We propose an unbiased estimator for $$P\left( X>Y\right) $$PX>Y and obtain an exact expression for its variance, based on judgement post stratification (JPS) sampling scheme. We then prove that the introduced estimator is consistent and establish its asymptotic normality. We show that the proposed estimator is at least as efficient asymptotically as its counterpart in simple random sampling (SRS), regardless of the quality of the rankings. For finite sample sizes, a Monte Carlo simulation study and a real data set are employed to show the preference of the JPS estimator to its SRS competitor in a wide range of settings.

中文翻译：

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基于判断后分层的$$P\left( X > Y \right) $$ PX > Y 估计

我们为 $$P\left(X>Y\right)$$PX>Y 提出了一个无偏估计，并基于判断后分层 (JPS) 抽样方案获得其方差的精确表达式。然后我们证明引入的估计量是一致的并建立其渐近正态性。我们表明，无论排名的质量如何，所提出的估计量至少与简单随机抽样 (SRS) 中的对应估计量一样有效。对于有限样本量，采用蒙特卡罗模拟研究和真实数据集来显示 JPS 估计器在各种设置中对其 SRS 竞争对手的偏好。