Rank, Order-Statistic, and U-Statistic Methods in Nonparametric Inference

It is known that the maximum possible correlation between two order statistics from a simple random sample of size n occurs when the distribution is uniform. Motivated by the judgment rankings used in ranked-set sampling and related sample schemes, we study what correlations are possible if one looks at judgment order statistics rather than actual order statistics. In other words, we look for bounds on the correlation when we consider all possible distributions and all possible ranking schemes. We find that the interval of possible correlations becomes much wider; indeed, for sample sizes of at least four, all correlations except a perfect negative correlation are possible. We also study the intermediate case where the judgment rankings are based on a covariate, in which case the judgment order statistics are concomitants of order statistics. We explore whether our results can be used to improve statistical inference based on ranked-set sampling data. 

A common question when deciding whether to consider a type of forensic evidence is the degree at which trace objects from different sources can be distinguished. The degree is referred to as the individuality of the forensic modality, with respect to a specified metric for comparing a large number of samples from multiple sources. Regardless of the sample size, if the metric does not give rise to an exclusionary difference between two sources, then two sources are said to have indistinguishable profiles, with respect to the metric. We use the score developed in Davis et al. (2012) for sparse categorical data arising from the FlashID system©. We will define a new level alpha test for the equality of scores, based on all pairwise comparisons within and between documents from two different writers. In this presentation, we will derive the limiting distribution of our statistic. This test compares average within-writer scores with the between-writer scores. Using the corresponding p-values between all pairs of writers, we use a mixture model to make a statement about the confidence that a given p-value arose from two writers with indistinguishable profiles. 

Distance correlation and Chatterjee's correlation are two highly representative measures of dependence. Their simplicity and desirable properties have attracted much attention over the past years, leading to numerous extensions and in-depth discussions. The theoretical connections between these two measures, however, have remained largely underexplored, despite some empirical comparisons of their statistical power. In this work, we investigate some connections between Chatterjee's correlation and the rank-based distance correlation. We show that both belong to a class of dependence measures based on non-negative weighted rank differences, but with distinct weight allocations resulting in different asymptotic null distributions. Motivated by this finding, we construct a new rank distance correlation using Gaussian kernel that, quite interestingly, exhibits Chatterjee-like behavior under finite sample and specific bandwidth conditions. 

Recent theory by Kim, Balakrishnan, and Wasserman (2022) introduces a non-asymptotic framework for the minimax power of permutation tests. This study empirically validates these theoretical guarantees for U-statistic based permutation tests through extensive simulations.

We consider two-sample testing and categorical independence testing. In the two-sample setting, we compare the Mann–Whitney U-test based on normal approximations with its permutation-based version. For independence testing, we compare the classical chi-squared test, a bootstrapped chi-squared test, and the multinomial U-statistic proposed by Kim, Balakrishnan, and Wasserman.

Our results demonstrate that while traditional tests suffer from power collapse and poor Type I error control as dimensionality increases relative to sample size, the permutation-based U-statistic consistently achieves predicted minimax separation rates. By characterizing these error rates across diverse probability distributions and real-world data, we show that permutation tests offer a practically relevant and theoretically grounded alternative for small sample testing, particularly for high-dimensional multinomial settings. 

We study the analog of Fisher's exact test in the case where the data come from two independent samples drawn using ranked-set sampling (RSS) rather than simple random sampling. The exactness property is preserved not just under perfect rankings, but also under imperfect rankings as long as the same ranking scheme is used for both samples. We compare the new test to the simple random sampling Fisher's exact test and to existing RSS competitors in terms of power, and we also study what happens under imperfect rankings when the ranking schemes differ between the two samples. Fisher's exact test is famous for being conservative. We show that in the RSS case, the conservativeness of the test can be significantly reduced by using a data-based tie-breaking procedure that does not require randomization.