The shortest confidence interval for Poisson mean

Parameter estimation for fractional Poisson processes

The paper proposes a formal estimation procedure for parameters of the fractional Poisson process (fPp). Such procedures are needed to make the fPp model usable in applied situations. The basic idea of fPp, motivated by experimental data with long memory is to make the standard Poisson model more flexible by permitting non-exponential, heavy-tailed distributions of interarrival times and different scaling properties. We establish the asymptotic normality of our estimators for the two parameters appearing in our fPp model. This fact permits construction of the corresponding confidence intervals…

通过有效先验无关推断模型构建受限参数的置信区间

Constructing confidence intervals for constrained parameters via valid prior-free inferential models

本文提出先验无关推断模型方法，用于构建正态和泊松分布中受限参数的置信区间，解决传统方法在处理未知干扰参数时覆盖概率不准确的问题，通过随机加权技术提升性能，验证了其在高能物理实验数据中的有效性。

Nonparametric inference on Lévy measures of compound Poisson-driven Ornstein-Uhlenbeck processes under macroscopic discrete observations

This study examines a nonparametric inference on a stationary Lévy-driven Ornstein-Uhlenbeck (OU) process $X = (X_{t})_{t \geq 0}$ with a compound Poisson subordinator. We propose a new spectral estimator for the Lévy measure of the Lévy-driven OU process $X$ under macroscopic observations. We also derive, for the estimator, multivariate central limit theorems over a finite number of design points, and high-dimensional central limit theorems in the case wherein the number of design points increases with an increase in the sample size. Built on these asymptotic results, we develop methods to co…

基于Neyman-Pearson引理的随机置信区间方法

A Fuzzy Approach for Randomized Confidence Intervals

本文提出基于Neyman-Pearson引理的随机置信区间方法，用于更广泛地适用于不满足正则条件的分布，并在高方差情况下表现出更好的性能。

On split sample and randomized confidence intervals for binomial proportions

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…

Best estimator for bivariate Poisson regression

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…

Calculation of sample size guaranteeing the required width of the empirical confidence interval with predefined probability

The goal of any estimation study is an interval estimation of a the parameter(s) of interest. These estimations are mostly expressed using empirical confidence intervals that are based on sample point estimates of the corresponding parameter(s). In contrast, calculations of the necessary sample size usually use expected confidence intervals that are based on the expected value of the parameter(s). The approach that guarantees the required probability of the required width of empirical confidence interval is known at least since 1989. However, till now, this approach is not implemented for most…

Confidence interval for correlation estimator between latent processes

Kimura and Yoshida treated a model in which the finite variation part of a two-dimensional semimartingale is expressed by time-integration of latent processes. They proposed a correlation estimator between the latent processes and proved its consistency and asymptotic mixed normality. In this paper, we discuss the confidence interval of the correlation estimator to detect the correlation. %between latent processes. We propose two types of estimators for asymptotic variance of the correlation estimator and prove their consistency in a high frequency setting. Our model includes doubly stochastic…

Exact confidence intervals and hypothesis tests for parameters of discrete distributions

We study exact confidence intervals and two-sided hypothesis tests for univariate parameters of stochastically increasing discrete distributions, such as the binomial and Poisson distributions. It is shown that several popular methods for constructing short intervals lack strict nestedness, meaning that accepting a lower confidence level not always will lead to a shorter confidence interval. These intervals correspond to a class of tests that are shown to assign differing $p$-values to indistinguishable models. Finally, we show that among strictly nested intervals, fiducial intervals, includin…

Change-point model on nonhomogeneous Poisson processes with application in copy number profiling by next-generation DNA sequencing

We propose a flexible change-point model for inhomogeneous Poisson Processes, which arise naturally from next-generation DNA sequencing, and derive score and generalized likelihood statistics for shifts in intensity functions. We construct a modified Bayesian information criterion (mBIC) to guide model selection, and point-wise approximate Bayesian confidence intervals for assessing the confidence in the segmentation. The model is applied to DNA Copy Number profiling with sequencing data and evaluated on simulated spike-in and real data sets.

计数数据的去偏机器学习：基于神经网络的部分线性泊松模型

Debiased Machine Learning for Count Data: a Partially Linear Poisson Model Based on Neural Networks

本文提出一种基于神经网络和Neyman-正交得分的去偏机器学习估计器，用于部分线性泊松模型中的计数数据处理，以减少偏差并实现有效推断。

一种利用泊松-二项分布计算偏似然的精确计算方法

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。

Central limit theorems for smoothed extreme value estimates of Poisson point processes boundaries

In this paper, we give sufficient conditions to establish central limit theorems for boundary estimates of Poisson point processes. The considered estimates are obtained by smoothing some bias corrected extreme values of the point process. We show how the smoothing leads Gaussian asymptotic distributions and therefore pointwise confidence intervals. Some new unidimensional and multidimensional examples are provided.

高维 Poisson 回归中的统计推断及其在中介分析中的应用

Statistical Inference in High-dimensional Poisson Regression with Applications to Mediation Analysis

本文研究高维稀疏 Poisson 回归的统计推断，提出回归向量线性与二次变换的偏差校正估计量、置信区间和检验方法，并用于计数结果的高维中介分析。

Resampling-free bootstrap inference for quantiles

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 …

粒子物理实验中效率估计量的偏差、方差与置信区间

Bias, variance, and confidence intervals for efficiency estimators in particle physics experiments

本文研究粒子物理实验中效率估计的偏差、方差和置信区间，针对泊松试验次数、加权样本及超泊松方差等特殊条件，提出广义Wilson区间并分析其覆盖率。

Exact Confidence Bounds in Discrete Models -- Algorithmic Aspects of Sterne's Method

In this manuscript we review two methods to construct exact confidence bounds for an unknown real parameter in a general class of discrete statistical models. These models include the binomial family, the Poisson family as well as distributions connected to odds ratios in two-by-two tables. In particular, we discuss Sterne's (1954) method in our general framework and present an explicit algorithm for the computation of the resulting confidence bounds. The methods are illustrated with various examples.

Chi-square Intervals for a Poisson Parameter - Bayes, Classical and Structural

The 'standard' confidence interval for a Poisson parameter is only one of a number of estimation intervals based on the chi-square distribution that may be used in the estimation of the mean or mean rate for a Poisson model. Other chi-square intervals are available for experimenters using Bayesian or structural inference methods. Exploring these intervals also leads to other alternate approximate chi-square intervals. Although coverage probability may not always be of interest for Bayesian or structural intervals, coverage probabilities are useful for validating 'objective' priors. Coverage pr…

Efficient nonparametric inference for discretely observed compound Poisson processes

A compound Poisson process whose parameters are all unknown is observed at finitely many equispaced times. Nonparametric estimators of the jump and Lévy distributions are proposed and functional central limit theorems using the uniform norm are proved for both under mild conditions. The limiting Gaussian processes are identified and efficiency of the estimators is established. Kernel estimators for the mass function, the intensity and the drift are also proposed, their asymptotic properties including efficiency are analysed, and joint asymptotic normality is shown. Inference tools such as conf…

Locally correct confidence intervals for a binomial proportion: A new criteria for an interval estimator

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…

学习线性混合随机最短路径的近似极小极大最优遗憾

Nearly Minimax Optimal Regret for Learning Linear Mixture Stochastic Shortest Path

针对线性混合随机最短路径问题，提出基于扩展值迭代与高阶矩递归估计方差感知置信集的新算法，消除成本下界等限制性假设，实现了近似极小极大最优的遗憾界。

泊松对数正态模型的复合似然推断

Composite likelihood inference for the Poisson log-normal model

本文提出了一种结合EM框架与复合似然和重要性采样估计的新型推断方法，用于泊松对数正态模型参数估计，解决了高维积分瓶颈问题，实现了计算可行性，同时保持了最大似然估计的渐近性质。

基于重要性采样的自动驾驶性能评估中速率估计的置信区间

Confidence Intervals for Rate Estimation with Importance Sampling in Autonomous Vehicle Evaluation

本文提出统一的复合泊松模型框架，通过Horvitz-Thompson估计器实现无偏速率估计，并引入新的单调性准则和指数自助法来构建置信区间，提升实际应用的可解释性。

Interval Estimation of Relative Risks for Combined Unilateral and Bilateral Correlated Data

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 …

使用图灵估计器的置信区间：模拟与应用

Confidence Intervals Using Turing's Estimator: Simulations and Applications

研究使用图灵估计器估计罕见结果概率，通过模拟研究相关置信区间有限样本性能，引入选合适CI方法，应用于作者归属问题并推导其对于重要离散分布的理论结果。

Filtered Poisson Process Bandit on a Continuum

We consider a version of the continuum armed bandit where an action induces a filtered realisation of a non-homogeneous Poisson process. Point data in the filtered sample are then revealed to the decision-maker, whose reward is the total number of revealed points. Using knowledge of the function governing the filtering, but without knowledge of the Poisson intensity function, the decision-maker seeks to maximise the expected number of revealed points over T rounds. We propose an upper confidence bound algorithm for this problem utilising data-adaptive discretisation of the action space. This a…

通过置信分布进行两两表的最优推断：建模为泊松对的固定效应和随机效应

Optimal inference via confidence distributions for two-by-two tables modelled as Poisson pairs: fixed and random effects

本文提出基于置信分布的2×2表元分析方法，适用于稀有事件，提供治疗效应和比率的最优推断。

The case for balanced hypothesis tests and equal-tailed confidence intervals

Introduction: there is an ongoing debate about directional inference of two-sided hypothesis tests for which some authors argue that rejecting $θ= θ_0$ does not allow to conclude that $θ> θ_0$ or $θ< θ_0$ but only that $θ\neq θ_0$, while others argue that this is a minor error without practical consequence. Discussion: new elements are brought to the debate. It is shown that the directional interpretation of some non-directional hypothesis tests about Receiver Operating Characteristic (ROC) and survival curves may lead to inflated type III error rates with a probability of concluding that a di…

二项比例的稳健置信区间：局部最优性和适应性

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。

Estimating Traffic Parameters with Rigorous Error Control

To perform a queuing analysis or design in a communications context, we need to estimate the values of the input parameters, specifically the mean of the arrival rate and service time. In this paper, we propose an approach for estimating the arrival rate of Poisson processes and the average service time for servers under the assumption that the service time is exponential. In particular, we derive sample size (i.e., the number of i.i.d. observations) required to obtain an estimate satisfying a pre-specified relative accuracy with a given confidence level. A remarkable feature of this approach …

比例差的最优置信区间

Optimal confidence interval for the difference of proportions

针对小样本下两比例差的最优置信区间构建问题，证明单样本局部优化加反演的方法失效，提出通过全局优化求解，样本量小于15时用Gurobi得到近优解，区间长度较现有最优方法缩短1.5%-5%。

椭圆最优置信区间：Wilson 得分方法的双变量扩展

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。

基于置信度的报童问题优化

Confidence-based Optimization for the Newsvendor Problem

针对单周期单品种随机库存优化中的需求估计问题，提出一种结合置信区间分析与库存优化的策略，以指定置信概率确定包含真实最优订货量的候选范围，并给出成本上下界。

Approximate Negative-Binomial Confidence Intervals: Asbestos Fiber Counts

The negative-binomial distribution is adopted for analyzing asbestos-fiber counts so as to account for both the sampling errors in capturing only a finite number of fibers as well as the inevitable human variation in identifying and counting sampled fibers. A simple approximation to this distribution is developed for the derivation of quantiles and approximate confidence limits. The success of the approximation depends critically on the use of the Stirling expansion to sufficient order, on exact normalization of the approximating distribution, on reasonable perturbation of quantities from the …

Sample Efficient Model Evaluation

Labelling data is a major practical bottleneck in training and testing classifiers. Given a collection of unlabelled data points, we address how to select which subset to label to best estimate test metrics such as accuracy, $F_1$ score or micro/macro $F_1$. We consider two sampling based approaches, namely the well-known Importance Sampling and we introduce a novel application of Poisson Sampling. For both approaches we derive the minimal error sampling distributions and how to approximate and use them to form estimators and confidence intervals. We show that Poisson Sampling outperforms Impo…

Rgbp: An R Package for Gaussian, Poisson, and Binomial Random Effects Models with Frequency Coverage Evaluations

Rgbp is an R package that provides estimates and verifiable confidence intervals for random effects in two-level conjugate hierarchical models for overdispersed Gaussian, Poisson, and Binomial data. Rgbp models aggregate data from k independent groups summarized by observed sufficient statistics for each random effect, such as sample means, possibly with covariates. Rgbp uses approximate Bayesian machinery with unique improper priors for the hyper-parameters, which leads to good repeated sampling coverage properties for random effects. A special feature of Rgbp is an option that generates synt…

可教学的二项分布及相关概率或置信区间的正态近似

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。

Poisson-MNL Bandit: 近似最优的动态联合 assortment 与定价 with 决策依赖的顾客到达

Poisson-MNL Bandit: Nearly Optimal Dynamic Joint Assortment and Pricing with Decision-Dependent Customer Arrivals

Poisson-MNL Bandit通过结合上下文MNL选择模型与泊松到达模型，实现近似最优的动态联合assortment与定价，考虑决策依赖的顾客到达对提升决策效果至关重要。

Estimating the number of classes

Estimating the unknown number of classes in a population has numerous important applications. In a Poisson mixture model, the problem is reduced to estimating the odds that a class is undetected in a sample. The discontinuity of the odds prevents the existence of locally unbiased and informative estimators and restricts confidence intervals to be one-sided. Confidence intervals for the number of classes are also necessarily one-sided. A sequence of lower bounds to the odds is developed and used to define pseudo maximum likelihood estimators for the number of classes.

Practical targeted learning from large data sets by survey sampling

We address the practical construction of asymptotic confidence intervals for smooth (i.e., path-wise differentiable), real-valued statistical parameters by targeted learning from independent and identically distributed data in contexts where sample size is so large that it poses computational challenges. We observe some summary measure of all data and select a sub-sample from the complete data set by Poisson rejective sampling with unequal inclusion probabilities based on the summary measures. Targeted learning is carried out from the easier to handle sub-sample. We derive a central limit theo…

经典与贝叶斯统计方法在低水平计量中的应用

Classical and Bayesian statistical methods for low-level metrology

本文提出一种处理低水平噪声测量的统计方法，通过干扰参数框架和Bolstad准则定义显著性，适用于经典与贝叶斯范式，并首次正式推导出常用置信区间，性能优于ISO 11929标准。

Stable marked point processes

In many contexts such as queuing theory, spatial statistics, geostatistics and meteorology, data are observed at irregular spatial positions. One model of this situation involves considering the observation points as generated by a Poisson process. Under this assumption, we study the limit behavior of the partial sums of the marked point process $\{(t_i,X(t_i))\}$, where X(t) is a stationary random field and the points t_i are generated from an independent Poisson random measure $\mathbb{N}$ on $\mathbb{R}^d$. We define the sample mean and sample variance statistics and determine their joint a…

On the relation between frequentist and Bayesian approaches for the case of Poisson statistics

We propose modified frequentist definition for the determination of confidence intervals for the case of Poisson statistics. Namely, we require that 1-β' \geq \sum_{n=o}^{n_{obs}+k} P(n|λ) \geq α'. We show that this definition is equivalent to the Bayesian method with prior π(λ) \sim λ^{k}. We also propose modified frequentist definition for the case of nonzero background.

随机向量均值的时间一致置信球

Time-Uniform Confidence Spheres for Means of Random Vectors

本文基于 PAC-Bayesian 理论推导随机向量序贯均值估计的时间一致置信球，覆盖次高斯、次-ψ、对数凹及重尾分布，并支持时变均值与非独立同分布鞅设定。