Construction Of Simultaneous Confidence Bands Research Articles

Functional index coefficient models for locally stationary time series

In the analysis of nonlinear time series, we propose a novel functional index coefficient model for the locally stationary data. The proposed model can effectively capture the dynamic interaction effects between variables and the nature of data evolution. Drawing the idea from the spline-backfitted method, we propose a three-step estimation procedure and establish the asymptotic properties of the resulting nonparametric estimators. We further construct simultaneous confidence bands for the time-varying functions to explore the global variation of the original data. We also develop a test statistic to check the time-varying properties based on a bootstrap procedure. Simulation studies have been conducted to investigate the finite sample performance of the proposed methods. Two real applications in the finance market and the Hong Kong respiratory and circulatory disease data are analysed for illustration.

Simultaneous confidence bands in a zero-inflated regression model for binary data

Abstract The logistic regression model has become a standard tool to investigate the relationship between a binary outcome and a set of potential predictors. When analyzing binary data, it often arises, however, that the observed proportion of zeros is greater than expected under the postulated logistic model. Zero-inflated binomial (ZIB) models have been developed to fit binary data that contain too many zeros. Maximum likelihood estimators in these models have been proposed, and their asymptotic properties were recently established. In this paper, we use these asymptotic properties to construct simultaneous confidence bands for the probability of a positive outcome in a ZIB regression model. Simultaneous confidence bands are especially attractive since they allow inference to be made over the whole regressor space. We construct two types of confidence bands, based on: (i) the Scheffé method for the linear regression model; (ii) Monte Carlo simulations to approximate the distribution of the supremum of a Gaussian field indexed by the regressor. The finite-samples properties of these two types of bands are investigated and compared in a simulation study.

Construction of simultaneous confidence bands using conditional Monte Carlo

A method based on conditional Monte Carlo is introduced to construct the one-sided and two-sided simultaneous confidence bands of the constant width and hyperbolic shapes.

Predictive generalized varying-coefficient longitudinal model.

We propose a nonparametric bivariate varying coefficient generalized linear model to predict a mean response trajectory in the future given an individual's characteristics at present or an earlier time point in a longitudinal study. Given the measurement time of the predictors, the coefficients vary as functions of the future time over which the prediction of the mean response is concerned and illustrate the dynamic association between the future response and the earlier measured predictors. We use a nonparametric approach that takes advantage of features of both the kernel and the spline methods for estimation. The resulting coefficient estimator is asymptotically consistent under mild regularity conditions. We also develop a new bootstrap approach to construct simultaneous confidence bands for statistical inference about the coefficients and the predicted response trajectory based on the coverage rate of bootstrap estimates. We use the Framingham Heart Study to illustrate the methodology. The proposed procedure is applied to predict the probability trajectory of hypertension risk given individuals' health condition in early adulthood and to examine the impact of risk factors in early adulthood on a long-term risk of hypertension over several decades.

Simultaneous confidence bands for functional data using the Gaussian Kinematic formula

We propose a construction of simultaneous confidence bands (SCBs) for functional parameters over arbitrary dimensional compact domains using the Gaussian Kinematic formula of t-processes (tGKF). Although the tGKF relies on Gaussianity, we show that a central limit theorem (CLT) for the parameter of interest is enough to obtain asymptotically precise covering even if the observations are non-Gaussian processes. As a proof of concept we study the functional signal-plus-noise model and derive a CLT for an estimator of the Lipshitz–Killing curvatures, the only data-dependent quantities in the tGKF. We further discuss extensions to discrete sampling with additive observation noise using scale space ideas from regression analysis. Our theoretical work is accompanied by a simulation study comparing different methods to construct SCBs for the population mean. We show that the tGKF outperforms state-of-the-art methods with precise covering for small sample sizes, and only a Rademacher multiplier-t bootstrap performs similarly well. A further benefit is that our SCBs are computational fast even for domains of dimension greater than one. Applications of SCBs to diffusion tensor imaging (DTI) fibers (1D) and spatio-temporal temperature data (2D) are discussed.

Network and panel quantile effects via distribution regression

This paper provides a method to construct simultaneous confidence bands for quantile functions and quantile effects in nonlinear network and panel models with unobserved two-way effects, strictly exogenous covariates, and possibly discrete outcome variables. The method is based upon projection of simultaneous confidence bands for distribution functions constructed from fixed effects distribution regression estimators. These fixed effects estimators are debiased to deal with the incidental parameter problem. Under asymptotic sequences where both dimensions of the data set grow at the same rate, the confidence bands for the quantile functions and effects have correct joint coverage in large samples. An empirical application to gravity models of trade illustrates the applicability of the methods to network data.

Mean-Structure and Autocorrelation Consistent Covariance Matrix Estimation

We consider estimation of the asymptotic covariance matrix in nonstationary time series. A nonparametric estimator that is robust against unknown forms of trends and possibly a divergent number of change points (CPs) is proposed. It is algorithmically fast because neither a search for CPs, estimation of trends, nor cross-validation is required. Together with our proposed automatic optimal bandwidth selector, the resulting estimator is both statistically and computationally efficient. It is, therefore, useful in many statistical procedures, for example, CPs detection and construction of simultaneous confidence bands of trends. Empirical studies on four stock market indices are also discussed.

Simultaneous confidence bands for growth incidence curves in weighted sup-norm metrics

ABSTRACTThe so-called growth incidence curve (GIC) is a popular way to evaluate the distributional pattern of economic growth and pro-poorness of growth in development economics. The log-transformation of the the GIC is related to the sum of empirical quantile processes which allows for constructions of simultaneous confidence bands for the GIC. However, standard constructions of these bands tend to be too wide at the extreme points 0 and 1 because the estimator of the quantile function can be very volatile at the extreme points. In order to construct simultaneous confidence bands which are narrower at the ends, we consider the convergence of quantile processes with weight functions. In particular, we investigate the asymptotic convergence under specific weighted sup-norm metrics and compare different kinds of qualified weight functions. This implies simultaneous confidence bands that are narrower at the boundaries 0 and 1. We show in simulations that these bands have a more regular shape. Finally, we evaluate real data from Uganda with the improved confidence bands.

Generic Inference on Quantile and Quantile Effect Functions for Discrete Outcomes

Quantile and quantile effect (QE) functions are important tools for descriptive and causal analysis due to their natural and intuitive interpretation. Existing inference methods for these functions do not apply to discrete random variables. This article offers a simple, practical construction of simultaneous confidence bands for quantile and QE functions of possibly discrete random variables. It is based on a natural transformation of simultaneous confidence bands for distribution functions, which are readily available for many problems. The construction is generic and does not depend on the nature of the underlying problem. It works in conjunction with parametric, semiparametric, and nonparametric modeling methods for observed and counterfactual distributions, and does not depend on the sampling scheme. We apply our method to characterize the distributional impact of insurance coverage on health care utilization and obtain the distributional decomposition of the racial test score gap. We find that universal insurance coverage increases the number of doctor visits across the entire distribution, and that the racial test score gap is small at early ages but grows with age due to socio-economic factors especially at the top of the distribution. Supplementary materials (additional results, R package, replication files) for this article are available online.

UNIFORMLY VALID POST-REGULARIZATION CONFIDENCE REGIONS FOR MANY FUNCTIONAL PARAMETERS IN Z-ESTIMATION FRAMEWORK.

In this paper, we develop procedures to construct simultaneous confidence bands for potentially infinite-dimensional parameters after model selection for general moment condition models where is potentially much larger than the sample size of available data, n. This allows us to cover settings with functional response data where each of the parameters is a function. The procedure is based on the construction of score functions that satisfy Neyman orthogonality condition approximately. The proposed simultaneous confidence bands rely on uniform central limit theorems for high-dimensional vectors (and not on Donsker arguments as we allow for ). To construct the bands, we employ a multiplier bootstrap procedure which is computationally efficient as it only involves resampling the estimated score functions (and does not require resolving the high-dimensional optimization problems). We formally apply the general theory to inference on regression coefficient process in the distribution regression model with a logistic link, where two implementations are analyzed in detail. Simulations and an application to real data are provided to help illustrate the applicability of the results.

Simultaneous confidence bands for a percentile hyper-plane with covariates constrained in a restricted range

A simultaneous confidence band provides useful information about the plausible range of the unknown regression model. In several recent papers, confidence bands have been used for various inferential purposes; see, for example, Liu et al. (2004, 2005, 2009, 2010, 2012, 2014), Liu and Hayter (2007) [ 1 ], Liu and Lin (2009), Piegorsch et al. (2005), Han et al. (2015) and Zhou (2017). The construction of simultaneous confidence bands has a long history, going back to Working and Hotelling (1929) [ 2 ], and is often difficult, when the region over which a confidence band is required is restricted and the number of predictor variables is greater than one. In this paper we have developed methods to construct simultaneous confidence bands for a percentile hyper-plane. These methods allow us to construct the bands over arbitrary intervals, either finite or infinite; they also allow us to construct the Type I and Type II bands. In order to get an idea of the efficiency of these bands, we have also developed the confidence sets for the bands and have compared the bands under the minimum volume confidence set optimality criterion. The proposed methods are illustrated by an example.

Adaptive Learning Hybrid Model for Solar Intensity Forecasting

Energy management is indispensable in the smart grid, which integrates more renewable energy resources, such as solar and wind. Because of the intermittent power generation from these resources, precise power forecasting has become crucial to achieve efficient energy management. In this paper, we propose a novel adaptive learning hybrid model (ALHM) for precise solar intensity forecasting based on meteorological data. We first present a time-varying multiple linear model (TMLM) to capture the linear and dynamic property of the data. We then construct simultaneous confidence bands for variable selection. Next, we apply the genetic algorithm back propagation neural network (GABP) to learn the nonlinear relationships in the data. We further propose ALHM by integrating TMLM, GABP, and the adaptive learning online hybrid algorithm. The proposed ALHM captures the linear, temporal, and nonlinear relationships in the data, and keeps improving the predicting performance adaptively online as more data are collected. Simulation results show that ALHM outperforms several benchmarks in both short-term and long-term solar intensity forecasting.

Construction of simultaneous confidence bands for a percentile hyper-plane with predictor variables constrained in an ellipsoidal region

This paper provides the method of constructing the simultaneous confidence band for a percentile hyper-plane with predictor variables constrained in an ellipsoidal region. The band is compared with that on a rectangular region. It is shown that the band on the ellipsoidal region has a big advantage. The proposed method allows to construct the Type I band and the Type II band, and also allows to constrain the predictor variables either in finite intervals or in infinite intervals. In order to evaluate the efficiency of the band, the average width criterion is introduced, under which the Type I band and the Type II band are compared. All stated above are illustrated by a real example of a leaching rate study.

Simultaneous inference for the mean of repeated functional data

Motivated by recent works studying the longitudinal diffusion tensor imaging (DTI) studies, we develop a novel procedure to construct simultaneous confidence bands for mean functions of repeatedly observed functional data. A fully nonparametric method is proposed to estimate the mean function and variance–covariance function of the repeated trajectories via polynomial spline smoothing. The proposed confidence bands are shown to be asymptotically correct by taking into account the correlation of trajectories within subjects. The procedure is also extended to the two-sample case in which we focus on comparing the mean functions from two populations of functional data. We show the finite-sample properties of the proposed confidence bands by simulation studies, and compare the performance of our approach with the “naive” method that assumes the independence within the repeatedly observed trajectories. The proposed method is applied to the DTI study.

Multiple treatment comparisons in analysis of covariance with interaction

When multiple treatments are analyzed together with a covariate, a treatment-covariate interaction complicates the interpretation of the treatment effects. The construction of simultaneous confidence bands for differences of the treatment specific regression lines is one option to proceed. The application of these methods is difficult because they are described as a collection of special cases and the implementation requires additional programming or relies on non-standard or proprietary software. If inferential interest can be restricted to a pre-specified set of covariate values, a flexible alternative is to compute simultaneous confidence intervals for multiple contrasts of the treatment effects over this grid. This approach is available in the R software: next to treatment differences in the linear model, approximate simultaneous confidence intervals for ratios of expected values and asymptotic extensions to generalized linear models are straightforward. The paper summarizes the available methodology and presents three case studies to illustrate the application to different models, differences and ratios, as well as different types of between treatment comparisons. Simulation studies in the general linear model, for different parameters and different types of comparisons are provided. The R code to reproduce the case studies and a hint to a related R package is provided.

Comparing canopy leaf temperature of three Central European tree species based on simultaneous confidence bands for penalized splines

Temperature is an important physical factor that is known to strongly affect biodiversity as well as ecosystems and their functioning. However, research in this area is still relatively limited; this may also be attributed to the multitude of influencing factors and the complexity of the statistics involved. This study analyzes the differences between the surface temperature of three Central European broadleaf tree species. A better understanding of these differences may help to elucidate the role of microclimate in biodiversity. We consider a time series of high-resolution thermal images taken from a meteorological observation tower and calculate mean canopy leaf temperatures for beech, ash and maple (Fagus silvatica, Fraxinus excelsior and Acer pseudoplatanus). In a first step, comparable image areas are extracted from the thermal image sections of the crown of each tree species avoiding shadow areas, branches, etc. We used an automatic segmentation technique, the Otsu thresholding. Extracted canopy leaf temperature values were then processed and the resulting temperature profiles estimated by O’Sullivan penalized splines. For comparing the differences in canopy leaf temperature over time, we propose the construction of simultaneous confidence bands. The analyses show that there are significant—though small—differences in canopy surface temperature between the three tree species.

A penalized spline estimator for fixed effects panel data models

Estimating nonlinear effects of continuous covariates by penalized splines is well established for regressions with cross-sectional data as well as for panel data regressions with random effects. Penalized splines are particularly advantageous since they enable both the estimation of unknown nonlinear covariate effects and inferential statements about these effects. The latter are based, for example, on simultaneous confidence bands that provide a simultaneous uncertainty assessment for the whole estimated functions. In this paper, we consider fixed effects panel data models instead of random effects specifications and develop a first-difference approach for the inclusion of penalized splines in this case. We take the resulting dependence structure into account and adapt the construction of simultaneous confidence bands accordingly. In addition, the penalized spline estimates as well as the confidence bands are also made available for derivatives of the estimated effects which are of considerable interest in many application areas. As an empirical illustration, we analyze the dynamics of life satisfaction over the life span based on data from the German Socio-Economic Panel. An open-source software implementation of our methods is available in the R package pamfe.

A Penalized Spline Estimator for Fixed Effects Panel Data Models

Estimating nonlinear effects of continuous covariates by penalized splines is well established for regressions with cross-sectional data as well as for panel data regressions with random effects. Penalized splines are particularly advantageous since they enable both the estimation of unknown nonlinear covariate effects and inferential statements about these effects. The latter are based, for example, on simultaneous confidence bands that provide a simultaneous uncertainty assessment for the whole estimated functions. In this paper, we consider fixed effects panel data models instead of random effects specifications and develop a first-difference approach for the inclusion of penalized splines in this case. We take the resulting dependence structure into account and adapt the construction of simultaneous confidence bands accordingly. In addition, the penalized spline estimates as well as the confidence bands are also made available for derivatives of the estimated effects which are of considerable interest in many application areas. As an empirical illustration, we analyze the dynamics of life satisfaction over the life span based on data from the German Socio-Economic Panel (SOEP). An open source software implementation of our methods is available in the R package pamfe.

Simultaneous confidence bands for the mean of functional data

The mean function is a central object of inquiry in the analysis of functional data. Typical questions related to the mean function include quantifying estimation uncertainty, testing parametric models, and making comparisons between populations. To make probabilistic statements about the mean function over its entire domain, rather than at a single location, it is necessary to infer all of its values simultaneously. Pointwise inference is not appropriate for this task and indeed produces anticonservative results, i.e., the coverage level of confidence regions is too low and the significance level of hypothesis tests too high. In contrast, simultaneous confidence bands (SCB) provide a flexible framework for conducting simultaneous inference on the mean function and other functional parameters. They also offer powerful visualization tools for communicating analytic results to interdisciplinary audiences. The construction ofSCBin the context of functional data requires specific theory and methods. In particular, it is not addressed by the nonparametric regression literature. Although software is available to perform individual steps of anSCBprocedure, resources that provide end‐to‐end computations are scarce. Applications ofSCBto one‐ and two‐sample inferences are illustrated here with the R package SCBmeanfd.WIREs Comput Stat2017, 9:e1397. doi: 10.1002/wics.1397This article is categorized under:Statistical and Graphical Methods of Data Analysis > Nonparametric MethodsStatistical and Graphical Methods of Data Analysis > Statistical Graphics and VisualizationData: Types and Structure > Time Series, Stochastic Processes, and Functional Data

Simultaneous confidence bands for restricted logistic regression models

ABSTRACTThe hyperbolic confidence bands for one logistic regression model with restricted predictors have been considered in the statistical literature. At times, one wishes to construct simultaneous confidence bands for comparing several logistic regression models. It seems that Liu's book [Simultaneous Inference in Regression, Chapman & Hall, 2010, Chapter 8] is the only published work that has addressed this problem. Liu suggested simulation-based methods for constructing simultaneous confidence bands for comparing several logistic models, but further research was warranted to assess the conservativeness of the bands. In this paper, we propose a dimension-wise partitioning method to construct a set of simultaneous confidence bands for the comparisons of several logistic regression functions with a pre-specified function in a stepwise fashion. In addition, simulation studies cast new light on the assumption of predetermined testing order for the stepwise procedures presented in this paper and by Hsu and Berger [Stepwise confidence intervals without multiplicity adjustment for dose–response and toxicity studies, J. Amer. Statist. Assoc. 94 (1999), pp. 468–482]. As an illustration, we include an example on the success rate of thrombolysis associated with patient characteristics regarding post-thrombotic syndrome.