Inference on a Distribution Function from Ranked Set Samples

Consider independent observations (X1, R1), (X2, R2),..., (Xn, Rn) with random or fixed ranks Ri ∈ {1, 2,..., k}, while conditional on Ri = r, the random variable Xi has the same distribution as the r-th order statistic within a random sample of size k from an unknown continuous distribution function F. Such observation schemes are utilized in situ-ations in which ranking observations is much easier than obtaining their precise values. Two wellknown special cases are ranked set sampling (McIntyre 1952) with k = n and Ri = i, and judgement post-stratification (MacEachern et al. 2004) with Ri ∼ Unif({1, 2,..., k}). Within a rather general setting we analyze and compare the asymptotic distribution of three different estimators of the distribution function F. The asymptotics are for fixed number k and n → ∞. The three estimators under consideration are the stratified esti-mator of Stokes and Sager (1988), a new moment-based estimator, and the nonparametric maximum-likelihood estimator of Kvam and Samaniego (1994). Our results on the latter estimator generalize and refine the analysis of Huang (1997). It turns out that it is asymp-totically more efficient than the former two estimators. The efficiency gain over the new estimator is typically rather small, whereas the inefficiency of the stratified estimator may be quite severe in imbalanced settings. Finally we describe briefly pointwise and simultaneous confidence intervals for the dis-tribution function F with guaranteed coverage probability for finite sample sizes. This may be accomplished by considering the conditional distribution of the data, given the ranks R1, R2,..., Rn. In particular, pointwise confidence intervals for F may be obtained by adapting a method of Terpstra and Miller (2006).