Inference on a Distribution Function from Ranked Set Samples

Abstract

Consider independent observations $(X_1,R_1)$, $(X_2,R_2)$, \ldots, $(X_n,R_n)$ with random or fixed ranks $R_i \in \{1,2,\ldots,k\}$, while conditional on $R_i = r$, the random variable $X_i$ 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 situations in which ranking observations is much easier than obtaining their precise values. Two well-known special cases are ranked set sampling (McIntyre 1952) and judgement post-stratification (MacEachern et al. 2004). Within a general setting including unbalanced ranked set sampling we derive and compare the asymptotic distributions of three different estimators of the distribution function $F$ as $n \to \infty$ with fixed $k$: The stratified estimator (SE) of Stokes and Sager (1988), the nonparametric maximum-likelihood estimator (LE) of Kvam and Samaniego (1994) and the moment-based estimator (ME) of Chen (2001). Our functional central limit theorems generalize and refine the asymptotic analyses of Stokes and Sager (1988), Huang (1997) and Chen (2001). In particular we show that LE is always more efficient than SE and ME. But the efficiency loss of ME versus LE is bounded and often rather small, whereas the efficiency loss of SE versus LE can be arbitrarily large in unbalanced settings. We also show that all three estimators are asymptotically nearly equivalent in the tails of $F$. Finally we describe briefly pointwise and simultaneous confidence intervals for the distribution function $F$ with guaranteed coverage probability for finite sample sizes. Here the ME approach turns out to be most efficient computationally.