Let $\hat f_n$ be the nonparametric maximum likelihood estimator of a decreasing density. Grenander characterized this as the left-continuous slope of the least concave majorant of the empirical distribution function. For a sample from the uniform distribution, the asymptotic distribution of the $L_2$-distance of the Grenander estimator to the uniform density was derived in Groeneboom and Pyke (1983) by using a representation of the Grenander estimator in terms of conditioned Poisson and gamma random variables. This representation was also used in Groeneboom and Lopuhaa (1993) to prove a central limit result of Sparre Andersen on the number of jumps of the Grenander estimator. Here we extend this to the proof of a general result on integrals of the Grenander estimator. We also correct Groeneboom and Pyke (1983), where the limit distribution of the sums of gamma and Poisson variables on which the conditioning was done did not have the right form. Saddle point methods and Cauchy's formula are important tools in our development.

中文翻译：

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格林纳德泛函和柯西公式

令 $\hat f_n$ 是密度递减的非参数最大似然估计量。格瑞南德将此描述为经验分布函数的最小凹主项的左连续斜率。对于来自均匀分布的样本，Grenander 估计量到均匀密度的 $L_2$-距离的渐近分布是在 Groeneboom 和 Pyke (1983) 中通过使用 Grenander 估计量在条件泊松和伽玛方面的表示得出的随机变量。Groeneboom 和 Lopuhaa (1993) 也使用这种表示来证明 Sparre Andersen 对 Grenander 估计量的跳跃次数的中心极限结果。在这里，我们将其扩展到格瑞南德估计量积分的一般结果的证明。我们还纠正了 Groeneboom 和 Pyke (1983)，其中进行条件处理的 gamma 和 Poisson 变量总和的极限分布没有正确的形式。鞍点法和柯西公式是我们开发中的重要工具。