Inference on a distribution function from ranked set samples

• Published in: Annals of the Institute of Statistical Mathematics Vol. 72; no. 1; pp. 157 - 185
• Main Authors: Dümbgen, Lutz, Zamanzade, Ehsan
• Format: Journal Article
• Language: English
• Published: Tokyo Springer Japan 01.02.2020; Springer Nature B.V
• Subjects: Asymptotic properties; Confidence intervals; Distribution functions; Economic models; Economics; Finance; Insurance; Management; Mathematics; Mathematics and Statistics; Maximum likelihood estimators; Random variables; Sampling methods; Statistical analysis; Statistical methods; Statistics; Statistics for Business; Stratification; Empirical process; Confidence band; Conditional inference; Moment equations; Functional limit theorem; Relative asymptotic efficiency; Unbalanced samples; Imperfect ranking
• ISSN: 0020-3157; 1572-9052
• DOI: 10.1007/s10463-018-0680-y

Consider independent observations ( X i , R i ) with random or fixed ranks R i , while conditional on R i , the random variable X i has the same distribution as the R i -th order statistic within a random sample of size k from an unknown distribution function F . Such observation schemes are well known from ranked set sampling and judgment post-stratification. Within a general, not necessarily balanced setting we derive and compare the asymptotic distributions of three different estimators of the distribution function F : a stratified estimator, a nonparametric maximum-likelihood estimator and a moment-based estimator. Our functional central limit theorems generalize and refine previous asymptotic analyses. In addition, we discuss briefly pointwise and simultaneous confidence intervals for the distribution function with guaranteed coverage probability for finite sample sizes. The methods are illustrated with a real data example, and the potential impact of imperfect rankings is investigated in a small simulation experiment.