Bounds on coverage probabilities of the empirical likelihood ratio confidence regions
Min Tsao
基于估计方程，研究了经验似然比置信区域覆盖概率的上确界。经验似然推断的界限的影响也进行了讨论。
This paper studies the least upper bounds on coverage probabilities of the empirical likelihood ratio confidence regions based on estimating equations. The implications of the bounds on empirical likelihood inference are also discussed.
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We provide Buehler-optimal one-sided and some valid two-sided confidence intervals for the average success probability of a possibly inhomogeneous fixed length Bernoulli chain, based on the number of observed successes. Contrary to some claims in the literature, the one-sided Clopper-Pearson intervals for the homogeneous case are not completely robust here, not even if applied to hypergeometric estimation problems.
Optimal Explicit Binomial Confidence Interval with Guaranteed Coverage Probability
在本文中，我们开发了一种方法来优化显式二项置信区间最近导出的Chen等人。优化降低保守性，同时保证规定的覆盖概率。
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.
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.
Consider a linear regression model with regression parameter beta=(beta_1,..., beta_p) and independent normal errors. Suppose the parameter of interest is theta = a^T beta, where a is specified. Define the s-dimensional parameter vector tau = C^T beta - t, where C and t are specified. Suppose that we carry out a preliminary F test of the null hypothesis H_0: tau = 0 against the alternative hypothesis H_1: tau not equal to 0. It is common statistical practice to then construct a confidence interval for theta with nominal coverage 1-alpha, using the same data, based on the assumption that the se…
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…
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…
First and second kind modifications of usual confidence intervals for estimating the expectation and of usual local alternative parameter choices are introduced in a way such that the asymptotic behavior of the true non-covering probabilities and the covering probabilities under the modified local non-true parameter assumption can be asymptotically exactly controlled. The orders of convergence to zero of both types of probabilities are assumed to be suitably bounded below according to an Osipov-type condition and the sample distribution is assumed to satisfy a corresponding tail condition due …
Improving Coverage Accuracy of Block Bootstrap Confidence Intervals
基于相关数据的块引导置信区间可以优于计算上更方便的正态近似，只有非平凡的学生化，在复杂的统计情况下，需要高度专业的治疗。我们提出了两种不同的方法来提高块引导置信区间在非常一般的条件下的准确性。第一个通过迭代块引导来校准覆盖水平。第二个计算Studentizing因素直接从块bootstrap系列，不需要非平凡的分析处理。这两种方法都涉及两个嵌套级别的块自举复位，并产生高阶精度与简单的调整块长度在两个复位级别。一个模拟研究报告，我们的理论提供实证支持。
Simultaneous confidence bands for the integrated hazard function
考虑了综合风险函数的同时置信带的构造。使用Nelson-Aalen估计量。给出了基于Bootstrap方法的联合置信带。两种方法的建设，这样的置信带的建议。奇怪的bootstrap方法被用于恢复。仿真比较了Bootstrap和渐近同步置信带的实际覆盖概率。结果表明，等尾bootstrap置信带的覆盖概率最接近标称值。我们还提出了应用我们的置信带的数据，心脏移植后的生存。
An important problem in statistics is the construction of confidence regions for unknown parameters. In most cases, asymptotic distribution theory is used to construct confidence regions, so any coverage probability claims only hold approximately, for large samples. This paper describes a new approach, using random sets, which allows users to construct exact confidence regions without appeal to asymptotic theory. In particular, if the user-specified random set satisfies a certain validity property, confidence regions obtained by thresholding the induced data-dependent plausibility function are…
Explicit Formula for Constructing Binomial Confidence Interval with Guaranteed Coverage Probability
本文给出了在保证覆盖概率下构造二项参数置信区间的显式公式。该公式克服了正态近似在性质上是渐近的限制，从而在应用中不可避免地引入未知误差。此外，从区间宽度的角度来看，与经典的Clopper-Pearson方法相比，该公式非常严格。在严格公式的基础上，我们还得到了性能优良的覆盖概率近似公式。
Upper bounds on the minimum coverage probability of confidence intervals in regression after variable selection
我们考虑一个线性回归模型，其中所关注的参数是回归参数向量的指定线性组合。我们假设，作为第一步，使用基于数据的模型选择（例如，通过初步假设检验或最小化AIC）来选择模型。通常的统计实践是，基于所选模型已先验地提供给我们的假设，为感兴趣的参数构建置信区间。该假设是错误的，并且其可导致具有较差覆盖属性的置信区间。我们提供了一个易于计算的有限样本上界（计算重复数值评估的二重积分），这个置信区间的最小覆盖概率。该界限适用于通过以下任何方法进行的模型选择：最小AIC、最小BIC、最大调整R平方、最小Mallow ' Cp和t检验。该上界的重要性在于，它描述了设计矩阵和模型选择程序的一般类别，对于这些设计矩阵和模型选择程序，该置信区间具有较差的覆盖特性。这个上界被证明是一个有限的样本模拟早期的大样本上界由于Kabaila和Leeb。
We generalize standard credal set models for imprecise probabilities to include higher order credal sets -- confidences about confidences. In doing so, we specify how an agent's higher order confidences (credal sets) update upon observing an event. Our model begins to address standard issues with imprecise probability models, like Dilation and Belief Inertia. We conjecture that when higher order credal sets contain all possible probability functions, then in the limiting case the highest order confidences converge to form a uniform distribution over the first order credal set, where we define …
In this article, weak convergence of the general non-Markov state transition probability estimator by Titman (2015) is established which, up to now, has not been verified yet for other general non-Markov estimators. A similar theorem is shown for the bootstrap, yielding resampling-based inference methods for statistical functionals. Formulas of the involved covariance functions are presented in detail. Particular applications include the conditional expected length of stay in a specific state, given occupation of another state in the past, as well as the construction of time-simultaneous confi…
This paper studies higher-order inference properties of nonparametric local polynomial regression methods under random sampling. We prove Edgeworth expansions for $t$ statistics and coverage error expansions for interval estimators that (i) hold uniformly in the data generating process, (ii) allow for the uniform kernel, and (iii) cover estimation of derivatives of the regression function. The terms of the higher-order expansions, and their associated rates as a function of the sample size and bandwidth sequence, depend on the smoothness of the population regression function, the smoothness ex…
For the zone of moderate deviation probabilities the local asymptotic minimax lower bound of asymptotic efficiency of estimators is established. The estimation parameter is multidimensional. The lower bound admits the interpretation as the lower bound of asymptotic efficiency in confidence estimation.
The general problem of constructing confidence regions is unsolved in the sense that there is no algorithm that provides such a region with guaranteed coverage for an arbitrary parameter $ψ\inΨ.$ Moreover, even when such a region exists, it may be absurd in the sense that either the set $Ψ$ or the null set $ϕ$ is reported with positive probability. An approach to the construction of such regions with guaranteed coverage and which avoids absurdity is applied here to several problems that have been discussed in the recent literature and for which some standard approaches produce absurd regions.
Iterated smoothed bootstrap confidence intervals for population quantiles
本文研究了平滑Bootstrap迭代对总体分位数的平滑Bootstrap和Bootstrap t置信区间的覆盖概率的影响，并建立了平滑过程各阶段的最优核带宽。已知传统的平滑bootstrap和bootstrap-t方法对于基于大小为n的随机样本的样本分位数的区间分别产生O（n^{-1/2}）和O（n^{-2/3}）阶的单侧覆盖误差。通过在自举和学生化步骤中适当选择带宽，我们将后一个结果锐化为O（n^{-5/6}）。我们进一步表明，通过迭代引导的标称覆盖水平的校准成功地减少覆盖误差的平滑引导百分位间隔的顺序为O（n^{-2/3}）和平滑引导-t间隔的顺序为O（n^{-58/57}），只要带宽被选择适当的顺序。仿真结果证实了我们的渐近研究结果，表明迭代平滑bootstrap-t方法产生最准确的覆盖。另一方面，迭代平滑Bootstrap百分位数方法区间具有比Bootstrap-t区间更短且更稳定的优点。
We describe a new recentered confidence sphere for the mean, theta, of a multivariate normal distribution. This sphere is centred on the positive-part James-Stein estimator, with radius that is a piecewise cubic Hermite interpolating polynomial function of the norm of the data vector. This radius function is determined by numerically minimizing the scaled expected volume, at theta = 0, of this confidence sphere, subject to the coverage constraint. We use the computationally-convenient formula, derived by Casella and Hwang [3], for the coverage probability of a recentered confidence sphere. Cas…
Model-Agnostic Bounds for Augmented Inverse Probability Weighted Estimators' Wald-Confidence Interval Coverage in Randomized Controlled Trials
本文推导了AIPW估计量Wald置信区间覆盖率的非渐近Berry-Esseen型界，并分析了方差估计量的偏差，解释了交叉拟合为何能改善覆盖率。
In this paper, we develop a general theory on the coverage probability of random intervals defined in terms of discrete random variables with continuous parameter spaces. The theory shows that the minimum coverage probabilities of random intervals with respect to corresponding parameters are achieved at discrete finite sets and that the coverage probabilities are continuous and unimodal when parameters are varying in between interval endpoints. The theory applies to common important discrete random variables including binomial variable, Poisson variable, negative binomial variable and hypergeo…
Confidence intervals for median absolute deviations
本文针对中位数绝对偏差（MAD）提出区间估计方法，用于单总体离散度推断及两总体MAD比值与差值的比较，并通过模拟和部分影响函数验证了其覆盖概率与稳健性。
Confidence nets, that is, collections of confidence intervals that fill out the parameter space and whose exact parameter coverage can be computed, are familiar in nonparametric statistics. Here, the distributional assumptions are based on invariance under the action of a finite reflection group. Exact confidence nets are exhibited for a single parameter, based on the root system of the group. The main result is a formula for the generating function of the coverage interval probabilities. The proof makes use of the theory of "buildings" and the Chevalley factorization theorem for the length di…
We investigate the frequentist coverage properties of Bayesian credible sets in a general, adaptive, nonparametric framework. It is well known that the construction of adaptive and honest confidence sets is not possible in general. To overcome this problem we introduce an extra assumption on the functional parameters, the so called "general polished tail" condition. We then show that under standard assumptions both the hierarchical and empirical Bayes methods results in honest confidence sets for sieve type of priors in general settings and we characterize their size. We apply the derived abst…
Selective and marginal selective inference for exceptional groups
针对多群体研究中选择性推断导致置信区间期望宽度无穷大的问题，提出选择调整的经验贝叶斯方法，在保证有限期望宽度的同时实现近似选择性覆盖控制。
When computing a confidence interval for a binomial proportion p one must choose between using an exact interval, which has a coverage probability of at least 1-α for all values of p, and a shorter approximate interval, which may have lower coverage for some p but that on average has coverage equal to 1-α. We investigate the cost of using the exact one and two-sided Clopper--Pearson confidence intervals rather than shorter approximate intervals, first in terms of increased expected length and then in terms of the increase in sample size required to obtain a desired expected length. Using asymp…
We study the problem of estimating finite sample confidence intervals of the mean of a normal population under the constraint of differential privacy. We consider both the known and unknown variance cases and construct differentially private algorithms to estimate confidence intervals. Crucially, our algorithms guarantee a finite sample coverage, as opposed to an asymptotic coverage. Unlike most previous differentially private algorithms, we do not require the domain of the samples to be bounded. We also prove lower bounds on the expected size of any differentially private confidence set showi…
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…
Teachable normal approximations to binomial and related probabilities or confidence bounds
本文提出了可教学的二项分布及相关概率的正态近似方法，提供误差界和置信区间界，适用于不同层次的学生理解。
We introduce a new adjusted residual maximum likelihood method (REML) in the context of producing an empirical Bayes (EB) confidence interval for a normal mean, a problem of great interest in different small area applications. Like other rival empirical Bayes confidence intervals such as the well-known parametric bootstrap empirical Bayes method, the proposed interval is second-order correct, that is, the proposed interval has a coverage error of order $O(m^{-{3}/{2}})$. Moreover, the proposed interval is carefully constructed so that it always produces an interval shorter than the correspondi…
We propose confidence regions with asymptotically correct uniform coverage probability of parameters whose Fisher information matrix can be singular at important points of the parameter set. Our work is motivated by the need for reliable inference on scale parameters close or equal to zero in mixed models, which is obtained as a special case. The confidence regions are constructed by inverting a continuous extension of the score test statistic standardized by expected information, which we show exists at points of singular information under regularity conditions. Similar results have previousl…
Smoothed weighted empirical likelihood ratio confidence intervals for quantiles
到目前为止，基于似然的区间估计的分位数还没有研究在文献中的区间删失情况2数据和部分区间删失数据，在这种情况下，使用平滑没有被认为是任何类型的删失数据。本文在统一的框架下，针对不同类型的删失数据，包括右删失数据、双删失数据、区间删失数据和部分区间删失数据，构造了分位数的平滑加权经验似然比置信区间（WELRCI）。推导了加权经验对数似然比的四阶展开式，建立了WELRCI的理论覆盖精度方程，该方程一般保证至少一阶精度。特别是，对于右删失数据，我们表明，覆盖精度至少是$O(n^{-1/2})$和我们的模拟研究表明，与经验似然为基础的方法相比，在WELRCI中使用的平滑一般提供了一个较短的置信区间具有可比的覆盖精度。对于区间删失数据，有趣的是发现在调整率$n^{-1/3}$下，加权经验对数似然比的渐近分布完全不同于经验似然法所得的渐近分布，并且所得的WELRCI在现有的比较模拟研究中表现良好.
The most frequently taught confidence intervals for a proportion are the classical Wald (Ws) and the Clopper-Pearson (CP) ones because of the simplicity of their definition. However, their actual coverage probability of the parameter p is erratic, often quite far from the nominal probability which is aimed at. Other confidence intervals are clearly preferable to the former, but their expression is generally complex and they are difficult to interpret. But nevertheless, through a simple modification of the definition of the Ws and CP intervals, we obtain some confidence intervals with much bett…
Confidences in Hypotheses
该研究将假设评估方法拓展至两个竞争简单假设的场景，明确了条件选择对推断相关性的作用，探讨了容许性与一致性定义，可生成假设的最小最大置信度，还讨论了其与贝叶斯后验概率的关系。
Fast and reliable confidence intervals for a variance component
本研究提出通过对受限似然得分的检验统计量求反构造方差分量置信区间的方法，该方法适用于边界情形，小样本下仍有良好表现，在空间分辨转录组学应用中速度远超现有方法。
Simultaneous Frequentist Calibration of Confidence Regions for Multiple Functionals in Constrained Inverse Problems
本文通过统计检验反演框架，统一并扩展了线性逆问题中基于优化的置信域构造方法，在结构约束下实现了多泛函多维置信域的同时频率学派校准，有效缩小了置信域体积并推广至一般对数凹误差分布。
Measurements are generally collected as unilateral or bilateral data in clinical trials or observational studies. For example, in ophthalmology studies, the primary outcome is often obtained from one eye or both eyes of an individual. In medical studies, the relative risk is usually the parameter of interest and is commonly used. In this article, we develop three confidence intervals for the relative risk for combined unilateral and bilateral correlated data under the equal dependence assumption. The proposed confidence intervals are based on maximum likelihood estimates of parameters derived …
Coverage errors for Student's t confidence intervals comparable to those in Hall (1988)
本文推导了缺失的 Student's t 置信区间渐近覆盖误差公式，并指出 Hall (1988) 表中的公式实际上与双侧覆盖误差成正比。
Confidence on the Focal: Conformal Prediction with Selection-Conditional Coverage
针对焦点单元选择偏差导致保形预测边际覆盖失效问题，提出适用于任意排列不变选择规则的有限样本精确条件覆盖框架，推广Mondrian方法并实现多种规则的高效计算，在药物发现和健康风险预测中验证了性能。
For many causal effect parameters of interest, doubly robust machine learning (DRML) estimators $\hatψ_{1}$ are the state-of-the-art, incorporating the good prediction performance of machine learning; the decreased bias of doubly robust estimators; and the analytic tractability and bias reduction of sample splitting with cross fitting. Nonetheless, even in the absence of confounding by unmeasured factors, the nominal $(1 - α)$ Wald confidence interval $\hatψ_{1} \pm z_{α/ 2} \widehat{\mathsf{se}} [\hatψ_{1}]$ may still undercover even in large samples, because the bias of $\hatψ_{1}$ may be of…
Nonparametric Empirical Bayes Confidence Intervals
本文提出非参数经验贝叶斯置信区间，用于正常均值模型中不可观测的个体效应，通过非参数先验构造置信区间，具有渐近精确性，尽管存在统计成本，但模拟显示其在非高斯先验下保持名义覆盖概率。
An Optimal Transportation Approach for Improved Confidence Intervals
提出基于最优传输的置信区间构建方法，通过最优耦合最小化覆盖偏差，理论证明一致性并给出误差界，模拟显示优于标准方法。
In a linear regression model of fixed dimension $p \leq n$, we construct confidence regions for the unknown parameter vector based on the Lasso estimator that uniformly and exactly hold the prescribed in finite samples as well as in an asymptotic setup. We thereby quantify estimation uncertainty as well as the "post-model selection error" of this estimator. More concretely, in finite samples with Gaussian errors and asymptotically in the case where the Lasso estimator is tuned to perform conservative model selection, we derive exact formulas for computing the minimal coverage probability over …
Confidence intervals centred on bootstrap smoothed estimators: an impossibility result
本文基于决策论性能界限证明，在嵌套线性回归模型测试床中，不存在以自助法平滑估计量为中心且基于数据宽度的置信区间，能同时满足最小覆盖率与理想的缩放期望长度要求。
Signed-Perturbed Sums Estimation of ARX Systems: Exact Coverage and Strong Consistency (Extended Version)
本文研究ARX系统的符号扰动和（SPS）方法，证明其置信域在有限数据下具有用户选择的精确覆盖概率，并具备强一致性，同时渐近性能逼近传统置信椭球。
Functional data analysis, FDA, is now a well established discipline of statistics, with its core concepts and perspectives in place. Despite this, there are still fundamental statistical questions which have received relatively little attention. One of these is the systematic construction of confidence regions for functional parameters. This work is concerned with developing, understanding, and visualizing such regions. We provide a general strategy for constructing confidence regions in a real separable Hilbert space using hyper-ellipsoids and hyper-rectangles. We then propose specific implem…
Rectifying Conformity Scores for Better Conditional Coverage
本文提出一种在分裂共形预测框架下对一致性分数进行可训练变换的新方法，通过估计条件分位数改善条件覆盖并保持精确边际覆盖，在多输出问题中优于现有方法。
In complicated/nonlinear parametric models, it is generally hard to know whether the model parameters are point identified. We provide computationally attractive procedures to construct confidence sets (CSs) for identified sets of full parameters and of subvectors in models defined through a likelihood or a vector of moment equalities or inequalities. These CSs are based on level sets of optimal sample criterion functions (such as likelihood or optimally-weighted or continuously-updated GMM criterions). The level sets are constructed using cutoffs that are computed via Monte Carlo (MC) simulat…
The Elliptically Optimal Confidence Interval: A Bivariate Extension of Wilson's Score Method
本文提出椭圆最优置信区间，通过闭式优化投影椭圆区域构造二项比例差的最短得分区间，保证覆盖且不欠覆盖，并精确量化极端比例下的过度覆盖。
The proposed approach extends the confidence posterior distribution to the semi-parametric empirical Bayes setting. Whereas the Bayesian posterior is defined in terms of a prior distribution conditional on the observed data, the confidence posterior is defined such that the probability that the parameter value lies in any fixed subset of parameter space, given the observed data, is equal to the coverage rate of the corresponding confidence interval. A confidence posterior that has correct frequentist coverage at each fixed parameter value is combined with the estimated local false discovery ra…
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