Alex D Gottlieb
We show that that the jackknife variance estimator $v_{jack}$ and the the infinitesimal jackknife variance estimator are asymptotically equivalent if the functional of interest is a smooth function of the mean or a smooth trimmed L-statistic. We calculate the asymptotic variance of $v_{jack}$ for these functionals.
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The jackknife variance estimator and the the infinitesimal jackknife variance estimator are shown to be asymptotically equivalent if the functional of interest is a smooth function of the mean or a trimmed L-statistic with Hoelder continuous weight function.
Efron [J. Roy. Statist. Soc. Ser. B 54 (1992) 83--111] proposed a computationally efficient method, called the jackknife-after-bootstrap, for estimating the variance of a bootstrap estimator for independent data. For dependent data, a version of the jackknife-after-bootstrap method has been recently proposed by Lahiri [Econometric Theory 18 (2002) 79--98]. In this paper it is shown that the jackknife-after-bootstrap estimators of the variance of a bootstrap quantile are consistent for both dependent and independent data. Results from a simulation study are also presented.
A general jackknife estimator for the asymptotic covariance of moment estimators is considered in the case when the sample is taken from a mixture with varying concentrations of components. Consistency of the estimator is demonstrated. A fast algorithm for its calculation is described. The estimator is applied to construction of confidence sets for regression parameters in the linear regression with errors in variables. An application to sociological data analysis is considered.
Neyman Jackknife: Design-Based Variance Estimation for Causal Inference under Interference
本文提出Neyman Jackknife框架，提供一种通用方法，用于在存在干扰的情况下保守估计因果推断中的变异数。该方法在经典设置中恢复与Neyman估计器和Newey-West HAC方差估计器相关的估计器，并通过数值实验表明其性能可与专门设计的基准方法相媲美。
Jackknife Variance Estimation for Hájek-Dominated Generalized U-Statistics
针对一类广义U统计量，证明Jackknife方差估计量的比率相合性，并应用于两尺度分布最近邻回归估计器，在更弱条件下得到一致方差估计。
Samples with a common mean but possibly different, ordered variances arise in various fields such as interlaboratory experiments, field studies or the analysis of sensor data. Estimators for the common mean under ordered variances typically employ random weights, which depend on the sample means and the unbiased variance estimators. They take different forms when the sample estimators are in agreement with the order constraints or not, which complicates even basic analyses such as estimating their variance. We propose to use the jackknife, whose consistency is established for general smooth tw…
Though introduced nearly 50 years ago, the infinitesimal jackknife (IJ) remains a popular modern tool for quantifying predictive uncertainty in complex estimation settings. In particular, when supervised learning ensembles are constructed via bootstrap samples, recent work demonstrated that the IJ estimate of variance is particularly convenient and useful. However, despite the algebraic simplicity of its final form, its derivation is rather complex. As a result, studies clarifying the intuition behind the estimator or rigorously investigating its properties have been severely lacking. This wor…
When is it worthwhile to jackknife? Breaking the quadratic barrier for Z-estimators
本研究证明刀切法可将高维Z估计量的相合性门槛从n≥d²降至n≥d^{3/2}，为维度适中时使用刀切法提供理论依据。
We analyze bias correction methods using jackknife, bootstrap, and Taylor series. We focus on the binomial model, and consider the problem of bias correction for estimating $f(p)$, where $f \in C[0,1]$ is arbitrary. We characterize the supremum norm of the bias of general jackknife and bootstrap estimators for any continuous functions, and demonstrate the in delete-$d$ jackknife, different values of $d$ may lead to drastically different behaviors in jackknife. We show that in the binomial model, iterating the bootstrap bias correction infinitely many times may lead to divergence of bias and va…
The V-fold jackknife for semiparametric inference: variance estimation, confidence intervals, and simultaneous confidence bands
研究针对半参数推断开发V折刀切法，它计算高效且理论合理，仅需V次留一法重拟合，利用伪值经验离散度量化不确定性，给出有效置信区间和同时置信带，扩展理论到特定估计量，模拟证实其可靠推断。
We study the properties of a leave-node-out jackknife procedure for network data. Under the sparse graphon model, we prove an Efron-Stein-type inequality, showing that the network jackknife leads to conservative estimates of the variance (in expectation) for any network functional that is invariant to node permutation. For a general class of count functionals, we also establish consistency of the network jackknife. We complement our theoretical analysis with a range of simulated and real-data examples and show that the network jackknife offers competitive performance in cases where other resam…
The Bayesian Infinitesimal Jackknife for Variance
针对贝叶斯估计量频率学派协方差近似方法计算不便的问题，提出基于influence function的infinitesimal jackknife (IJ)方法，证明其可从单组后验样本高效计算且与经典方法渐近等价，并通过实验验证了其准确性与计算优势。
We study the implications of including many covariates in a first-step estimate entering a two-step estimation procedure. We find that a first order bias emerges when the number of \textit{included} covariates is "large" relative to the square-root of sample size, rendering standard inference procedures invalid. We show that the jackknife is able to estimate this "many covariates" bias consistently, thereby delivering a new automatic bias-corrected two-step point estimator. The jackknife also consistently estimates the standard error of the original two-step point estimator. For inference, we …
Empirical Likelihood Based Inference for a Divergence Measure Based on Survival Extropy
本文提出基于生存熵的分歧度度量及其非参数估计方法，并通过模拟研究验证其在实际应用中的有效性，用于检测图像数据集中的细微差异。
In estimation a parameter $θ\in{\mathbb R}$ from a sample $(x_1,\ldots,x_n)$ from a population $P_θ$ a simple way of incorporating a new observation $x_{n+1}$ into an estimator $\tildeθ_{n} = \tildeθ_{n}(x_1,\ldots,x_n)$ is transforming $\tildeθ_n$ to what we call the {\it jackknife extension} $\tildeθ_{n+1}^{(e)} = \tildeθ_{n+1}^{(e)}(x_1,\ldots,x_n,x_{n+1})$, \[\tildeθ_{n+1}^{(e)} = \{\tildeθ_n (x_1 ,\ldots,x_n)+ \tildeθ_n (x_{n+1},x_2 ,\ldots,x_n) + \ldots + \tildeθ_n (x_1 ,\ldots,x_{n-1},x_{n+1})\}/(n+1).\] Though $\tildeθ_{n+1}^{(e)}$ lacks an innovation the statistician could expect from…
Jackknife Empirical Likelihood Method for U Statistics Based on Multivariate Samples and its Applications
本文提出了一种基于多元三样本U统计量的jackknife经验似然方法，用于构建参数的置信区间，通过蒙特卡洛实验验证其在覆盖概率和计算效率上的优越性。
Estimation of smooth functionals of covariance operators: jackknife bias reduction and bounds in terms of effective rank
本文研究了在可分巴拿赫空间中基于刀切法偏差缩减估计协方差算子光滑泛函的问题，证明了估计误差率取决于有效秩、样本量与Hölder光滑度，并在有效秩受限时实现了经典$\sqrt{n}$速率与渐近正态性。
Variance Inference Beyond the Sandwich for Asymptotically Linear Estimators with Second-Order Remainders
针对半参数估计中二阶余项贡献不可忽略方差的情况，提出有限样本方差分解，并证明留一法Jackknife和配对聚类Bootstrap可估计总方差，通过阶梯楔形聚类随机试验验证改进覆盖。
Cross validation (CV) and the bootstrap are ubiquitous model-agnostic tools for assessing the error or variability of machine learning and statistical estimators. However, these methods require repeatedly re-fitting the model with different weighted versions of the original dataset, which can be prohibitively time-consuming. For sufficiently regular optimization problems the optimum depends smoothly on the data weights, and so the process of repeatedly re-fitting can be approximated with a Taylor series that can be often evaluated relatively quickly. The first-order approximation is known as t…
Let $\bx_j = \btheta +\bep_j, j=1,...,n$, be observations of an unknown parameter $\btheta$ in a Euclidean or separable Hilbert space $\scrH$, where $\bep_j$ are noises as random elements in $\scrH$ from a general distribution. We study the estimation of $f(\btheta)$ for a given functional $f:\scrH\rightarrow \RR$ based on $\bx_j$'s. The key element of our approach is a new method which we call High-Order Degenerate Statistical Expansion. It leverages the use of classical multivariate Taylor expansion and degenerate $U$-statistic and yields an elegant explicit formula. In the univariate case o…
Refined Inference for Asymptotically Linear Estimators with Non-Negligible Second-Order Remainders
针对半参数模型中渐近线性估计量的二阶余项方差不可忽略的情形，提出了留一法刀切法和配对自助法的有效推断方法，并推导了聚类数据下的解析表达式。
An Extension of Greenwood's Formula to Variances
本文针对Greenwood方差估计量提出其渐近方差的估计量，并证明该渐近方差显著小于Kaplan-Meier方差估计量，从而说明Greenwood估计量更稳健。
An admissible estimator of the eigenvalues of the variance-covariance matrix is given for multivariate normal distributions with respect to the scale-invariant squared error loss.
Singh and Kumar (2011) suggested estimators for calculating population variance using auxiliary attributes. This paper proposes a family of estimators based on an adaptation of the estimators presented by Kadilar and Cingi (2004) and Singh et al. (2007), and introduces a new family of estimators using auxiliary attributes. The expressions of the mean square errors (MSEs) of the adapted and proposed families are derived. It is shown that adapted estimators and suggested estimators are more efficient than Singh and Kumar (2011) estimators. The theoretical findings are supported by a numerical ex…
We propose a procedure to handle the problem of Gaussian regression when the variance is unknown. We mix least-squares estimators from various models according to a procedure inspired by that of Leung and Barron (2007). We show that in some cases the resulting estimator is a simple shrinkage estimator. We then apply this procedure in various statistical settings such as linear regression or adaptive estimation in Besov spaces. Our results provide non-asymptotic risk bounds for the Euclidean risk of the estimator.
An Unbiased Variance Estimator with Denominator $N$
本文提出均值调整的无偏方差（AAUV）方法，证明通过构造特定均值估计量可在分母为 $N$ 时获得无偏方差估计量，并揭示其与经典 $N-1$ 公式的关系。
Markov chain Monte Carlo (MCMC) algorithms are used to estimate features of interest of a distribution. The Monte Carlo error in estimation has an asymptotic normal distribution whose multivariate nature has so far been ignored in the MCMC community. We present a class of multivariate spectral variance estimators for the asymptotic covariance matrix in the Markov chain central limit theorem and provide conditions for strong consistency. We examine the finite sample properties of the multivariate spectral variance estimators and its eigenvalues in the context of a vector autoregressive process …
For fixed size sampling designs with high entropy it is well known that the variance of the Horvitz-Thompson estimator can be approximated by the Hájek formula. The interest of this asymptotic variance approximation is that it only involves the first order inclusion probabilities of the statistical units. We extend this variance formula when the variable under study is functional and we prove, under general conditions on the regularity of the individual trajectories and the sampling design, that we can get a uniformly convergent estimator of the variance function of the Horvitz-Thompson estima…
Minimax Simple Bayes Estimators of a Normal Variance
本文提出了一种简洁的贝叶斯估计器，解决了正态方差在熵损失下的最小最大性问题，提供了闭式结果。
New variances for various kappa coefficients based on the unbiased estimator of the expected index of agreements
针对 Martín Andrés 和 Álvarez Hernández（2024）基于期望一致性指数无偏估计量得到的 kappa 系数新估计量，本文将其方差推导由单变量 delta 法扩展为多变量 delta 法并给出新公式。
Control variates for variance-reduced ratio of means estimators
研究比率形式的量，提出将控制变量应用于分子分母并联合优化系数的方差缩减估计器，经仿真研究及多保真度应用验证，该方法能显著缩减方差，具有实用价值。
We consider the problem of empirical Bayes estimation of multiple variances when provided with sample variances. Assuming an arbitrary prior on the variances, we derive different versions of the Bayes estimators using different loss functions. For one particular loss function, the resulting Bayes estimator relies on the marginal cumulative distribution function of the sample variances only. When replacing it with the empirical distribution function, we obtain an empirical Bayes version called F-modeling based empirical Bayes estimator of variances. We provide theoretical properties of this est…
In many epidemiological contexts, disease occurrences and their rates are naturally modelled by counting processes and their intensities, allowing an analysis based on martingale methods. These methods lend themselves to extensions of nested case-control sampling designs where general methods of control selection can be easily incorporated. This same methodology allows for extensions of the Mantel-Haenszel estimator in two main directions. First, a variety of new sampling designs can be incorporated which can yield substantial efficiency gains over simple random sampling. Second, the extension…
Estimating ratio of variance components in a balanced one-way random effect model
本文针对平衡单向随机效应模型方差比率估计问题，在尺度不变损失下推导改进估计量与多种区间估计方法，经模拟和实际数据分析验证了其性能。
We introduce a class of dimension reduction estimators based on an ensemble of the minimum average variance estimates of functions that characterize the central subspace, such as the characteristic functions, the Box--Cox transformations and wavelet basis. The ensemble estimators exhaustively estimate the central subspace without imposing restrictive conditions on the predictors, and have the same convergence rate as the minimum average variance estimates. They are flexible and easy to implement, and allow repeated use of the available sample, which enhances accuracy. They are applicable to bo…
The surface area of a set which is only observed as a binary pixel image is often estimated by a weighted sum of pixel configurations counts. In this paper we examine these estimators in a design based setting -- we assume that the observed set is shifted uniformly randomly. Bounds for the difference between the essential supremum and the essential infimum of such an estimator are derived, which imply that the variance is in $O(t^2)$ as the lattice distance $t$ tends to zero. In particular, it is asymptotically neglectable compared to the bias. A simulation study shows that the theoretically d…
We consider the problem of estimating the error variance in a general linear model when the error distribution is assumed to be spherically symmetric, but not necessary Gaussian. In particular we study the case of a scale mixture of Gaussians including the particularly important case of the multivariate-t distribution. Under Stein's loss, we construct a class of estimators that improve on the usual best unbiased (and best equivariant) estimator. Our class has the interesting double robustness property of being simultaneously generalized Bayes (for the same generalized prior) and minimax over t…
The Shapley effects are global sensitivity indices: they quantify the impact of each input variable on the output variable in a model. In this work, we suggest new estimators of these sensitivity indices. When the input distribution is known, we investigate the already existing estimator and suggest a new one with a lower variance. Then, when the distribution of the inputs is unknown, we extend these estimators. Finally, we provide asymptotic properties of the estimators studied in this article.
This paper proposes a class of ratio type estimators of finite population variance, when the population variance of an auxiliary character is known. Asymptotic expression for mean square error (MSE) is derived and compared with the mean square errors of some existing estimators. An empirical study is carried out to illustrate the performance of the constructed estimator over others.
Calculating a Monte Carlo standard error (MCSE) is an important step in the statistical analysis of the simulation output obtained from a Markov chain Monte Carlo experiment. An MCSE is usually based on an estimate of the variance of the asymptotic normal distribution. We consider spectral and batch means methods for estimating this variance. In particular, we establish conditions which guarantee that these estimators are strongly consistent as the simulation effort increases. In addition, for the batch means and overlapping batch means methods we establish conditions ensuring consistency in t…
Detection of Long Range Dependence in the Time Domain for (In)Finite-Variance Time Series
本文为方差图估计器在长程依赖性检测中提供严格的渐近窗口条件，适用于平方可积及经变换的无限方差时间序列，模拟显示其优于谱域GPH估计器。
The batch means estimator of the MCMC variance is a simple and effective measure of accuracy for MCMC based ergodic averages. Under various regularity conditions, the estimator has been shown to be consistent for the true variance. However, the estimator can be unstable in practice as it depends directly on the raw MCMC output. A measure of accuracy of the batch means estimator itself, ideally in the form of a confidence interval, is therefore desirable. The asymptotic variance of the batch means estimator is known; however, without any knowledge of asymptotic distribution, asymptotic variance…
The asymptotic normality of conditional least squares estimators for the offspring variance in critical branching processes with non-homogeneous immigration is established, under moment assumptions on both reproduction and immigration. The proofs use martingale techniques and weak convergence results in Skorokhod spaces.
We revisit resampling procedures for error estimation in binary classification in terms of U-statistics. In particular, we exploit the fact that the error rate estimator involving all learning-testing splits is a U-statistic. Thus, it has minimal variance among all unbiased estimators and is asymptotically normally distributed. Moreover, there is an unbiased estimator for this minimal variance if the total sample size is at least the double learning set size plus two. In this case, we exhibit such an estimator which is another U-statistic. It enjoys, again, various optimality properties and yi…
The classical Cramér-Rao inequality gives a lower bound for the variance of a unbiased estimator of an unknown parameter, in some statistical model of a random process. In this note we rewrite the statment and proof of the bound using contemporary geometric language.
This note describes non-asymptotic variance and tail bounds for order statistics of samples of independent identically distributed random variables. Those bounds are checked to be asymptotically tight when the sampling distribution belongs to a maximum domain of attraction. If the sampling distribution has non-decreasing hazard rate (this includes the Gaussian distribution), we derive an exponential Efron-Stein inequality for order statistics: an inequality connecting the logarithmic moment generating function of centered order statistics with exponential moments of Efron-Stein (jackknife) est…
In a completely randomized experiment, the variances of treatment effect estimators in the finite population are usually not identifiable and hence not estimable. Although some estimable bounds of the variances have been established in the literature, few of them are derived in the presence of covariates. In this paper, the difference-in-means estimator and the Wald estimator are considered in the completely randomized experiment with perfect compliance and noncompliance, respectively. Sharp bounds for the variances of these two estimators are established when covariates are available. Further…
Analytic evaluation of heteroskedasticity consistent covariance matrix estimates (HCCME) is difficult because of the complexity of the formulae currently available. We obtain new analytic formulae for the bias of a class of estimators of the covariance matrix of OLS in a standard linear regression model. These formulae provide substantial insight into the properties and performance characteristics of these estimators. In particular, we find a new estimator which minimizes the maximum possible bias and improves substantially on the standard Eicker-White estimate.
We consider the derivation of data-dependent simultaneous bandwidths for double kernel heteroskedasticity and autocorrelation consistent (DK-HAC) estimators. In addition to the usual smoothing over lagged autocovariances for classical HAC estimators, the DK-HAC estimator also applies smoothing over the time direction. We obtain the optimal bandwidths that jointly minimize the global asymptotic MSE criterion and discuss the trade-off between bias and variance with respect to smoothing over lagged autocovariances and over time. Unlike the MSE results of Andrews (1991), we establish how nonstatio…
We consider batch size selection for a general class of multivariate batch means variance estimators, which are computationally viable for high-dimensional Markov chain Monte Carlo simulations. We derive the asymptotic mean squared error for this class of estimators. Further, we propose a parametric technique for estimating optimal batch sizes and discuss practical issues regarding the estimating process. Vector auto-regressive, Bayesian logistic regression, and Bayesian dynamic space-time examples illustrate the quality of the estimation procedure where the proposed optimal batch sizes outper…
In this paper, we propose a random projection approach to estimate variance in kernel ridge regression. Our approach leads to a consistent estimator of the true variance, while being computationally more efficient. Our variance estimator is optimal for a large family of kernels, including cubic splines and Gaussian kernels. Simulation analysis is conducted to support our theory.
At the heart of non-parametric spectral estimation, lies the dilemma known as the bias-variance trade-off: low biased estimators tend to have high variance and low variance estimators tend to have high bias. In 1982, Thomson introduced a multitaper method where this trade-off is made explicit by choosing a target bias resolution and obtaining a corresponding variance reduction. The method became the standard in many applications. Its favorable bias-variance trade-off is due to an empirical fact, conjectured by Thomson based on numerical evidence: assuming bandwidth W and N time domain observat…
The two-level normal hierarchical model (NHM) has played a critical role in the theory of small area estimation (SAE), one of the growing areas in statistics with numerous applications in different disciplines. In this paper, we address major well-known shortcomings associated with the empirical best linear unbiased prediction (EBLUP) of a small area mean and its mean squared error (MSE) estimation by considering an appropriate model variance estimator that satisfies multiple properties. The proposed model variance estimator simultaneously (i) improves on the estimation of the related shrinkag…
Equivariant Variance Estimation for Multiple Change-point Model
本文针对多变点模型提出等变方差估计框架，刻画了等变无偏二次方差估计器的集合，并建立了相应的极小极大理论。
Order-Induced Variance in the Moving-Range Sigma Estimator: A Total-Variance Decomposition
本文通过总方差分解揭示了移动范围西格玛估计器中顺序诱导的方差成分，证明了排列均值与吉尼均差的关系，并在正态分布下推导了相邻分量的闭式表达式。
Variance-reduced extreme value index estimators using control variates in a semi-supervised setting
针对极值指数（EVI）估计因依赖少量极端观测值导致方差高的问题，提出半监督框架下基于控制变量的迁移学习方法，通过联合优化系数对Hill估计器的分子和分母应用近似控制变量，实现无偏方差减少，理论和实验验证了其有效性。
Conditional Variance Estimation (CVE) is a novel sufficient dimension reduction (SDR) method for additive error regressions with continuous predictors and link function. It operates under the assumption that the predictors can be replaced by a lower dimensional projection without loss of information. In contrast to the majority of moment based sufficient dimension reduction methods, Conditional Variance Estimation is fully data driven, does not require the restrictive linearity and constant variance conditions, and is not based on inverse regression. CVE is shown to be consistent and its objec…
High-Dimensional Variance Estimation for the Generalized Regression Estimator
研究调查抽样中高维情形下广义回归估计器的方差估计问题，推导常用方差估计器高维渐近偏差的新理论结果，证明在合适假设下交叉验证方差估计器自然渐近无偏。
Inference Optimal Long Run Variance Estimation with Lugsail Kernels
针对具有平稳序列依赖的数据，提出基于非标准固定平滑极限分布的Lugsail估计器最优带宽规则，改善偏差校正并优化推断。
We analyse the properties of an unbiased gradient estimator of the ELBO for variational inference, based on the score function method with leave-one-out control variates. We show that this gradient estimator can be obtained using a new loss, defined as the variance of the log-ratio between the exact posterior and the variational approximation, which we call the $\textit{log-variance loss}$. Under certain conditions, the gradient of the log-variance loss equals the gradient of the (negative) ELBO. We show theoretically that this gradient estimator, which we call $\textit{VarGrad}$ due to its co…
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