Simultaneous Confidence Regions Research Articles

Empirical likelihood M‐estimation for the varying‐coefficient model with functional response

AbstractThis work is motivated by a gap in the functional data analysis literature, particularly in the context of neuroimaging, regarding the ability of functional models to robustly accommodate intra‐observation dependence. In response, we propose an M‐estimator based on generalized empirical likelihood for the varying‐coefficient model with a functional response. We develop statistical inference procedures, simultaneous confidence regions, and a global general linear hypothesis test for the model's functional coefficient. Our theoretical results establish the weak convergence of the log‐likelihood ratio process, a nonparametric version of Wilks' theorem for the log‐likelihood ratio, and asymptotic properties of the proposed estimator. Through a simulation study, we show that the proposed confidence sets have close‐to‐nominal coverage probabilities. In a real‐world application to a neuroimaging dataset, we show that mini‐mental state examination score and apolipoprotein E genotype have significant associations with fractional anisotropy, while associations with gender and age are only present at high quantile levels.

Simultaneous confidence region of an embedded one-dimensional curve in multi-dimensional space

This paper focuses on the simultaneous confidence region of a one-dimensional curve embedded in multi-dimensional space. Local linear regression is applied component-wise to each variable in multi-dimensional data, which yields an estimator of the one-dimensional curve. A simultaneous confidence region of the curve is proposed based on this estimator and theoretical results for the estimator and the region are developed under some reasonable assumptions. Practically efficient algorithms to determine the thickness of the region are also addressed. The effectiveness of the region is investigated through simulation studies and applications to artificial and real datasets, which reveal that the proposed simultaneous confidence region works well.

Uniform inference in high-dimensional Gaussian graphical models

Summary Graphical models have become a popular tool for representing dependencies within large sets of variables and are crucial for representing causal structures. We provide results for uniform inference on high-dimensional graphical models, in which the number of target parameters $d$ is potentially much larger than the sample size, under approximate sparsity. Our results highlight how graphical models can be estimated and recovered using modern machine learning methods in high-dimensional complex settings. To construct simultaneous confidence regions on many target parameters, it is crucial to have sufficiently fast estimation rates of the nuisance functions. In this context, we establish uniform estimation rates and sparsity guarantees for the square-root lasso estimator in a random design under approximate sparsity conditions. These might be of independent interest for related problems in high dimensions. We also demonstrate in a comprehensive simulation study that our procedure has good small sample properties in comparison to existing methods, and we present two empirical applications.

Analyzing differences between restricted mean survival time curves using pseudo-values

Hazard ratios are ubiquitously used in time to event analysis to quantify treatment effects. Although hazard ratios are invaluable for hypothesis testing, other measures of association, both relative and absolute, may be used to fully elucidate study results. Restricted mean survival time (RMST) differences between groups have been advocated as useful measures of association. Recent work focused on model-free estimates of the difference in restricted mean survival through follow-up times, instead of focusing on a single time horizon. The resulting curve can be used to quantify the association in time units with a simultaneous confidence band. In this work a model-based estimate of the curve is proposed using pseudo-values allowing for possible covariate adjustment. The method is easily implementable with available software and makes possible to compute a simultaneous confidence region for the curve. The pseudo-values regression using multiple restriction times is in good agreement with the estimates obtained by standard direct regression models fixing a single restriction time. Moreover, the proposed method is flexible enough to reproduce the results of the non-parametric approach when no covariates are considered. Examples where it is important to adjust for baseline covariates will be used to illustrate the different methods together with some simulations.

High-Dimensional MANOVA Via Bootstrapping and Its Application to Functional and Sparse Count Data

We propose a new approach to the problem of high-dimensional multivariate ANOVA via bootstrapping max statistics that involve the differences of sample mean vectors. The proposed method proceeds via the construction of simultaneous confidence regions for the differences of population mean vectors. It is suited to simultaneously test the equality of several pairs of mean vectors of potentially more than two populations. By exploiting the variance decay property that is a natural feature in relevant applications, we are able to provide dimension-free and nearly parametric convergence rates for Gaussian approximation, bootstrap approximation, and the size of the test. We demonstrate the proposed approach with ANOVA problems for functional data and sparse count data. The proposed methodology is shown to work well in simulations and several real data applications.

Simultaneous inference of the partially linear model with a multivariate unknown function

In this paper, we conduct simultaneous inference of the non-parametric part of a partially linear model when the non-parametric component is a multivariate unknown function. Based on semi-parametric estimates of the model, we construct a simultaneous confidence region of the multivariate function for simultaneous inference. The developed methodology is applied to perform simultaneous inference for the U.S. gasoline demand where the income and price variables are contaminated by Berkson errors. The empirical results strongly suggest that the linearity of the U.S. gasoline demand is rejected. The results are also used to propose an alternative form for the demand.

Spectral Inference under Complex Temporal Dynamics

We develop a unified theory and methodology for the inference of evolutionary Fourier power spectra for a general class of locally stationary and possibly nonlinear processes. In particular, simultaneous confidence regions (SCR) with asymptotically correct coverage rates are constructed for the evolutionary spectral densities on a nearly optimally dense grid of the joint time-frequency domain. A simulation based bootstrap method is proposed to implement the SCR. The SCR enables researchers and practitioners to visually evaluate the magnitude and pattern of the evolutionary power spectra with asymptotically accurate statistical guarantee. The SCR also serves as a unified tool for a wide range of statistical inference problems in time-frequency analysis ranging from tests for white noise, stationarity and time-frequency separability to the validation for non-stationary linear models.

Simultaneous Inference of the Partially Linear Model with a Multivariate Unknown Function

In this paper, we conduct simultaneous inference of the non-parametric part of a partially linear model when the non-parametric component is a multivariate unknown function. Based on semi-parametric estimates of the model, we construct a simultaneous confidence region of the multivariate function for simultaneous inference. The developed methodology is applied to perform simultaneous inference for the U.S. gasoline demand where the income and price variables are contaminated by Berkson errors. The empirical results strongly suggest that the linearity of the U.S. gasoline demand is rejected. The results are also used to propose an alternative form for the demand.

On nonparametric ridge estimation for multivariate long-memory processes

We consider nonparametric estimation of the ridge of a probability density function for multivariate linear processes with long-range dependence. We derive functional limit theorems for estimated eigenvectors and eigenvalues of the Hessian matrix. We use these results to obtain the weak convergence for the estimated ridge and asymptotic simultaneous confidence regions.

Comparison among simultaneous confidence regions for nonlinear diffusion models

Accuracy measures for parameter estimates represent a tricky issue in nonlinear models. Practitioners often use separate marginal confidence intervals for each parameter in place of a simultaneous confidence region (sCR). However, this can be extremely misleading due to the curvature of the parameter space of the nonlinear model. For low parameter dimensions, routines for evaluating approximate sCRs are available in the most common software programs; however, the degree of accuracy depends on the curvature of the parameter space and the sample size. Exact sCRs are computationally intensive, and for this reason, in the past, they did not receive much attention. In this paper, we perform a comparison among exact, asymptotic exact, approximate sCRs, and marginal confidence intervals. More modern regions based on bootstrap are also examined as an alternative approach (both parametric and nonparametric). Their degree of accuracy is compared with both real data and simulation results. Among the nonlinear models, in this paper, the focus is on two of the most widespread diffusion models of products and technologies, that is, the Bass and Generalized Bass models. Three different empirical studies are analyzed here. Simulation studies are also performed for lifecycles with the same diffusion characteristics as those of the empirical studies. Our results show that, as the parameter dimension increases, overlapping among the alternative sCRs reduces. The approximate sCR shows inadequate values of overlapping with the exact sCR, even for moderate parameter dimension. Bootstrap regions also exhibit good performance in describing the shape of the exact region when curvature is present, but they fail to spread up to its boundary. The coverage probability of each region is assessed with simulations. We observe that the coverage probability of the approximate sCR decreases rapidly, even for moderate parameter dimension, and it is smaller than the nominal level for bootstrap regions.

On minimum volume properties of some confidence regions for multiple multivariate normal means

In a multi-sample model of multivariate normal distributions with covariance matrices being known or known except for unknown multipliers, simultaneous confidence regions for the mean vectors are provided with minimum volume properties. The univariate case with unknown variances is included.

Confidence regions for Pareto parameters from a single and independent samples

[] Based on a single and on two independent samples, joint confidence regions for parameters of Pareto distributions are proposed with minimum volume properties and without assigning the confidence level to dimensions. In the one-sample case, comparisons are made to former simultaneous confidence sets for Pareto parameters by means of simulation and a real data set. The two-sample case is studied in various set-ups and comprises simultaneous confidence regions for the shape parameters, the scale parameters, and higher-dimensional vectors of these parameters, where common shape and common scale models are also considered.

Confidence regions for entries of a large precision matrix

We consider the statistical inference for high-dimensional precision matrices. Specifically, we propose a data-driven procedure for constructing a class of simultaneous confidence regions for a subset of the entries of a large precision matrix. The confidence regions can be applied to test for specific structures of a precision matrix, and to recover its nonzero components. We first construct an estimator for the precision matrix via penalized node-wise regression. We then develop the Gaussian approximation to approximate the distribution of the maximum difference between the estimated and the true precision coefficients.A computationally feasible parametric bootstrap algorithm is developed to implement the proposed procedure. The theoretical justification is established under the setting which allows temporal dependence among observations. Therefore the proposed procedure is applicable to both independent and identically distributed data and time series data. Numerical results with both simulated and real data confirm the good performance of the proposed method.

Estimation of the simultaneous confidence regions for the ratios of quantile residual lifetimes

ABSTRACTIn many clinical studies, especially when the associated diseases can be incurable, it is critical to evaluate the effect of treatments on mean residual lifetime or quantile residual lifetime. In this article, for right-censored survival data, we concern the construction of simultaneous confidence regions for the ratios of quantile residual lifetimes between treatment and control groups. We propose two approaches: the first is based on the score-type test of Jeong et al. [Nonparametric inference on median residual life function. Biometrics. 2008;64:157–163], while the second is based on the empirical likelihood ratio test of Zhou and Jeong [Empirical likelihood ratio test for median and mean residual lifetime. Statist. Med. 2011;30:152–159]. The performance of the associated coverage probability and the expected coverage area are investigated via a simulation study. The proposed method is illustrated using a real data set.

Generalized simultaneous confidence regions for regression coefficients and their ratios

ABSTRACTThis study constructs a simultaneous confidence region for two combinations of coefficients of linear models and their ratios based on the concept of generalized pivotal quantities. Many biological studies, such as those on genetics, assessment of drug effectiveness, and health economics, are interested in a comparison of several dose groups with a placebo group and the group ratios. The Bonferroni correction and the plug-in method based on the multivariate-t distribution have been proposed for the simultaneous region estimation. However, the two methods are asymptotic procedures, and their performance in finite sample sizes has not been thoroughly investigated. Based on the concept of generalized pivotal quantity, we propose a Bonferroni correction procedure and a generalized variable (GV) procedure to construct the simultaneous confidence regions. To address a genetic concern of the dominance ratio, we conduct a simulation study to empirically investigate the probability coverage and expected length of the methods for various combinations of sample sizes and values of the dominance ratio. The simulation results demonstrate that the simultaneous confidence region based on the GV procedure provides sufficient coverage probability and reasonable expected length. Thus, it can be recommended in practice. Numerical examples using published data sets illustrate the proposed methods.

Empirical likelihood confidence tubes for functional parameters in plug-in estimation

We consider the infinite-dimensional inference problem in which the parameter of interest is a multivariate trajectory that can be written as an explicit functional T of a number of probability distributions. We propose an empirical likelihood procedure to build simultaneous confidence regions for these trajectories. Our main assumption is the Hadamard differentiability of T under norms adapted to empirical measures, i.e., supremum norms indexed by Donsker classes of functions. In order to handle practical computational issues, the proposed method, which we prove to be consistent, is based on a first order expansion of T . We also prove a general result of independent interest in empirical likelihood theory. Three applications are provided.

Empirical likelihood confidence bands for mean functions of recurrent events with competing risks and a terminal event

In this paper, we consider recurrent events with competing risks in the presence of a terminal event and a censorship. We focus our attention on the mean functions which give the expected number of events of a specific type that have occurred up to a time t . Using heuristics from empirical likelihood theory, we propose a method to build simultaneous (in t ) confidence regions for these functions. To establish the consistency of this estimation method (as well as its bootstrap calibration), we prove a weak convergence (as stochastic processes) of the associated empirical likelihood ratio processes. Our approach almost entirely relies on empirical process methods. In the proofs, we also establish some results in empirical processes theory that may present some independent interest. Then we carry out a simulation study of our confidence bands, we compare those obtained by empirical likelihood to the ones obtained by bootstrap. Finally, our procedure is applied on a real data set of nosocomial infections in an intensive care unit of a French hospital.

Simultaneous Tests with Combining Functions under Normality

We propose simultaneous tests for mean and variance under the normality assumption. After formulating the null hypothesis and its alternative, we construct test statistics based on the individual p-values for the partial tests with combining functions and derive the null distributions for the combining functions. We then illustrate our procedure with industrial data and compare the efficiency among the combining functions with individual partial ones by obtaining empirical powers through a simulation study. A discussion then follows on the intersection-union test with a combining function and simultaneous confidence region as a simultaneous inference; in addition, we discuss weighted functions and applications to the statistical quality control. Finally we comment on nonparametric simultaneous tests.

Inference in a model of successive failures with shape-adjusted hazard rates

For successive failure times of components in a technical system, a flexible model based on sequential order statistics is proposed. Beyond the common assumption of proportionality, this model allows for structural adjustments of the hazard rates of the underlying lifetime distributions in situations, where failures have an impact on the entire shape of the hazard rate of remaining components. The hazard rates may be chosen, e.g., as strictly ordered bathtub curves. The general structure is analysed, and maximum likelihood estimators are stated for both, unrestricted and order restricted model parameters, as well as for parameters connected by a linear link function. Several properties of the estimators are obtained. Utilizing the maximum likelihood estimators, simultaneous confidence regions based on the Jeffreys–Kullback–Leibler distance and the Hellinger distance are examined.

Statistical analysis of a questionnaire: voluntary health insurance implementation among patients suffering from allergy and asthma

We consider statistical analysis of multiple answers in a questionnaire. We propose a new method of calculating simultaneous confidence regions. In a communication presented at the European Academy of Allergy and Clinical Immunology the authors (Borowicz et al. (2009)) reported the proportions of respondents which gave one of three possible exclusive answers in a questionnaire concerning the role of voluntary health insurance. There were three possible answers. Apart from percentages of answers confidence intervals of every single answer have been reported. Unfortunately inference about the population based on such intervals may lead to imprecise conclusions. The inference about the respective population suffering from allergy and asthma proportions requires the construction of two-dimensional confidence region. We propose the use of a simultaneous confidence intervals to inference about true population proportions. Most of our attention is given to the case of three possible answers but the results may be generalized to any questionnaire with more than two excluding answers.