A Nonparametric Mean Estimator for Judgment Poststratified Data

Abstract

MacEachern, Stasny, and Wolfe (2004, Biometrics60, 207-215) introduced a data collection method, called judgment poststratification (JPS), based on ideas similar to those in ranked set sampling, and proposed methods for mean estimation from JPS samples. In this article, we propose an improvement to their methods, which exploits the fact that the distributions of the judgment poststrata are often stochastically ordered, so as to form a mean estimator using isotonized sample means of the poststrata. This new estimator is strongly consistent with similar asymptotic properties to those in MacEachern et al. (2004). It is shown to be more efficient for small sample sizes, which appears to be attractive in applications requiring cost efficiency. Further, we extend our method to JPS samples with imprecise ranking or multiple rankers. The performance of the proposed estimators is examined on three data examples through simulation.

Constrained estimation using judgment post-stratification

“…The types of rankings used were (i) perfect rankings, (ii) random rankings, and (iii) perfectly wrong rankings obtained by ranking the units in each set in an order exactly opposite to the true ordering. For the first two types of rankings, the stochastic ordering assumption used by Ozturk (2007) and Wang et al (2008) holds, but for the third ranking type, that The three types of rankings are perfect rankings, random rankings, and perfectly wrong rankings assumption fails. The set size was fixed at m = 2, and 100,000 samples were simulated for each combination of parent distribution, type of ranking, and average sample size.…”

“…While Ozturk (2007) and Wang et al (2008) used a stochastic-ordering assumption to obtain more efficient inference, the gains offered by our methods do not require any additional assumptions. Nonetheless, our new methods are sometimes every bit as effective as methods that do make a stochastic-ordering assumption.…”

“…We demonstrate this increased efficiency by presenting results from two related simulation studies. The first study focused on assessing the performance of the restricted mean estimator relative to that of the standard JPS estimator, and the second focused on comparing the performance of the restricted estimator to that of the isotonic JPS mean estimator developed by Wang et al (2008).…”

“…In addition, recent work by Ozturk (2007) and Wang et al (2008) has shown that if one is willing to assume that the in-stratum distributions are stochastically ordered, then even greater gains in efficiency can be obtained using in-stratum estimators that satisfy the same stochastic-ordering constraints. The stochastic-ordering assumption is certainly reasonable in many settings.…”

Some nonparametric tests of perfect judgment ranking for judgment post stratification

“…Frey and Feeman (2012) showed that the standard mean estimator in the JPS setting is inadmissible under mean square loss function, they then proposed an alternate mean estimator. Another nonparametric mean esti-mator is proposed by Wang et al (2008), by imposing an order constraint on judgment sample means. Frey and Feeman (2013) proposed a conditionally unbiased variance estimator assuming that the post stratum sample sizes are known.…”