Statistical inference on the cumulative distribution function using judgment post stratification

In this work, we discuss a general class of estimators for the cumulative distribution function (CDF) based on judgment post stratification (JPS) sampling scheme, which includes both empirical and kernel distribution functions. Specifically, we obtain the expectation of the estimators in this class and show that they are asymptotically more efficient than their competitors in simple random sampling (SRS), as long as the rankings are better than random guessing. We find a mild condition that is necessary and sufficient for them to be asymptotically unbiased. We also prove that given the same condition, the estimators in this class are strongly uniformly consistent estimators of the true CDF, and converge in distribution to a normal distribution when the sample size approaches infinity. We then focus on the kernel distribution function (KDF) in the JPS design and obtain the optimal bandwidth. We next carry out a comprehensive Monte Carlo simulation to compare the performance of the KDF in the JPS design for different choices of sample size, set size, ranking quality, parent distribution, kernel function, as well as both perfect and imperfect rankings set-ups, with its counterpart in the SRS design. We find that the JPS estimator dramatically improves the efficiency of the KDF compared to its SRS competitor across a wide range of the settings. Finally, we apply the described procedure to a real dataset from a medical context to show its usefulness and applicability in practice.