Construct Simultaneous Confidence Research Articles

Two-Sample Distribution Tests in High Dimensions via Max-Sliced Wasserstein Distance and Bootstrapping

Summary Two-sample hypothesis testing is a fundamental statistical problem for inference about two populations. In this paper, we construct a novel test statistic to detect high-dimensional distributional differences based on the max-sliced Wasserstein distance to mitigate the curse of dimensionality. By exploiting an intriguing link between the distance and suprema of empirical processes, we develop an effective bootstrapping procedure to approximate the null distribution of the test statistic. One distinctive feature of the proposed test is the ability to construct simultaneous confidence intervals for the max-sliced Wasserstein distances of projected distributions of interest. This enables not only the detection of global distributional differences but also the identification of significantly different marginal distributions between two populations, without the need for additional tests. We establish the convergence of Gaussian and bootstrap approximations of the proposed test, based on which we show that the test is asymptotically valid and powerful as long as the considered max-sliced Wasserstein distance is adequately large. The merits of our approach are illustrated via simulated and real data examples.

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.

Tests under simple order in one‐way ANCOVA

AbstractAnalysis of covariance (ANCOVA) models are used when apart from the main treatments, some covariates also affect the response variable. In this article, the problem of testing the homogeneity of treatment effects against ordered alternatives is addressed in a fixed effects one‐way ANCOVA model when error variances are heterogeneous. The likelihood ratio test (LRT) and two approximate tests based on the asymptotic distributions of some union‐intersection type test statistics are proposed. A parametric bootstrap technique has been used to implement the LRT and its asymptotic validity is proved. A method to construct simultaneous confidence intervals is proposed. All the test procedures are further extended to the case of more than one covariate. The robustness of tests is also studied under departure from normality. Extensive simulation studies show that the proposed test procedures perform well in terms of achieving the nominal size value and good power values.

Simultaneous confidence intervals from randomization tests with application in testing bioequivalence with multiple endpoints.

We propose a method to construct simultaneous confidence intervals for a parameter vector from inverting a series of randomization tests (RT). The randomization tests are facilitated by an efficient multivariate Robbins-Monro procedure that takes the correlation information of all components into account. The estimation method does not require any distributional assumption of the population other than the existence of the second moments. The resulting simultaneous confidence intervals are not necessarily symmetric about the point estimate of the parameter vector but possess the property of equal tails in all dimensions. In particular, we present the constructing the mean vector of one population and the difference between two mean vectors of two populations. Extensive simulation is conducted to show numerical comparison with four methods. We illustrate the application of the proposed method to test bioequivalence with multiple endpoints on some realdata.

Empirical likelihood for the analysis of experimental designs

Empirical likelihood enables a nonparametric, likelihood-driven style of inference without restrictive assumptions routinely made in parametric models. We develop a framework for applying empirical likelihood to the analysis of experimental designs, addressing issues that arise from blocking and multiple hypothesis testing. In addition to popular designs such as balanced incomplete block designs, our approach allows for highly unbalanced, incomplete block designs. We derive an asymptotic multivariate chi-square distribution for a set of empirical likelihood test statistics and propose two single-step multiple testing procedures: asymptotic Monte Carlo and nonparametric bootstrap. Both procedures asymptotically control the generalised family-wise error rate and efficiently construct simultaneous confidence intervals for comparisons of interest without explicitly considering the underlying covariance structure. A simulation study demonstrates that the performance of the procedures is robust to violations of standard assumptions of linear mixed models. We also present an application to experiments on a pesticide.

Debiased and thresholded ridge regression for linear models with heteroskedastic and correlated errors

AbstractHigh-dimensional linear models with independent errors have been well-studied. However, statistical inference on a high-dimensional linear model with heteroskedastic, dependent (and possibly nonstationary) errors is still a novel topic. Under such complex assumptions, the paper at hand introduces a debiased and thresholded ridge regression estimator that is consistent, and is able to recover the model sparsity. Moreover, we derive a Gaussian approximation theorem for the estimator, and apply a dependent wild bootstrap algorithm to construct simultaneous confidence interval and hypothesis tests for linear combinations of parameters. Numerical experiments with both real and simulated data show that the proposed estimator has good finite sample performance. Of independent interest is the development of a new class of heteroscedastic, (weakly) dependent, and nonstationary random variables that can be used as a general model for regression errors.

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.

Two-Sample Inference for High-Dimensional Markov Networks

Abstract Markov networks are frequently used in sciences to represent conditional independence relationships underlying observed variables arising from a complex system. It is often of interest to understand how an underlying network differs between two conditions. In this paper, we develop methods for comparing a pair of high-dimensional Markov networks where we allow the number of observed variables to increase with the sample sizes. By taking the density ratio approach, we are able to learn the network difference directly and avoid estimating the individual graphs. Our methods are thus applicable even when the individual networks are dense as long as their difference is sparse. We prove finite-sample Gaussian approximation error bounds for the estimator we construct under significantly weaker assumptions than are typically required for model selection consistency. Furthermore, we propose bootstrap procedures for estimating quantiles of a max-type statistics based on our estimator, and show how they can be used to test the equality of two Markov networks or construct simultaneous confidence intervals. The performance of our methods is demonstrated through extensive simulations. The scientific usefulness is illustrated with an analysis of a new fMRI data set.

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.

On the Admissibility of Simultaneous Bootstrap Confidence Intervals

Simultaneous confidence intervals are commonly used in joint inference of multiple parameters. When the underlying joint distribution of the estimates is unknown, nonparametric methods can be applied to provide distribution-free simultaneous confidence intervals. In this note, we propose new one-sided and two-sided nonparametric simultaneous confidence intervals based on the percentile bootstrap approach. The admissibility of the proposed intervals is established. The numerical results demonstrate that the proposed confidence intervals maintain the correct coverage probability for both normal and non-normal distributions. For smoothed bootstrap estimates, we extend Efron’s (2014) nonparametric delta method to construct nonparametric simultaneous confidence intervals. The methods are applied to construct simultaneous confidence intervals for LASSO regression estimates.

Simultaneous confidence intervals for all pairwise differences between the coefficients of variation of rainfall series in Thailand

The delta-lognormal distribution is a combination of binomial and lognormal distributions, and so rainfall series that include zero and positive values conform to this distribution. The coefficient of variation is a good tool for measuring the dispersion of rainfall. Statistical estimation can be used not only to illustrate the dispersion of rainfall but also to describe the differences between rainfall dispersions from several areas simultaneously. Therefore, the purpose of this study is to construct simultaneous confidence intervals for all pairwise differences between the coefficients of variation of delta-lognormal distributions using three methods: fiducial generalized confidence interval, Bayesian, and the method of variance estimates recovery. Their performances were gauged by measuring their coverage probabilities together with their expected lengths via Monte Carlo simulation. The results indicate that the Bayesian credible interval using the Jeffreys’ rule prior outperformed the others in virtually all cases. Rainfall series from five regions in Thailand were used to demonstrate the efficacies of the proposed methods.

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.

Simultaneous confidence interval for assessing non-inferiority with assay sensitivity in a three-arm trial with binary endpoints.

A three-arm trial including an experimental treatment, an active reference treatment and a placebo is often used to assess the non-inferiority (NI) with assay sensitivity of an experimental treatment. Various hypothesis-test-based approaches via a fraction or pre-specified margin have been proposed to assess the NI with assay sensitivity in a three-arm trial. There is little work done on confidence interval in a three-arm trial. This paper develops a hybrid approach to construct simultaneous confidence interval for assessing NI and assay sensitivity in a three-arm trial. For comparison, we present normal-approximation-based and bootstrap-resampling-based simultaneous confidence intervals. Simulation studies evidence that the hybrid approach with the Wilson score statistic performs better than other approaches in terms of empirical coverage probability and mesial-non-coverage probability. An example is used to illustrate the proposed approaches.

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.

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.

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.

Bias Reduction for Nonparametric and Semiparametric Regression Models

Nonparametric and semiparametric regression models are useful statistical regression models to discover nonlinear relationships between the response variable and predictor variables. However, optimal efficient estimators for the nonparametric components in the models are biased which hinders the development of methods for further statistical inference. In this paper, based on the local linear fitting, we propose a simple bias reduction approach for the estimation of the nonparametric regression model. Our approach does not need to use higher-order local polynomial regression to estimate the bias, and hence avoids the double bandwidth selection and design sparsity problems suffered by higher-order local polynomial fitting. It also does not inflate the variance. Hence it can be easily applied to complex statistical inference problems. We extend our approach to varying coefficient models, to estimate the variance function, and to construct simultaneous confidence band for the nonparametric regression function. Simulations are carried out for comparisons with existing methods, and a data example is used to investigate the performance of the proposed method.

Confidence intervals for the means of the selected populations

Consider an experiment in which $p$ independent populations $\pi_{i}$ with corresponding unknown means $\theta_{i}$ are available, and suppose that for every $1\leq i\leq p$, we can obtain a sample $X_{i1},\ldots,X_{in}$ from $\pi_{i}$. In this context, researchers are sometimes interested in selecting the populations that yield the largest sample means as a result of the experiment, and then estimate the corresponding population means $\theta_{i}$. In this paper, we present a frequentist approach to the problem and discuss how to construct simultaneous confidence intervals for the means of the $k$ selected populations, assuming that the populations $\pi_{i}$ are independent and normally distributed with a common variance $\sigma^{2}$. The method, based on the minimization of the coverage probability, obtains confidence intervals that attain the nominal coverage probability for any $p$ and $k$, taking into account the selection procedure.