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Keywords:

  • Cumulative distribution function;
  • Distribution free;
  • Imperfect ranking;
  • Isotonic regression;
  • MaxMin;
  • MinMax;
  • Nonparametric;
  • Ranked set sampling;
  • Relative efficiency;
  • Stochastic ordering

Summary In applications that require cost efficiency, sample sizes are typically small so that the problem of empty strata may often occur in judgment poststratification (JPS), an important variant of balanced ranked set sampling. In this article, we consider estimation of population cumulative distribution functions (CDF) from JPS samples with empty strata. In the literature, the standard and restricted CDF estimators (Stokes and Sager, 1988, Journal of the American Statistical Association83, 374381; Frey and Ozturk, 2011, Annals of the Institute of Statistical Mathematics, to appear) do not perform well when simply ignoring empty strata. In this article, we show that the original isotonized estimator (Ozturk, 2007, Journal of Nonparametric Statistics19, 131–144) can handle empty strata automatically through two methods, MinMax and MaxMin. However, blindly using them can result in undesirable results in either tail of the CDF. We thoroughly examine MinMax and MaxMin and find interesting results about their behaviors and performance in the presence of empty strata. Motivated by these results, we propose modified isotonized estimators to improve estimation efficiency. Through simulation and empirical studies, we show that our estimators work well in different regions of the CDF, and also improve the overall performance of estimating the whole function.