不变测度作为估计量：二阶随机展开与偏差缩减
Invariant Measures as Estimators: Second-Order Stochastic Expansions and Bias Reduction
浏览论文内容
中文总结 AI 辅助
本文提出一个基于不变测度的二阶渐近框架，实现无需重采样的通用偏差校正，在似然设定下对光滑泛函达到偏差$o(a_n^2)$，并通过伽马模型数值验证。
中文摘要 AI 辅助
我们为基于数据依赖马尔可夫过程的不变概率测度所构造的估计量，发展了一个一般的二阶渐近框架。该框架为光滑泛函提供了一种通用的偏差校正，该校正由用于构造估计量的同一不变测度计算得出，无需像自助法和刀切法这类偏差校正方法那样进行重采样和重复估计。统计输入由一个随机一次形式表示，使得对数似然的微分、一般随机准则以及估计函数能够在统一框架下处理。在局部正则性和局部化条件下，我们推导了关于这些不变测度的平均值以及通过一般损失函数定义的决策规则的二阶随机展开。这些展开利用预对比几何和由损失诱导的O-导数，分离了统计输入、温度、漂移和损失的贡献。该框架包含最大似然估计量和基于后验的估计量，并恢复了基于先验选择或估计方程调整的现有偏差缩减方法。在似然设定下，对于Jeffreys后验$\mu_n$及其Fisher--Rao Fréchet均值$\widehat z_n$，校正估计量$2\gamma(\widehat z_n)-\mu_n(\gamma)$对每个固定光滑泛函$\gamma$具有频率偏差$o(a_n^2)$，其中$a_n$表示估计速率。对于坐标平方损失，该理论产生一个仅由似然和Fisher信息决定的扩散过程。其不变概率测度的均值以偏差$o(a_n^2)$估计参数。即使不存在偏差缩减先验时，该构造仍然可用。伽马形状-尺度模型中的数值结果展示了这种偏差缩减。
英文摘要
We develop a general second-order asymptotic framework for estimators constructed from invariant probability measures of data-dependent Markov processes. The framework yields a universal bias correction for smooth functionals, computed from the same invariant measure used to construct the estimator, without the resampling and repeated estimation involved in bias-correction methods such as the bootstrap and jackknife. The statistical input is represented by a random one-form, allowing differentials of log-likelihoods and general random criteria, as well as estimating functions, to be treated in a common framework. Under local regularity and localization conditions, we derive second-order stochastic expansions for averages with respect to these invariant measures and for decision rules defined through general loss functions. The expansions separate the contributions of the statistical input, temperature, drift, and loss, using precontrast geometry and an O-derivative induced by the loss. The framework includes maximum likelihood and posterior-based estimators and recovers existing bias-reduction methods based on the choice of prior or adjustments to estimating equations. In the likelihood setting, for the Jeffreys posterior $μ_n$ and its Fisher--Rao Fréchet mean $\widehat z_n$, the corrected estimator $2γ(\widehat z_n)-μ_n(γ)$ has frequentist bias $o(a_n^2)$ for every fixed smooth functional $γ$, where $a_n$ denotes the estimation rate. For coordinate squared loss, the theory yields a diffusion determined only by the likelihood and Fisher information. The mean of its invariant probability measure estimates the parameter with bias $o(a_n^2)$. This construction remains available even when no bias-reducing prior exists. Numerical results in the gamma shape--scale model illustrate this bias reduction.
