Quantile estimation for discrete distributions has not been well studied, although discrete data are common in practice. Under the assumption that data are drawn from a discrete distribution, we examine the consistency of the maximum empirical likelihood estimator (MELE) of the pth population quantile µp, with the assistance of a jittering method and results for continuous distributions. The MELE and the sample quantile estimator are closely related, and they may or may not be consistent for µp, depending on whether or not the underlying distribution has a plateau at the level of p. We pro- pose an empirical likelihood-based categorization procedure which not only helps in determining the shape of the true distribution at level p, but also provides a way of formulating a new estimator that is consistent in any case. Analogous to confidence intervals in the continuous case, the probability of a correct estimate (PCE) accompanies the point estimator. Simulation results show that PCE can be estimated using the simple bootstrap method.
Quantile Estimation for Discrete Data via Empirical Likelihood