Under left truncation, data $(X_i,Y_i)$ are observed only when $Y_i\le X_i$.Usually, the distribution function $F$ of the $X_i$ is the target of interest.In this paper, we study linear functionals $\int\varphi \mathrm{d}F_n$ of thenonparametric maximum likelihood estimator (MLE) of $F$, the Lynden-Bellestimator $F_n$. A useful representation of $\int \varphi \mathrm{d}F_n$ isderived which yields asymptotic normality under optimal moment conditions onthe score function $\varphi$. No continuity assumption on $F$ is required. As aby-product, we obtain the distributional convergence of the Lynden-Bellempirical process on the whole real line.
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机译：在左截断下，仅当$ Y_i \ le X_i $时才观察到数据$（X_i，Y_i）$。通常，$ X_i $的分布函数$ F $是关注的目标。本文研究线性函数非参数最大似然估计器（MLE）的$ \ int \ varphi \ mathrm {d} F_n $，即Lynden-Bellestimator $ F_n $。推导了$ \ int \ varphi \ mathrm {d} F_n $的有用表示形式，该函数在最佳矩条件下在得分函数$ \ varphi $上产生渐近正态性。不需要$ F $的连续性假设。作为副产品，我们在整个实线上获得了Lynden-Bellempirical过程的分布收敛性。
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