雪球抽样厄尔多斯-雷尼网络的精确似然推断
Exact Likelihood Inference for Snowball-Sampled Erdős-Rényi Networks
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中文总结 AI 辅助
针对雪球抽样厄尔多斯-雷尼网络的偏差问题，本文推导了其精确似然，得到了校正抽样设计的无偏最大似然估计量及有效置信区间，模拟验证了方法的有效性。
中文摘要 AI 辅助
通过链接追踪设计（如雪球抽样）获取的网络数据，其收集机制依赖于分析所要估计的网络结构本身。若忽略这种依赖性，将观测到的样本当作完整网络处理，会导致推断结果出现严重偏差。虽然这类选择问题在一般情况下难以处理，但本文证明，对于从厄尔多斯-雷尼总体中抽取的、采用全邻域招募的r波雪球抽样，该问题存在精确解。我们推导了此类样本的精确似然，证明其在边概率π下定义了一个具有低维充分统计量的弯曲指数族。基于此结果，我们得到了π的最大似然估计量，该估计量能正确考虑抽样设计，且作为最小充分统计量的函数可充分利用样本中的信息。模拟研究显示，相较于朴素估计量，这种校正方法大幅降低了偏差，即使样本仅覆盖网络的0.1%，仍基本保持无偏。我们还通过基于精确抽样分布构建检验并反转该检验，构造了π的有效置信区间，其中精确抽样分布通过蒙特卡洛模拟近似。模拟研究证实，在一系列边概率和波数范围内，这些置信区间在蒙特卡洛误差内达到了标称覆盖率。
英文摘要
Network data obtained through link-tracing designs, such as snowball sampling, are collected through a mechanism that depends on the very structure the analysis seeks to estimate. Ignoring this dependence and treating the observed sample as though it were itself a complete network can lead to substantially biased inference. While the resulting selection problem is intractable in general, we show that it admits an exact solution for $r$-wave snowball samples, with full-neighbourhood recruitment, drawn from an Erdős--Rényi population. We derive the exact likelihood of such a sample and show that it defines a curved exponential family in the edge probability $π$, with a low-dimensional sufficient statistic. Building on this result, we obtain the maximum likelihood estimator of $π$ that correctly accounts for the sampling design and, as a function of the minimal sufficient statistic, makes full use of the information in the sample. Simulation studies show that this correction substantially reduces bias relative to the naive estimator, remaining effectively unbiased even when the sample covers as little as 0.1\% of the network. We further construct valid confidence intervals for $π$ by inverting a test built on the exact sampling distribution, approximated via Monte Carlo simulation. Simulation studies confirm that these confidence intervals attain the nominal coverage level within Monte Carlo error across a range of edge probabilities and numbers of waves.
