测量误差下的分布函数、其跳跃及区间概率的估计
Estimation of distribution functions, their jumps and interval probabilities under measurement error
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中文总结 AI 辅助
该研究针对加性测量误差模型，开发了无需$F_Y$满足密度等强假设的分布函数、区间概率及跳跃大小的直接估计量，通过模拟验证了其有限样本性能。
中文摘要 AI 辅助
我们考虑经典的加性测量误差模型$X=Y+Z$，其中潜在随机变量$Y$具有未知分布$F_Y$，误差$Z$具有已知分布。我们针对$F_Y$的三个泛函开发了直接估计量：(i) 连续点处的$F_Y(x)$；(ii) 当$x<y$为连续点时的区间概率$F_Y(y)-F_Y(x)$；(iii) 预先指定的不连续点处的跳跃大小。我们推导了非渐近的偏差和方差界，并建立了渐近无偏性和一致性。与以往工作不同，我们不要求$F_Y$具有密度、混合表示或满足全局Sobolev光滑性假设。该框架适用于任意潜在分布，包括同时具有离散和连续分量的分布，以及具有多个跳跃的分布。这些结果依赖于傅里叶逆定理与Mynbaev、Martins-Filho和Henderson（2022）提出的一类估计量的代数结构之间的联系。一项模拟研究评估了可行的调优程序，并在可用时将所提出估计量的有限样本性能与现有方法进行了比较。
英文摘要
We consider the classical additive measurement-error model $X=Y+Z$, where the latent random variable $Y$ has unknown distribution $F_Y$ and the error $Z$ has a known distribution. We develop direct estimators for three functionals of $F_Y$: (i) $F_Y(x)$ at continuity points; (ii) interval probabilities $F_Y(y)-F_Y(x)$ when $x<y$ are continuity points; and (iii) the size of a jump at a prespecified discontinuity. We derive non-asymptotic bias and variance bounds, and establish asymptotic unbiasedness and consistency. Unlike previous work, we do not require $F_Y$ to admit a density, have a mixture representation, or satisfy global Sobolev smoothness assumptions. The framework accommodates arbitrary latent distributions, including those with both discrete and continuous components, and distributions with multiple jumps. These results rely on a link between Fourier inversion theorems and the algebraic structure of a class of estimators proposed in Mynbaev, Martins-Filho and Henderson (2022). A simulation study evaluates feasible tuning procedures and, where available, compares the finite-sample performance of the proposed estimators with existing methods.
