  In this work, we discuss a general class of the 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 goes to 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 SRS design. It is found that the JPS estimator
dramatically improves the efficiency of the KDF as compared to its SRS
competitor for a wide range of the settings. Finally, we apply the described
procedure on a real dataset from medical context to show their usefulness and
applicability in practice.
