The block bootstrap confidence interval based on dependent data can outperform the computationally more convenient normal approximation only with non-trivial Studentization which, in the case of complicated statistics, calls for highly specialist treatment. We propose two different approaches to improving the accuracy of the block bootstrap confidence interval under very general conditions. The first calibrates the coverage level by iterating the block bootstrap. The second calculates Studentizing factors directly from block bootstrap series and requires no non-trivial analytic treatment. Both approaches involve two nested levels of block bootstrap resampling and yield high-order accuracy with simple tuning of block lengths at the two resampling levels. A simulation study is reported to provide empirical support for our theory.
In this paper we study the problem of constructing bootstrap confidence intervals for the Lévy density of the driving Lévy process based on high-frequency observations of a Lévy-driven moving average processes. Using a spectral estimator of the Lévy density, we propose a novel implementations of multiplier and empirical bootstraps to construct confidence bands on a compact set away from the origin. We also provide conditions under which the confidence bands are asymptotically valid.
In this paper, we propose to construct confidence bands by bootstrapping the debiased kernel density estimator (for density estimation) and the debiased local polynomial regression estimator (for regression analysis). The idea of using a debiased estimator was recently employed by Calonico et al. (2018b) to construct a confidence interval of the density function (and regression function) at a given point by explicitly estimating stochastic variations. We extend their ideas of using the debiased estimator and further propose a bootstrap approach for constructing simultaneous confidence bands. T…
The asymptotic behaviour of the commonly used bootstrap percentile confidence interval is investigated when the parameters are subject to linear inequality constraints. We concentrate on the important one- and two-sample problems with data generated from general parametric distributions in the natural exponential family. The focus of this paper is on quantifying the coverage probabilities of the parametric bootstrap percentile confidence intervals, in particular their limiting behaviour near boundaries. We propose a local asymptotic framework to study this subtle coverage behaviour. Under this…
This paper develops bootstrap methods to construct uniform confidence bands for nonparametric spectral estimation of Lévy densities under high-frequency observations. We assume that we observe $n$ discrete observations at frequency $1/Δ> 0$, and work with the high-frequency setup where $Δ= Δ_{n} \to 0$ and $nΔ\to \infty$ as $n \to \infty$. We employ a spectral (or Fourier-based) estimator of the Lévy density, and develop novel implementations of Gaussian multiplier (or wild) and empirical (or Efron's) bootstraps to construct confidence bands for the spectral estimator on a compact set that doe…
In this paper we study the consistency of different bootstrap procedures for constructing confidence intervals (CIs) for the unique jump discontinuity (change-point) in an otherwise smooth regression function in a stochastic design setting. This problem exhibits nonstandard asymptotics and we argue that the standard bootstrap procedures in regression fail to provide valid confidence intervals for the change-point. We propose a version of smoothed bootstrap, illustrate its remarkable finite sample performance in our simulation study, and prove the consistency of the procedure. The $m$ out of $n…
In the extreme value analysis of time series, not only the tail behavior is of interest, but also the serial dependence plays a crucial role. Drees and Rootzén (2010) established limit theorems for a general class of empirical processes of so-called cluster functionals which can be used to analyse various aspects of the extreme value behavior of mixing time series. However, usually the limit distribution is too complex to enable a direct construction of confidence regions. Therefore, we suggest a multiplier block bootstrap analog to the empirical processes of cluster functionals. It is shown t…
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…
This article is concerned with proving the consistency of Efron's (1981) bootstrap for the Kaplan-Meier estimator on the whole support of a survival function. While other works address the asymptotic Gaussianity of the estimator itself without restricting time (e.g. Gill, 1983, and Ying, 1989), we enable the construction of bootstrap-based time-simultaneous confidence bands for the whole survival function. Other practical applications include bootstrap-based confidence bands for the mean residual life-time function or the Lorenz curve as well as confidence intervals for the Gini index.
In this paper we introduce the concept of bootstrapped pivots for the sample and the population means. This is in contrast to the classical method of constructing bootstrapped confidence intervals for the population mean via estimating the cutoff points via drawing a number of bootstrap sub-samples. We show that this new method leads to constructing asymptotic confidence intervals with significantly smaller error in comparison to both of the traditional t-intervals and the classical bootstrapped confidence intervals. The approach taken in this paper relates naturally to super-population modeli…
The construction of confidence regions for parameter vectors is a difficult problem in the nonparametric setting, particularly when the sample size is not large. The bootstrap has shown promise in solving this problem, but empirical evidence often indicates that some bootstrap methods have difficulty in maintaining the correct coverage probability, while other methods may be unstable, often resulting in very large confidence regions. One way to improve the performance of a bootstrap confidence region is to restrict the shape of the region in such a way that the error term of an expansion is as…
This study aims to evaluate the performance of power in the likelihood ratio test for changepoint detection by bootstrap sampling, and proposes a hypothesis test based on bootstrapped confidence interval lengths. Assuming i.i.d normally distributed errors, and using the bootstrap method, the changepoint sampling distribution is estimated. Furthermore, this study describes a method to estimate a data set with no changepoint to form the null sampling distribution. With the null sampling distribution, and the distribution of the estimated changepoint, critical values and power calculations can be…
We compare the following two sources of poor coverage of post-model-selection confidence intervals: the preliminary data-based model selection sometimes chooses the wrong model and the data used to choose the model is re-used for the construction of the confidence interval.
Bootstrap smoothed (bagged) parameter estimators have been proposed as an improvement on estimators found after preliminary data-based model selection. The key result of Efron (2014) is a very convenient and widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. This approximation provides an easily computed guide to the accuracy of this estimator. In addition, Efron (2014) proposed a confidence interval centered on the bootstrap smoothed estimator, with width proportional to the estimate of this approximation to the standard d…
Impact of existence and nonexistence of pivot on the coverage of empirical best linear prediction intervals for small areas
Department of Mathematics & Statistics, Eastern Kentucky University（东肯塔基大学数学与统计系）；Institute of Mathematics for Industry, Kyushu University（九州大学产业数学研究所）；Joint Program in Survey Methodology, University of Maryland（马里兰大学联合调查方法项目）
We consider the classic problem of interval estimation of a proportion $p$ based on binomial sampling. The "exact" Clopper-Pearson confidence interval for $p$ is known to be unnecessarily conservative. We propose coverage-adjustments of the Clopper-Pearson interval using prior and posterior distributions of $p$. The adjusted intervals have improved coverage and are often shorter than competing intervals found in the literature. Using new heatmap-type plots for comparing confidence intervals, we find that the coverage-adjusted intervals are particularly suitable for $p$ close to 0 or 1.
Bootstrap inference is a powerful tool for obtaining robust inference for quantiles and difference-in-quantiles estimators. The computationally intensive nature of bootstrap inference has made it infeasible in large-scale experiments. In this paper, the theoretical properties of the Poisson bootstrap algorithm and quantile estimators are used to derive alternative resampling-free algorithms for Poisson bootstrap inference that reduce the computational complexity substantially without additional assumptions. These findings are connected to existing literature on analytical confidence intervals …
Constructing confidence intervals for the coefficients of high-dimensional sparse linear models remains a challenge, mainly because of the complicated limiting distributions of the widely used estimators, such as the lasso. Several methods have been developed for constructing such intervals. Bootstrap lasso+ols is notable for its technical simplicity, good interpretability, and performance that is comparable with that of other more complicated methods. However, bootstrap lasso+ols depends on the beta-min assumption, a theoretic criterion that is often violated in practice. Thus, we introduce a…
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…
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 …
Subsampling and block-based bootstrap methods have been used in a wide range of inference problems for time series. To accommodate the dependence, these resampling methods involve a bandwidth parameter, such as subsampling window width and block size in the block-based bootstrap. In empirical work, using different bandwidth parameters could lead to different inference results, but the traditional first order asymptotic theory does not capture the choice of the bandwidth. In this article, we propose to adopt the fixed-b approach, as advocated by Kiefer and Vogelsang (2005) in the heteroscedasti…
We focus on the construction of confidence corridors for multivariate nonparametric generalized quantile regression functions. This construction is based on asymptotic results for the maximal deviation between a suitable nonparametric estimator and the true function of interest which follow after a series of approximation steps including a Bahadur representation, a new strong approximation theorem and exponential tail inequalities for Gaussian random fields. As a byproduct we also obtain confidence corridors for the regression function in the classical mean regression. In order to deal with th…
Standard approaches to constructing nonparametric confidence bands for functions are frustrated by the impact of bias, which generally is not estimated consistently when using the bootstrap and conventionally smoothed function estimators. To overcome this problem it is common practice to either undersmooth, so as to reduce the impact of bias, or oversmooth, and thereby introduce an explicit or implicit bias estimator. However, these approaches, and others based on nonstandard smoothing methods, complicate the process of inference, for example, by requiring the choice of new, unconventional smo…
Bootstrap smoothed (bagged) estimators have been proposed as an improvement on estimators found after preliminary data-based model selection. Efron, 2014, derived a widely applicable formula for a delta method approximation to the standard deviation of the bootstrap smoothed estimator. He also considered a confidence interval centered on the bootstrap smoothed estimator, with width proportional to the estimate of this standard deviation. Kabaila and Wijethunga, 2019, assessed the performance of this confidence interval in the scenario of two nested linear regression models, the full model and …
In the analysis of panel data that includes a time-varying covariate, a Hausman pretest is commonly used to decide whether subsequent inference is made using the random effects model or the fixed effects model. We consider the effect of this pretest on the coverage probability of a confidence interval for the slope parameter. We prove three new finite sample theorems that make it easy to assess, for a wide variety of circumstances, the effect of the Hausman pretest on the minimum coverage probability of this confidence interval. Our results show that for the small levels of significance of the…
Let $\mathbf{X}=(X_1,X_2,X_3)$ be a spherically symmetric random vector of which only $(X_1,X_2)$ can be observed. We focus attention on estimating F, the distribution function of the squared radius $Z:=X_1^2+X_2^2+X_3^2$, from a random sample of $(X_1,X_2)$. Such a problem arises in astronomy where $(X_1,X_2,X_3)$ denotes the three dimensional position of a star in a galaxy but we can only observe the projected stellar positions $(X_1,X_2)$. We consider isotonic estimators of F and derive their limit distributions. The results are nonstandard with a rate of convergence $\sqrt{n/{\log n}}$. Th…
A general approach to selective inference is considered for hypothesis testing of the null hypothesis represented as an arbitrary shaped region in the parameter space of multivariate normal model. This approach is useful for hierarchical clustering where confidence levels of clusters are calculated only for those appeared in the dendrogram, thus subject to heavy selection bias. Our computation is based on a raw confidence measure, called bootstrap probability, which is easily obtained by counting how many times the same cluster appears in bootstrap replicates of the dendrogram. We adjust the b…
The multivariate Fay-Herriot model is quite effective in combining information through correlations among small area survey estimates of related variables or historical survey estimates of the same variable or both. Though the literature on small area estimation is already very rich, construction of second-order efficient confidence intervals from multivariate models have so far received very little attention. In this paper, we develop a parametric bootstrap method for constructing a second-order efficient confidence interval for a general linear combination of small area means using the multi…
One of the most commonly used methods for forming confidence intervals for statistical inference is the empirical bootstrap, which is especially expedient when the limiting distribution of the estimator is unknown. However, despite its ubiquitous role, its theoretical properties are still not well understood for non-asymptotically normal estimators. In this paper, under stability conditions, we establish the limiting distribution of the empirical bootstrap estimator, derive tight conditions for it to be asymptotically consistent, and quantify the speed of convergence. Moreover, we propose thre…
Structural equation models and Bayesian networks have been widely used to study causal relationships between continuous variables. Recently, a non-Gaussian method called LiNGAM was proposed to discover such causal models and has been extended in various directions. An important problem with LiNGAM is that the results are affected by the random sampling of the data as with any statistical method. Thus, some analysis of the statistical reliability or confidence level should be conducted. A common method to evaluate a confidence level is a bootstrap method. However, a confidence level computed by…
The Highly-Adaptive-LASSO Targeted Minimum Loss Estimator (HAL-TMLE) is an efficient plug-in estimator of a pathwise differentiable parameter in a statistical model that at minimal (and possibly only) assumes that the sectional variation norm of the true nuisance functional parameters (i.e., the relevant part of data distribution) are finite. It relies on an initial estimator (HAL-MLE) of the nuisance functional parameters by minimizing the empirical risk over the parameter space under the constraint that the sectional variation norm of the candidate functions are bounded by a constant, where …
In assessing prediction accuracy of multivariable prediction models, optimism corrections are essential for preventing biased results. However, in most published papers of clinical prediction models, the point estimates of the prediction accuracy measures are corrected by adequate bootstrap-based correction methods, but their confidence intervals are not corrected, e.g., the DeLong's confidence interval is usually used for assessing the C-statistic. These naive methods do not adjust for the optimism bias and do not account for statistical variability in the estimation of parameters in the pred…
Let $X_{1},\ldots,X_{n}$ be i.i.d. sample in $\mathbb{R}^{p}$ with zero mean and the covariance matrix $\mathbfΣ$. The problem of recovering the projector onto an eigenspace of $\mathbfΣ$ from these observations naturally arises in many applications. Recent technique from [Koltchinskii, Lounici, 2015] helps to study the asymptotic distribution of the distance in the Frobenius norm $\| \mathbf{P}_r - \widehat{\mathbf{P}}_r \|_{2}$ between the true projector $\mathbf{P}_r$ on the subspace of the $r$-th eigenvalue and its empirical counterpart $\widehat{\mathbf{P}}_r$ in terms of the effective ra…
The goal of this paper is to develop a practical and general-purpose approach to construct confidence intervals for differentially private parametric estimation. We find that the parametric bootstrap is a simple and effective solution. It cleanly reasons about variability of both the data sample and the randomized privacy mechanism and applies "out of the box" to a wide class of private estimation routines. It can also help correct bias caused by clipping data to limit sensitivity. We prove that the parametric bootstrap gives consistent confidence intervals in two broadly relevant settings, in…
We show that, when the double bootstrap is used to improve performance of bootstrap methods for bias correction, techniques based on using a single double-bootstrap sample for each single-bootstrap sample can be particularly effective. In particular, they produce third-order accuracy for much less computational expense than is required by conventional double-bootstrap methods. However, this improved level of performance is not available for the single double-bootstrap methods that have been suggested to construct confidence intervals or distribution estimators.
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…
Bootstrap is a useful tool for making statistical inference, but it may provide erroneous results under complex survey sampling. Most studies about bootstrap-based inference are developed under simple random sampling and stratified random sampling. In this paper, we propose a unified bootstrap method applicable to some complex sampling designs, including Poisson sampling and probability-proportional-to-size sampling. Two main features of the proposed bootstrap method are that studentization is used to make inference, and the finite population is bootstrapped based on a multinomial distribution…
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…
In this paper, we show an innovative way to construct bootstrap confidence interval of a signal estimated based on a univariate LSTM model. We take three different types of bootstrap methods for dependent set up. We prescribe some useful suggestions to select the optimal block length while performing the bootstrapping of the sample. We also propose a benchmark to compare the confidence interval measured through different bootstrap strategies. We illustrate the experimental results through some stock price data set.
In this paper, we investigate the (in)-consistency of different bootstrap methods for constructing confidence intervals in the class of estimators that converge at rate $n^{1/3}$. The Grenander estimator, the nonparametric maximum likelihood estimator of an unknown nonincreasing density function $f$ on $[0,\infty)$, is a prototypical example. We focus on this example and explore different approaches to constructing bootstrap confidence intervals for $f(t_0)$, where $t_0\in(0,\infty)$ is an interior point. We find that the bootstrap estimate, when generating bootstrap samples from the empirical…
Inference procedures in sequential trial emulation with survival outcomes: comparing confidence intervals based on the sandwich variance estimator, bootstrap and jackknife
Out-of-bag error is commonly used as an estimate of generalisation error in ensemble-based learning models such as random forests. We present confidence intervals for this quantity using the delta-method-after-bootstrap and the jackknife-after-bootstrap techniques. These methods do not require growing any additional trees. We show that these new confidence intervals have improved coverage properties over the naive confidence interval, in real and simulated examples.
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…
Upper Confidence Bound (UCB) method is arguably the most celebrated one used in online decision making with partial information feedback. Existing techniques for constructing confidence bounds are typically built upon various concentration inequalities, which thus lead to over-exploration. In this paper, we propose a non-parametric and data-dependent UCB algorithm based on the multiplier bootstrap. To improve its finite sample performance, we further incorporate second-order correction into the above construction. In theory, we derive both problem-dependent and problem-independent regret bound…
