Exact bounds of Spearman’s footrule in the presence of missing data with applications to independence testing
File(s)
Published version
Author(s)
Zeng, Yijin
Adams, Niall
Bodenham, Dean
Type
Journal Article
Abstract
This work studies exact bounds of Spearman’s footrule in the presence of missing data for two n-dimensional distinct real-valued vectors X and Y. The lower bound is obtained by sequentially constructing imputations of the partially observed vectors, each with a non-increasing value of Spearman’s footrule. The upper bound is found by first considering the set of all possible values of Spearman’s footrule for imputations of X and Y, and then the size of this set is gradually reduced using several constraints. Algorithms with computational complexities O(n²) and O(n³) are provided for computing these tight lower and upper bounds, respectively. As an application, we propose a novel two-sample independence testing method for data with missing values. Improving on all existing approaches, our method controls the Type I error under arbitrary missingness. Simulation results demonstrate that our method has good power, typically when the proportion of pairs containing missing data is below 15%. We illustrate our method on real-world data by comparing the dependence of yearly precipitation levels in the United Kingdom, Germany and the Unites States.
Date Issued
2025-12-29
Date Acceptance
2025-12-01
Citation
Electronic Journal of Statistics, 2025, 19 (2), pp.6103-6140
ISSN
1935-7524
Publisher
Institute of Mathematical Statistics
Start Page
6103
End Page
6140
Journal / Book Title
Electronic Journal of Statistics
Volume
19
Issue
2
Copyright Statement
Rights: Creative Commons Attribution 4.0 International License.
License URL
Identifier
10.1214/25-EJS2479
Subjects
MSC2020 subject classifications: Primary 62H20
62D10; secondary 62G10 Spearman's footrule
rank correlation
missing data
independence testing
Publication Status
Published
Date Publish Online
2025-12-29