We consider bootstrap-based testing for threshold effects in non-linear threshold autoregressive (TAR) models. It is well-known that classic tests based on asymptotic theory tend to be oversized in the case of small, or even moderate sample sizes, or when the estimated parameters indicate non-stationarity, as often witnessed in the analysis of financial or climate data. To address the issue we propose a supremum Lagrange Multiplier test statistic (sLMb), where the null hypothesis specifies a linear autoregressive (AR) model against the alternative of a TAR model. We consider a recursive bootstrap applied to the sLMb statistic and establish its validity. This result is new, and requires the proof of non-standard results for bootstrap analysis in time series models; this includes a uniform bootstrap law of large numbers and a bootstrap functional central limit theorem. These new results can also be used as a general theoretical framework that can be adapted to other situations, such as regime-switching processes with exogenous threshold variables, or testing for structural breaks. The Monte Carlo evidence shows that the bootstrap test has correct empirical size even for small samples, and also no loss of empirical power when compared to the asymptotic test. Moreover, its performance is not affected if the order of the autoregression is estimated based on information criteria. Finally, we analyse a panel of short time series to assess the effect of warming on population dynamics.
We consider hypothesis testing for the null hypothesis being represented as an arbitrary-shaped region in the parameter space. We compute an approximate p-value by counting how many times the null hypothesis holds in bootstrap replicates. This frequency, known as bootstrap probability, is widely used in evolutionary biology, but often reported as biased in the literature. Based on the asymptotic theory of bootstrap confidence intervals, there have been some new attempts for adjusting the bias via bootstrap probability without direct access to the parameter value. One such an attempt is the dou…
This paper provides conditions under which subsampling and the bootstrap can be used to construct estimators of the quantiles of the distribution of a root that behave well uniformly over a large class of distributions $\mathbf{P}$. These results are then applied (i) to construct confidence regions that behave well uniformly over $\mathbf{P}$ in the sense that the coverage probability tends to at least the nominal level uniformly over $\mathbf{P}$ and (ii) to construct tests that behave well uniformly over $\mathbf{P}$ in the sense that the size tends to no greater than the nominal level unifo…
It can be argued that optimal prediction should take into account all available data. Therefore, to evaluate a prediction interval's performance one should employ conditional coverage probability, conditioning on all available observations. Focusing on a linear model, we derive the asymptotic distribution of the difference between the conditional coverage probability of a nominal prediction interval and the conditional coverage probability of a prediction interval obtained via a residual-based bootstrap. Applying this result, we show that a prediction interval generated by the residual-based b…
In this paper, we propose a general method for testing inequality restrictions on nonparametric functions. Our framework includes many nonparametric testing problems in a unified framework, with a number of possible applications in auction models, game theoretic models, wage inequality, and revealed preferences. Our test involves a one-sided version of $L_{p}$ functionals of kernel-type estimators $(1\leq p <\infty )$ and is easy to implement in general, mainly due to its recourse to the bootstrap method. The bootstrap procedure is based on nonparametric bootstrap applied to kernel-based test …
We propose a novel test statistic for testing exogeneity in the functional linear regression model. In contrast to Hausman-type tests in finite dimensional linear regression setups, a direct extension to the functional linear regression model is not possible. Instead, we propose a test statistic based on the sum of the squared difference of projections of the two estimators for testing the null hypothesis of exogeneity in the functional linear regression model. We derive asymptotic normality under the null and consistency under general alternatives. Moreover, we prove bootstrap consistency res…
We construct new testing procedures for spherical and elliptical symmetry based on the characterization that a random vector $X$ with finite mean has a spherical distribution if and only if $\Ex[u^\top X | v^\top X] = 0$ holds for any two perpendicular vectors $u$ and $v$. Our test is based on the Kolmogorov-Smirnov statistic, and its rejection region is found via the spherically symmetric bootstrap. We show the consistency of the spherically symmetric bootstrap test using a general Donsker theorem which is of some independent interest. For the case of testing for elliptical symmetry, the Kolm…
The general philosophy for bootstrap or permutation methods for testing hypotheses is to simulate the variation of the test statistic by generating the sampling distribution which assumes both that the null hypothesis is true, and that the data in the sample is somehow representative of the population. This philosophy is inapplicable for testing hypotheses for a single parameter like the population mean, since the two assumptions are contradictory (e.g., how can we assume both that the mean of the population is $μ_0,$ and that the individuals in the sample with a mean $M \ne μ_0$ are represent…
The multivariate linear regression model is an important tool for investigating relationships between several response variables and several predictor variables. The primary interest is in inference about the unknown regression coefficient matrix. We propose multivariate bootstrap techniques as a means for making inferences about the unknown regression coefficient matrix. These bootstrapping techniques are extensions of those developed in Freedman (1981), which are only appropriate for univariate responses. Extensions to the multivariate linear regression model are made without proof. We forma…
We show that the full-sample bootstrap is asymptotically valid for constructing confidence intervals for high-quantiles, tail probabilities, and other tail parameters of a univariate distribution. This resolves the doubts that have been raised about the validity of such bootstrap methods. In our extensive simulation study, the overall performance of the bootstrap method was better than that of the standard asymptotic method, indicating that the bootstrap method is at least as good, if not better than, the asymptotic method for inference. This paper also lays the foundation for developing boots…
We provide a functional central limit theorem for a broad class of smooth functions for possibly noncausal multivariate linear processes with time-varying coefficients. Since the limiting processes depend on unknown quantities, we propose a local block bootstrap procedure to circumvent this inconvenience in practical applications. In particular, we prove bootstrap validity for a very broad class of processes. Our results are illustrated by some numerical examples.
In this paper, we develop uniform inference methods for the conditional mode based on quantile regression. Specifically, we propose to estimate the conditional mode by minimizing the derivative of the estimated conditional quantile function defined by smoothing the linear quantile regression estimator, and develop two bootstrap methods, a novel pivotal bootstrap and the nonparametric bootstrap, for our conditional mode estimator. Building on high-dimensional Gaussian approximation techniques, we establish the validity of simultaneous confidence rectangles constructed from the two bootstrap met…
Standard statistical methods that do not take proper account of the complexity of survey design can lead to erroneous inferences when applied to survey data due to unequal selection probabilities, clustering, and other design features. In particular, the actual type I error rates of tests of hypotheses using standard methods can be much bigger than the nominal significance level. Methods that take account of survey design features in testing hypotheses have been proposed, including Wald tests and quasi-score tests that involve the estimated covariance matrices of parameter estimates. In this p…
We introduce a bootstrap procedure for high-frequency statistics of Brownian semistationary processes. More specifically, we focus on a hypothesis test on the roughness of sample paths of Brownian semistationary processes, which uses an estimator based on a ratio of realized power variations. Our new resampling method, the local fractional bootstrap, relies on simulating an auxiliary fractional Brownian motion that mimics the fine properties of high frequency differences of the Brownian semistationary process under the null hypothesis. We prove the first order validity of the bootstrap method …
Split-plot or repeated measures designs are frequently used for planning experiments in the life or social sciences. Typical examples include the comparison of different treatments over time, where both factors may possess an additional factorial structure. For such designs, the statistical analysis usually consists of several steps. If the global null is rejected, multiple comparisons are usually performed. Usually, general factorial repeated measures designs are inferred by classical linear mixed models. Common underlying assumptions, such as normality or variance homogeneity are often not m…
We consider semiparametric transformation models, where after pre-estimation of a parametric transformation of the response the data are modeled by means of nonparametric regression. We suggest subsequent procedures for testing lack-of-fit of the regression function and for significance of covariables, which - in contrast to procedures from the literature - are asymptotically not influenced by the pre-estimation of the transformation. The test statistics are asymptotically pivotal and have the same asymptotic distribution as in regression models without transformation. We show validity of a mu…
Kronecker product covariance structure provides an efficient way to modeling the inter-correlations of matrix-variate data. In this paper, we propose testing statistics for Kronecker product covariance matrix based on linear spectral statistics of renormalized sample covariance matrices. Central limit theorem is proved for the linear spectral statistics with explicit formulas for mean and covariance functions, which fills the gap in the literature. We then theoretically justify that the proposed testing statistics have well-controlled sizes and strong powers. To facilitate practical usefulness…
The process comparing the empirical cumulative distribution function of the sample with a parametric estimate of the cumulative distribution function is known as the empirical process with estimated parameters and has been extensively employed in the literature for goodness-of-fit testing. The simplest way to carry out such goodness-of-fit tests, especially in a multivariate setting, is to use a parametric bootstrap. Although very easy to implement, the parametric bootstrap can become very computationally expensive as the sample size, the number of parameters, or the dimension of the data incr…
In this paper we consider a heteroscedastic transformation model, where the transformation belongs to a parametric family of monotone transformations, the regression and variance function are modelled nonparametrically and the error is independent of the multidimensional covariates. In this model, we first consider the estimation of the unknown components of the model, namely the transformation parameter, regression and variance function and the distribution of the error. We show the asymptotic normality of the proposed estimators. Second, we propose tests for the validity of the model, and es…
The paper considers a paired data framework and discuss the question of marginal homogeneity of bivariate high dimensional or functional data. The related testing problem can be endowed into a more general setting for paired random variables taking values in a general Hilbert space. To address this problem, a Cramer-von-Mises type test statistic is applied and a bootstrap procedure is suggested to obtain critical values and finally a consistent test. The desired properties of a bootstrap test can be derived, that are asymptotic exactness under the null hypothesis and consistency under alternat…
The bootstrap variance estimate is widely used in semiparametric inferences. However, its theoretical validity is a well known open problem. In this paper, we provide a {\em first} theoretical study on the bootstrap moment estimates in semiparametric models. Specifically, we establish the bootstrap moment consistency of the Euclidean parameter which immediately implies the consistency of $t$-type bootstrap confidence set. It is worth pointing out that the only additional cost to achieve the bootstrap moment consistency in contrast with the distribution consistency is to simply strengthen the $…
This paper investigates the theoretical underpinnings of two fundamental statistical inference problems, the construction of confidence sets and large-scale simultaneous hypothesis testing, in the presence of heavy-tailed data. With heavy-tailed observation noise, finite sample properties of the least squares-based methods, typified by the sample mean, are suboptimal both theoretically and empirically. In this paper, we demonstrate that the adaptive Huber regression, integrated with the multiplier bootstrap procedure, provides a useful robust alternative to the method of least squares. Our the…
In this paper, we consider possibly misspecified stochastic differential equation models driven by Lévy processes. Regardless of whether the driving noise is Gaussian or not, Gaussian quasi-likelihood estimator can estimate unknown parameters in the drift and scale coefficients. However, in the misspecified case, the asymptotic distribution of the estimator varies by the correction of the misspecification bias, and consistent estimators for the asymptotic variance proposed in the correctly specified case may lose theoretical validity. As one of its solutions, we propose a bootstrap method for …
Quantifying the uncertainty in penalized regression under group sparsity is an important open question. We establish, under a high-dimensional scaling, the asymptotic validity of a modified parametric bootstrap method for the group lasso, assuming a Gaussian error model and mild conditions on the design matrix and the true coefficients. Simulation of bootstrap samples provides simultaneous inferences on large groups of coefficients. Through extensive numerical comparisons, we demonstrate that our bootstrap method performs much better than popular competitors, highlighting its practical utility…
A common question being raised in automatic speech recognition (ASR) evaluations is how reliable is an observed word error rate (WER) improvement comparing two ASR systems, where statistical hypothesis testing and confidence interval (CI) can be utilized to tell whether this improvement is real or only due to random chance. The bootstrap resampling method has been popular for such significance analysis which is intuitive and easy to use. However, this method fails in dealing with dependent data, which is prevalent in speech world - for example, ASR performance on utterances from the same speak…
Both seasonal unit roots and periodic variation can be prevalent in seasonal data. When testing seasonal unit roots under periodic variation, the validity of the existing methods, such as the HEGY test, remains unknown. This paper analyzes the behavior of the augmented HEGY test and the unaugmented HEGY test under periodic variation. It turns out that the asymptotic null distributions of the HEGY statistics testing the single roots at $1$ or $-1$ when there is periodic variation are identical to the asymptotic null distributions when there is no periodic variation. On the other hand, the asymp…
We introduce and evaluate a new class of hypothesis testing procedures for moment structures. The methods are valid under weak assumptions and includes the well-known Satorra-Bentler adjustment as a special case. The proposed procedures applies also to difference testing among nested models. We prove the consistency of our approach. We introduce a bootstrap selection mechanism to optimally choose a p-value approximation for a given sample. Also, we propose bootstrap procedures for assessing the asymptotic robustness (AR) of the normal-theory maximum likelihood test, and for the key assumption …
A bootstrap approach for testing invariance under parameterized group actions: orthogonal reflections and axial symmetry
Centro de Matemáticas, Facultad de Ciencias, Universidad de la República（乌拉圭共和国大学理学院数学中心）；Departamento de Matemáticas, Estadística y Computación, Universidad de Cantabria（坎塔布里亚大学数学、统计与计算系）；Instituto de Estadística, Facultad de Ciencias Económicas y de Administración, Universidad de la República（乌拉圭共和国大学经济与管理学院统计研究所）
The asymptotic validity of a resampling method for two sequential processes constructed from non-degenerate $U$-statistics is established under mixing conditions. The resampling schemes, referred to as {\em dependent multiplier bootstraps}, result from an adaptation of the seminal approach of \cite{GomHor02} to mixing sequences. The proofs exploit recent results of \cite{DehWen10b} on degenerate $U$-statistics. A data-driven procedure for estimating a key bandwidth parameter involved in the resampling schemes is also suggested, making the use of the studied dependent multiplier bootstraps full…
Mack's distribution-free chain ladder reserving model belongs to the most popular approaches in non-life insurance mathematics. Proposed to determine the first two moments of the reserve, it does not allow to identify the whole distribution of the reserve. For this purpose, Mack's model is usually equipped with a tailor-made bootstrap procedure. Although widely used in practice to estimate the reserve risk, no theoretical bootstrap consistency results exist that justify this approach. To fill this gap in the literature, we adopt the framework proposed by Steinmetz and Jentsch (2022) to derive …
We study the weak convergence of conditional empirical copula processes, when the conditioning event has a nonzero probability. The validity of several bootstrap schemes is stated, including the exchangeable bootstrap. We define general - possibly conditional - multivariate dependence measures and their estimators. By applying our theoretical results, we prove the asymptotic normality of some estimators of such dependence measures.
Eigenspaces of covariance matrices play an important role in statistical machine learning, arising in variety of modern algorithms. Quantitatively, it is convenient to describe the eigenspaces in terms of spectral projectors. This work focuses on hypothesis testing for the spectral projectors, both in one- and two-sample scenario. We present new tests, based on a specific matrix norm developed in order to utilize the structure of the spectral projectors. A new resampling technique of independent interest is introduced and analyzed: it serves as an alternative to the well-known multiplier boots…
This article reviews recent progress in high-dimensional bootstrap. We first review high-dimensional central limit theorems for distributions of sample mean vectors over the rectangles, bootstrap consistency results in high dimensions, and key techniques used to establish those results. We then review selected applications of high-dimensional bootstrap: construction of simultaneous confidence sets for high-dimensional vector parameters, multiple hypothesis testing via stepdown, post-selection inference, intersection bounds for partially identified parameters, and inference on best policies in …
An algorithm is described that enables efficient deterministic approximate computation of the bootstrap distribution for any linear bootstrap method $T_n^*$, alleviating the need for repeated resampling from observations (resp. input-derived data). In essence, the algorithm computes the distribution function from a linear mixture of independent random variables each having a finite discrete distribution. The algorithm is applicable to elementary bootstrap scenarios (targetting the mean as parameter of interest), for block bootstrap, as well as for certain residual bootstrap scenarios. Moreover…
Cognitive Diagnosis Models (CDMs) are a special family of discrete latent variable models widely used in educational, psychological and social sciences. In many applications of CDMs, certain hierarchical structures among the latent attributes are assumed by researchers to characterize their dependence structure. Specifically, a directed acyclic graph is used to specify hierarchical constraints on the allowable configurations of the discrete latent attributes. In this paper, we consider the important yet unaddressed problem of testing the existence of latent hierarchical structures in CDMs. We …
The problem of testing for the presence of epidemic changes in random fields is investigated. In order to be able to deal with general changes in the marginal distribution, a Cramér-von Mises type test is introduced which is based on Hilbert space theory. A functional central limit theorem for $ρ$-mixing Hilbert space valued random fields is proven. In order to avoid the estimation of the long-run variance and obtain critical values, Shao's dependent wild bootstrap method is adapted to this context. For this, a joint functional central limit theorem for the original and the bootstrap sample is…
In this paper, we establish a uniform error rate of a Bahadur representation for local polynomial estimators of quantile regression functions. The error rate is uniform over a range of quantiles, a range of evaluation points in the regressors, and over a wide class of probabilities for observed random variables. Most of the existing results on Bahadur representations for local polynomial quantile regression estimators apply to the fixed data generating process. In the context of testing monotonicity where the null hypothesis is of a complex composite hypothesis, it is particularly relevant to …
The role played by the composite analogue of the log likelihood ratio in hypothesis testing and in setting confidence regions is not as prominent as it is in the canonical likelihood setting, since its asymptotic distribution depends on the unknown parameter. Approximate pivots based on the composite log likelihood ratio can be derived by using asymptotic arguments. However, the actual distribution of such pivots may differ considerably from the asymptotic reference, leading to tests and confidence regions whose levels are distant from the nominal ones. The use of bootstrap rather than asympto…
Testing Microbiome Community Differences in High Dimensions: A Bootstrap Approach for Compositional Data
Indian Institute of Technology Bombay（印度理工学院孟买分校）；Missouri University of Science and Technology（密苏里科技大学）；Washington University in St. Louis（圣路易斯华盛顿大学）；University of Missouri（密苏里大学）；University of Michigan（密歇根大学）
The consistency of a bootstrap or resampling scheme is classically validated by weak convergence of conditional laws. However, when working with stochastic processes in the space of bounded functions and their weak convergence in the Hoffmann-Jørgensen sense, an obstacle occurs: due to possible non-measurability, neither laws nor conditional laws are well-defined. Starting from an equivalent formulation of weak convergence based on the bounded Lipschitz metric, a classical circumvent is to formulate bootstrap consistency in terms of the latter distance between what might be called a \emph{cond…
In the problem of model selection for a given family of linear estimators, ordered by their variance, we offer a new "smallest accepted" approach motivated by Lepski's method and multiple testing theory. The procedure selects the smallest model which satisfies an acceptance rule based on comparison with all larger models. The method is completely data-driven and does not use any prior information about the variance structure of the noise: its parameters are adjusted to the underlying possibly heterogeneous noise by the so-called "propagation condition" using a wild bootstrap method. The validi…
The problem of quantifying uncertainty about the locations of multiple change points by means of confidence intervals is addressed. The asymptotic distribution of the change point estimators obtained as the local maximisers of moving sum statistics is derived, where the limit distributions differ depending on whether the corresponding size of changes is local, i.e. tends to zero as the sample size increases, or fixed. A bootstrap procedure for confidence interval generation is proposed which adapts to the unknown magnitude of changes and guarantees asymptotic validity both for local and fixed …
We devise a general result on the consistency of model-based bootstrap methods for U- and V-statistics under easily verifiable conditions. For that purpose, we derive the limit distributions of degree-2 degenerate U- and V-statistics for weakly dependent $\mathbb{R}^d$-valued random variables first. To this end, only some moment conditions and smoothness assumptions concerning the kernel are required. Based on this result, we verify that the bootstrap counterparts of these statistics have the same limit distributions. Finally, some applications to hypothesis testing are presented.
In this paper, we establish a high-dimensional CLT for the sample mean of $p$-dimensional spatial data observed over irregularly spaced sampling sites in $\mathbb{R}^d$, allowing the dimension $p$ to be much larger than the sample size $n$. We adopt a stochastic sampling scheme that can generate irregularly spaced sampling sites in a flexible manner and include both pure increasing domain and mixed increasing domain frameworks. To facilitate statistical inference, we develop the spatially dependent wild bootstrap (SDWB) and justify its asymptotic validity in high dimensions by deriving error b…
We consider the double functional nonparametric regression model $Y=r(X)+ε$, where the response variable $Y$ is Hilbert space-valued and the covariate $X$ takes values in a pseudometric space. The data satisfy an ergodicity criterion which dates back to Laib and Louani (2010) and are arranged in a triangular array. So our model also applies to samples obtained from spatial processes, e.g., stationary random fields indexed by the regular lattice $\mathbb{Z}^N$ for some $N\in\mathbb{N}_+$. We consider a kernel estimator of the Nadaraya--Watson type for the regression operator $r$ and study its l…
