wiener reproducing kernel Hilbert spaces. The starting point of the paper is a recently developed algorithm to which we propose three types of improvements. First, we relax the assumptions on the noises by replacing the symmetricity assumption with a weaker distributional invariance principle. Then, we propose a more efficient way to estimate the norm of the target function, and finally we enhance the construction of the confidence bands by tightening the constraints of the underlying convex optimization problems. The refinements are also illustrated through numerical experiments.

Simultaneous confidence bands for nonparametric regression with equal cluster size

Repeated/clustered data are becoming increasingly popular in many research studies. While there is a considerable literature in the estimation of the nonparametric component functions, the problem has been addressed by very few in statistical inference such as simultaneous confidence bands (SCBs). Based on the efficient kernel estimation incorporating within‐cluster error correlation, we provide asymptotic Wald‐type‐based SCBs for the nonparametric function when the repeated measures have the equal cluster size. The performance of the SCBs is evaluated by simulation studies which support the asymptotic theory. The proposed method is also applied to a dataset on the Junior School Project of test scores.

Uniform-in-bandwidth consistency results in the partially linear additive model components estimation

We are mainly concerned with the partially linear additive model defined for a measurable function ψ : R q → R , by ψ ( Y i ) : = Y i = Z i ⊤ β + ∑ l = 1 d m l ( X l , i ) + ε i for 1 ≤ i ≤ n , where Z i = ( Z 1 , i , … , Z p , i ) ⊤ and X i = ( X 1 , i , … , X i , d ) ⊤ are vectors of explanatory variables, β = ( β 1 , … , β p ) ⊤ is a vector of unknown parameters, m 1 , … , m d are unknown univariate real functions, and ε 1 , … , ε n are independent random errors with mean zero, finite variances σ ε and E ( ε | X , Z ) = 0 a.s. Under some mild conditions, we present a sharp uniform-in-bandwidth limit law for the nonlinear additive components of the model estimated by the marginal integration device with the kernel method. We allow the bandwidth to varying within the complete range for which the estimator is consistent. We provide the almost sure simultaneous asymptotic confidence bands for the regression functions.

A nonparametric simultaneous confidence band for biomarker effect on the restricted mean survival time.

Study of prognostic and predictive biomarkers plays an important role in the design and analysis of clinical trials. The Cox proportional hazards model is often used to study the biomarker main effect and the treatment-biomarker interaction effect for survival data. The estimated effects can be biased if the proportional hazards assumption is violated. The restricted mean survival time is becoming popular in clinical studies for having a clear intuitive interpretation. In this article, we first propose nonparametric methods to make statistical inference for the one-sample problem of the biomarker effect on the restricted mean survival time; we then extend the methods to the two-sample problem for studying the difference in the biomarker effects between treatment groups in clinical trials. For a given biomarker, the restricted mean survival time is estimated by kernel smoothing methods with the inverse probability of censoring weights. We prove the consistency for the estimates and develop simultaneous confidence bands for the biomarker effects on the restricted mean survival time. The simultaneous confidence bands are evaluated in extensive simulation studies and are found to have good finite sample performance. We then apply the proposed methods to a breast cancer study conducted by the Breast International Group (BIG) to illustrate how the Ki67 biomarker, a protein marker of cell proliferation, affects the survival time of patients, compared between the treatment groups.

Identifying alert concentrations using a model-based bootstrap approach.

The determination of alert concentrations, where a pre-specified threshold of the response variable is exceeded, is an important goal of concentration-response studies. The traditional approach is based on investigating the measured concentrations and attaining statistical significance of the alert concentration by using a multiple t-test procedure. In this paper, we propose a new model-based method to identify alert concentrations, based on fitting a concentration-response curve and constructing a simultaneous confidence band for the difference of the response of a concentration compared to the control. In order to obtain these confidence bands, we use a bootstrap approach which can be applied to any functional form of the concentration-response curve. This particularly offers the possibility to investigate also those situations where the concentration-response relationship is not monotone and, moreover, to detect alerts at concentrations which were not measured during the study, providing a highly flexible framework for the problem athand.

Use of clinical tolerance limits for assessing agreement.

In this study, we have further extended the methodology proposed, first, by Lin et al. (2002) and, later, extended by Stevens et al. (2017, 2018), on the coverage probability/probability of agreement, by relaxing the strong parametric assumptions regarding the distribution of the latent trait and developing inference methods allowing to compute both pointwise and simultaneous confidence bands. The methodology requires repeated measurements by at least one of the two measurement methods and accommodates heteroscedastic measurement errors. It performs often very well even when one has only one measurement by one of the two measurement methods and at least five repeated measurements from the other. It circumvents some of the deficiencies of the Bland & Altman limits of agreement method and provides a more direct assessment of the agreement level.

Generalized varying-coefficient additive model for locally stationary time series

ABSTRACT Recently, the varying-coefficient additive model (VCAM) has received much attention due to its flexibility and interpretability. In this paper, we consider its generalized form, called a generalized varying-coefficient additive model (GVCAM), which can model discrete and continuous responses for locally stationary covariates. Specifically, the conditional mean function is specified via a link function g and a nonparametric part given by a VCAM. We develop a pilot-estimation-based local linear maximum likelihood estimator (PEB-LLE) to approximate the univariate component function. We also show the proposed estimators are oracle efficient and present the pointwise asymptotic distribution as well as simultaneous confidence bands. Simulation studies investigating the finite-sample performance of the proposed methodology are considered, and a real data application illustrating our method is also given.

Convergence rate of plugin estimates for functional parameters with applications to locally-stationary time-series

Some distribution parameter estimates depend on nuisance parameters, and it is a common practice to plug in their estimates. The convergence properties of the resulting estimate are of interest. We consider the case where both estimators – the one plugged in, and its recipient – are random quantities, while literature covers mostly plugin to a deterministic function. We present a way of computing this final convergence rate, based on the convergence rate of a nonrandom, but perturbed quantity plugged in. The theoretical result is applied to the construction of simultaneous confidence band for auto-correlation functions of a locally-stationary time series.

Simultaneous confidence bands and global inferences for extended partially linear single-index models

Substantial efforts have been made for the popular semiparametric single-index models in the last two decades. Extended partially linear single-index models as generalized single-index semiparametric models effectively reduce the problem of model misspecification by allowing the data to automatically choose the principal linear component and the principal nonlinear component. In this paper, we construct an oracle-efficient simultaneous confidence band as a global inference tool for the extended partially linear single-index models. Specifically, we apply a LASSO penalized local linear smoothing method to estimate the coefficient parameters and conduct variable selection to avoid model overfitting caused by the two appearances of covariates. It is shown that the local linear estimator for the nonparametric link function by employing the penalized coefficient estimates is oracle-efficient in the sense that it is uniformly as efficient as the ideal one obtained by utilizing the true coefficient parameters. Then by applying the oracle efficiency and the extreme value distribution theory of the local linear regression, an asymptotically accurate simultaneous confidence band for the nonparametric link function in the extended partially linear single-index models is established. Simulation experiments with commonly encountered sample sizes corroborate our theoretical findings. The proposed method is applied to analyze two engineering datasets: motor trend car road tests dataset and concrete slump tests dataset.