Empirical Distributions in Selection Bias Models
选择偏差模型中的经验分布AI 解读 ↓
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AI 根据所提供的论文材料归类：摘要研究一般加权抽样下的分布恢复，未明确涉及微观分布效应或因果识别。
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选择偏差模型中的经验分布
- 研究问题
- 已知各样本抽样规则时，如何合并来自同一未知分布的多个非随机样本，并判断分布估计是否存在且唯一？
- 方法与思路
- 作者对已知权重函数形成的加权样本构造非参数极大似然估计，将计算化为少量非线性方程并用迭代求解。
- 材料陈述的贡献
- 摘要称给出可图形检验的存在唯一性充要条件、有效构造算法及充分统计量性质，并讨论删失与权重不完全已知的扩展。
- 阅读关注点
- 应核对图形条件、样本间覆盖关系、迭代收敛与充分性证明，尤其区分已知权重情形和扩展情形的条件。
- 阅读边界
- 摘要未列出充要条件或扩展细节，也未说明程序当前可用性；OpenAlex 索引版本未注明。
原文摘要
The following problem is treated: Given $s$ not-necessarily-random samples from an unknown distribution $F$, and assuming that we know the sampling rule of each sample, is it possible to combine the samples in order to estimate $F$, and if so what is the natural way of doing it? More formally, this translates to the problem of determining whether there exists a nonparametric maximum likelihood estimate (NPMLE) of $F$ on the basis of $s$ samples from weighted versions of $F$, with known weight functions, and if it exists, how to construct it? We give a simple necessary and sufficient condition, which can be checked graphically, for the existence and uniqueness of the NPMLE and, under this condition, we describe a simple method for constructing it. The method is numerically efficient and mathematically interesting because it reduces the problem to one of solving $s - 1$ nonlinear equations with $s - 1$ unknowns, the unique solution of which is easily obtained by the iterative, Gauss-Seidel type, scheme described in the paper. Extensions for the case where the weight functions are not completely specified and for censored samples, applications, numerical examples, and statistical properties of the NPMLE, are discussed. In particular, we prove under this condition that the NPMLE is a sufficient statistic for $F$. The technique has many potential applications, because it is not limited to the case where the sampled items are univariate. A FORTRAN program for the described algorithm is available from the author.