Xinjia Chen
In this paper, we develop an exact method for computing the minimum coverage probability of Wald interval for estimation of binomial parameters. Similar approach can be used for other type of confidence intervals.
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In this paper new test statistics are introduced and studied for the important problem of testing hypothesis that involves inequality constraint on proportions when the sample comes from independent binomial random variables: Wald type and phi-divergence based test-statistics. As a particular case of phi-divergence based test-statistics, the classical likelihood ratio test is considered. An illustrative example is given and the performance of all of them for small and moderate sample sizes is analyzed in an extensive simulation study.
In this paper, we develop an approach for optimizing the explicit binomial confidence interval recently derived by Chen et al. The optimization reduces conservativeness while guaranteeing prescribed coverage probability.
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。
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.
In this paper, we derive an explicit formula for constructing the confidence interval of binomial parameter with guaranteed coverage probability. The formula overcomes the limitation of normal approximation which is asymptotic in nature and thus inevitably introduce unknown errors in applications. Moreover, the formula is very tight in comparison with classic Clopper-Pearson's approach from the perspective of interval width. Based on the rigorous formula, we also obtain approximate formulas with excellent performance of coverage probability.
Penalization of the likelihood by Jeffreys' invariant prior, or by a positive power thereof, is shown to produce finite-valued maximum penalized likelihood estimates in a broad class of binomial generalized linear models. The class of models includes logistic regression, where the Jeffreys-prior penalty is known additionally to reduce the asymptotic bias of the maximum likelihood estimator; and also models with other commonly used link functions such as probit and log-log. Shrinkage towards equiprobability across observations, relative to the maximum likelihood estimator, is established theore…
Introduction: estimation of confidence intervals (CIs) of binomial proportions has been reviewed more than once but the directional interpretation, distinguishing the overestimation from the underestimation, was neglected while the sample size and theoretical proportion variances from experiment to experiment have not been formally taken in account. Herein, we define and apply new evaluation criteria, then give recommendations for the practical use of these CIs. Materials & methods: Google Scholar was used for bibliographic research. Evaluation criteria were (i) one-sided conditional errors, (…
Count data and recurrent events in clinical trials, such as the number of lesions in magnetic resonance imaging in multiple sclerosis, the number of relapses in multiple sclerosis, the number of hospitalizations in heart failure, and the number of exacerbations in asthma or in chronic obstructive pulmonary disease (COPD) are often modeled by negative binomial distributions. In this manuscript we study planning and analyzing clinical trials with group sequential designs for negative binomial outcomes. We propose a group sequential testing procedure for negative binomial outcomes based on Wald s…
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…
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…
Well-recommended methods of forming `confidence intervals' for a binomial proportion give interval estimates that do not actually meet the definition of a confidence interval, in that their coverages are sometimes lower than the nominal confidence level. The methods are favoured because their intervals have a shorter average length than the Clopper-Pearson (gold-standard) method, whose intervals really are confidence intervals. Comparison of such methods is tricky -- the best method should perhaps be the one that gives the shortest intervals (on average), but when is the coverage of a method s…
It is a common contention that it is an ``impossible mission'' to exactly determine the minimum sample size for the estimation of a binomial parameter with prescribed margin of error and confidence level. In this paper, we investigate such a very old but also extremely important problem and demonstrate that the difficulty for obtaining the exact solution is not insurmountable. Unlike the classical approximate sample size method based on the central limit theorem, we develop a new approach for computing the minimum sample size that does not require any approximation. Moreover, our approach over…
We consider interval estimation of the difference between two binomial proportions. Several methods of constructing such an interval are known. Unfortunately those confidence intervals have poor coverage probability: it is significantly smaller than the nominal confidence level. In this paper a new confidence interval is proposed. The construction needs only information on sample sizes and sample difference between proportions. The coverage probability of the proposed confidence interval is at least the nominal confidence level. The new confidence interval is illustrated by a medical example.
A Comprehensive Comparison of the Wald, Wilson, and adjusted Wilson Confidence Intervals for Proportions
本文通过彩虹色码和像素图全面比较 Wald、Wilson 及调整后 Wilson 置信区间，发现添加特定伪观测数（如 3、4、6）的调整后 Wilson 区间在多种样本量和比例下优于其他方法。
Split sample methods have recently been put forward as a way to reduce the coverage oscillations that haunt confidence intervals for parameters of lattice distributions, such as the binomial and Poisson distributions. We study split sample intervals in the binomial setting, showing that these intervals can be viewed as being based on adding discrete random noise to the data. It is shown that they can be improved upon by using noise with a continuous distribution instead, regardless of whether the randomization is determined by the data or an external source of randomness. We compare split samp…
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…
Two-sample inference for the difference of population means typically relies upon a Central Limit Theorem approximation. When data are drawn from a Negative Binomial distribution, previous work of Shilane et al. (2010) showed that a Normal approximation is often unreliable in one-sample inference and proposed alternative techniques. We seek to extend these methods to the problem of two-sample inference on the difference of sample means in highly dispersed Negative Binomial models. We demonstrate that the Normal approximation is considerably more robust in the two-sample setting and may often b…
There are over 55 different ways to construct a confidence respectively credible interval (CI) for the binomial proportion. Methods to compare them are necessary to decide which should be used in practice. The interval score has been suggested to compare prediction intervals. This score is a proper scoring rule that combines the coverage as a measure of calibration and the width as a measure of sharpness. We evaluate eleven CIs for the binomial proportion based on the expected interval score and propose a summary measure which can take into account different weighting of the underlying true pr…
Robust Confidence Intervals for a Binomial Proportion: Local Optimality and Adaptivity
本文研究了在未知污染比例下二项比例的稳健置信区间构造，通过局部最优性和适应性分析，提出了一种保证有效覆盖和高效计算的置信区间方法，并探讨了其在泊松数据和Erdős--Rényi模型中的应用。
INTRODUCTION: Wald's, the likelihood ratio (LR) and Rao's score tests and their corresponding confidence intervals (CIs), are the three most common estimators of parameters of Generalized Linear Models. On finite samples, these estimators are biased. The objective of this work is to analyze the coverage errors of the CI estimators in small samples for the log-Poisson model (i.e. estimation of incidence rate ratio) with innovative evaluation criteria, taking in account the overestimation/underestimation unbalance of coverage errors and the variable inclusion rate and follow-up in epidemiologica…
Inference for Median and a Generalization of HulC
本文提出HulC推广，对渐近正态估计量比Wald区间窄，且对非正态极限分布问题也有效。通过分析中位数无分布置信区间宽度特性，发现适当缩放后区间宽度收敛到非退化随机变量。
Penalized likelihood inference for beta-binomial meta-analysis of proportions of rare events
针对罕见事件比例元分析中常用最大似然估计器存在有限样本偏差的问题，提出最大惩罚似然估计器，通过最大化Jeffreys先验惩罚增强的对数似然得到，模拟显示其在多方面优于普通MLE，还引入新置信区间，为相关元分析提供有用替代方法。
Binomial confidence intervals for rare events: importance of defining margin of error relative to magnitude of proportion
本文针对稀有事件比例估计，提出以相对误差边际衡量置信区间精度，并评估四种常用区间方法在覆盖概率与相对误差边际下的表现，给出实践中可采用的相对误差边际取值范围。
We introduce a new framework for creating point-wise confidence intervals for the distribution of event times for current status data. Existing methods are based on asymptotics. Our framework is based on binomial properties and motivates confidence intervals that are very simple to apply and are valid, i.e., guarantee nominal coverage. Although these confidence intervals are necessarily conservative for small sample sizes, asymptotically their coverage rate approaches the nominal one. This binomial framework also motivates approximately valid confidence intervals, and simulations show that the…
Conformal Tradeoffs: Operational Profiles Beyond Coverage
本文提出SSBC方法，通过校准与审计机制，在有限样本下实现精确覆盖语义，并通过诱导划分揭示制度边界和权衡问题。
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…
Prediction Sets Adaptive to Unknown Covariate Shift
针对未知协变量偏移下预测集构建的难题，提出无分布方法PredSet-1Step，可高效构建具备渐近覆盖率保证的预测集，经理论与实验验证有效。
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 Negative Binomial distribution becomes highly skewed under extreme dispersion. Even at moderately large sample sizes, the sample mean exhibits a heavy right tail. The standard Normal approximation often does not provide adequate inferences about the data's mean in this setting. In previous work, we have examined alternative methods of generating confidence intervals for the expected value. These methods were based upon Gamma and Chi Square approximations or tail probability bounds such as Bernstein's Inequality. We now propose growth estimators of the Negative Binomial mean. Under high dis…
We develop a general framework for conducting inference on the mean of dependent random variables given constraints on their dependency graph. We establish the consistency of an oracle variance estimator of the mean when the dependency graph is known, along with an associated central limit theorem. We derive an integer linear program for finding an upper bound for the estimated variance when the graph is unknown, but topological and degree-based constraints are available. We develop alternative bounds, including a closed-form bound, under an additional homoskedasticity assumption. We establish…
Joint Coverage Regions: Simultaneous Confidence and Prediction Sets
本文提出联合覆盖区域（JCR），统一频率学派中的置信区间与预测区域，并给出有限样本有效构造方法及高效算法，适用于结构化参数空间下的预测集构造。
Maximum likelihood abundance estimation from capture-recapture data when covariates are missing at random
针对捕获-再捕获实验中协变量随机缺失问题，本文提出最大经验似然丰度估计方法，证明其渐近正态性与卡方极限分布，并通过模拟和香港黄腹山鹪莺数据验证其均方误差更小、置信区间覆盖率更准确。
Receiver operating characteristic (ROC) curves are widely used as a measure of accuracy of diagnostic tests and can be summarized using the area under the ROC curve (AUC). Often, it is useful to construct a confidence intervals for the AUC, however, since there are a number of different proposed methods to measure variance of the AUC, there are thus many different resulting methods for constructing these intervals. In this manuscript, we compare different methods of constructing Wald-type confidence interval in the presence of missing data where the missingness mechanism is ignorable. We find …
The bootstrap, based on resampling, has, for several decades, been a widely used method for computing confidence intervals for applications where no exact method is available and when sample sizes are not large enough to be able to rely on easy-to-compute large-sample approximate methods, such a Wald (normal-approximation) confidence intervals. Simulation based bootstrap intervals have been proven useful in that their actual coverage probabilities are close to the nominal confidence level in small samples. Small samples analytical approximations such as the Wald method, however, tend to have c…
An Accurate Computational Approach for Partial Likelihood Using Poisson-Binomial Distributions
针对Cox模型偏似然近似计算不精确的问题，提出基于泊松-二项分布精确计算条件概率的方法，统一处理有结与无结数据，减少偏差和均方误差，提高置信区间覆盖率。
On the Assessment of Bootstrap Intervals for Samples of Fixed Size
本文通过精确概率计算评估固定样本量下 bootstrap 区间的覆盖概率与期望长度，发现其置信系数不正确并导致不合理结论，进而探讨实践中如何适当使用 bootstrap 区间。
Our interest is whether two binomial parameters differ, which parameter is larger, and by how much. This apparently simple problem was addressed by Fisher in the 1930's, and has been the subject of many review papers since then. Yet there continues to be new work on this issue and no consensus solution. Previous reviews have focused primarily on testing and the properties of validity and power, or primarily on confidence intervals, their coverage, and expected length. Here we evaluate both. For example, we consider whether a p-value and its matching confidence interval are compatible, meaning …
Income inequality measures are often used as an indication of economic health. How to obtain reliable confidence intervals for these measures based on sampled data has been studied extensively in recent years. To preserve confidentiality, income data is often made available in summary form only (i.e. histograms, frequencies between quintiles, etc.). In this paper, we show that good coverage can be achieved for bootstrap and Wald-type intervals for quantile-based measures when only grouped (binned) data are available. These coverages are typically superior to those that we have been able to ach…
Likelihood confidence intervals for misspecified Cox models
本文针对误设定 Cox 模型提出稳健似然置信区间，通过证明似然比检验统计量收敛于加权卡方分布并反转稳健似然比检验来构造，解决了小事件数下 Wald 置信区间表现不佳的问题。
Quasi-Monte Carlo confidence intervals using quantiles of randomized nets
本文提出利用随机化网的分位数构造高维积分的渐近有效置信区间方法。
For any class of one-sided $1-α$ confidence intervals with a certain monotonicity ordering on the random confidence limit, the smallest interval, in the sense of the set inclusion for the difference of two proportions of two independent binomial random variables, is constructed based on a direct analysis of coverage probability function. A special ordering on the confidence limit is developed and the corresponding smallest confidence interval is derived. This interval is then applied to identify the minimum effective dose (MED) for binary data in dose-response studies, and a multiple test proc…
This paper concerns the construction of confidence intervals in standard seroprevalence surveys. In particular, we discuss methods for constructing confidence intervals for the proportion of individuals in a population infected with a disease using a sample of antibody test results and measurements of the test's false positive and false negative rates. We begin by documenting erratic behavior in the coverage probabilities of standard Wald and percentile bootstrap intervals when applied to this problem. We then consider two alternative sets of intervals constructed with test inversion. The firs…
Exact Gradient Evaluation for Adaptive Quadrature Approximate Marginal Likelihood in Mixed Models for Grouped Data
本文提出了一种在含单一分组因子的混合模型中，通过自适应高斯求积法计算近似对数边际似然精确梯度的算法，利用拟牛顿优化提升了计算效率，并在伯努利混合模型中通过逆对数Cholesky参数化实现了无需矩阵分解的高效推断。
New Confidence Regions for Linear Regression Parameters with Stationary-Ergodic Dependent Errors
本文提出了一种在回归变量和误差 jointly stationary and ergodic 且未指定序列依赖的情况下，利用随机平滑和辅助样本进行回归系数联合置信区域估计的方法，无需直接估计长期方差或参数依赖模型，且在实际应用中表现出良好的覆盖性能和区域体积。
Parametric modal regression for right-censored positive responses
本文提出了一种参数模态回归框架，用于处理右截断正响应，通过重新参数化将模式与线性预测变量直接关联，并通过实证数据验证了方法的有效性。
Profile Likelihood Inference for Anisotropic Hyperbolic Wrapped Normal Models on Hyperbolic Space
本文研究了超曲面上各向异性双曲包裹正态分布的基于似然的推断方法，提出通过优化似然函数获得参数估计，并证明了估计的强一致性及渐近正态性。
Score-based confidence intervals for variance-covariance parameters in linear mixed models
针对线性混合模型中方差-协方差参数，提出基于修正剖面得分统计量的置信区间，在参数接近边界时仍保持接近名义覆盖率，且比通用推断区间窄 1.7 至 3.1 倍。
This article describes the R package htmcglm implemented for performing hypothesis tests on regression and dispersion parameters of multivariate covariance generalized linear models (McGLMs). McGLMs provide a general statistical modeling framework for normal and non-normal multivariate data analysis along with a wide range of correlation structures. The proposed package considers the Wald statistics to perform general hypothesis tests and build tailored ANOVAs, MANOVAs and multiple comparison tests. The goal of the package is to provide tools to improve the interpretation of regression and dis…
Robust beta regression through the logit transformation
本文提出新的稳健估计方法，用于解决beta回归模型中最大似然估计的鲁棒性不足问题，通过渐近性质和稳健检验方法提升模型可靠性。
Confidence Intervals for the F1 Score: A Comparison of Four Methods
本文针对F1分数置信区间估计问题，提出Wilson直接法和Wilson间接法两种新方法，并通过理论分析与实验证明其在覆盖概率和区间长度上优于现有Clopper-Pearson和Wald方法，且不会出现越界和退化问题。
This paper reviews two main types of prediction interval methods under a parametric framework. First, we describe methods based on an (approximate) pivotal quantity. Examples include the plug-in, pivotal, and calibration methods. Then we describe methods based on a predictive distribution (sometimes derived based on the likelihood). Examples include Bayesian, fiducial, and direct-bootstrap methods. Several examples involving continuous distributions along with simulation studies to evaluate coverage probability properties are provided. We provide specific connections among different prediction…
Comparing Two Survival Functions at a Fixed Timepoint
本文针对右删失下固定时间点的两条生存函数比较问题，将单样本Greenwood方差调整纳入delta法两样本应用，扩展BPCP融合区间为中p版本，经模拟发现BPCP融合可保证全场景覆盖率，中p版本覆盖率最接近名义，还提供了bpcp R包函数。
Sequential Conformal Calibration under Recursive Selection
本文针对递归选择场景提出序列共形校准方法，推导了覆盖过程规律，刻画了新鲜度与数据复用的权衡，将其应用于多级库存问题，模拟显示其覆盖达标且成本低于Bonferroni基线。
Introductory texts on statistics typically only cover the classical "two sigma" confidence interval for the mean value and do not describe methods to obtain confidence intervals for other estimators. The present technical report fills this gap by first defining different methods for the construction of confidence intervals, and then by their application to a binomial proportion, the mean value, and to arbitrary estimators. Beside the frequentist approach, the likelihood ratio and the highest posterior density approach are explained. Two methods to estimate the variance of general maximum likel…
Profile likelihoods for parameters in Gaussian geostatistical models
针对高斯地质统计学模型中方差矩阵分解计算负担大导致剖面似然难以应用的问题，提出利用Matern相关函数族、重新参数化与GPU并行计算的剖面似然方法，证明其参数置信区间覆盖率优于传统Wald型区间。
For the quantification of QoE, subjects often provide individual rating scores on certain rating scales which are then aggregated into Mean Opinion Scores (MOS). From the observed sample data, the expected value is to be estimated. While the sample average only provides a point estimator, confidence intervals (CI) are an interval estimate which contains the desired expected value with a given confidence level. In subjective studies, the number of subjects performing the test is typically small, especially in lab environments. The used rating scales are bounded and often discrete like the 5-poi…
Finite-Sample Coverage Audits for High-Recall Candidate Generation: Certification and Learning-Theoretic Design
研究高召回率候选生成中认证错过相关质量所需的审核标签数，刻画其标签复杂度，证明排除池审核的最优性，开发有限样本工具包，可认证错过质量、转换为召回率等，还能与审查负担配对选合适候选生成器。
We propose a frequentist testing procedure that maintains a defined coverage and is optimal in the sense that it gives maximal power to detect deviations from a null hypothesis when the alternative to the null hypothesis is sampled from a pre-specified distribution (the prior distribution). Selecting a prior distribution allows to tune the decision rule. This leads to an increased power, if the true data generating distribution happens to be compatible with the prior. It comes at the cost of losing power, if the data generating distribution or the observed data are incompatible with the prior.…
Confidence Intervals for Evaluation of Data Mining
本文针对数据挖掘中二元预测性能度量的统计不确定性，提出基于渐近正态近似的个体与联合置信区间，并通过“模糊校正”改进有限样本表现，支持多规则多度量同时比较。
Sequential Probability Ratio Test using Z-Statistics (SPRT-z): A Practical Approach for Online Experimentation
针对在线实验中的重复窥视问题，提出SPRT-z方法，结合大样本正态近似、无标度视界校准和布朗中位无偏估计，实现早期有效性和无效性停止，控制错误率并减少样本量。
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