On a Test of Whether one of Two Random Variables is Stochastically Larger than the Other
1947/03/01 by H. B. Mann, D. R. Whitney, Douglas R. Whitney · 14,109 citations
Computer Science · Engineering · Mathematics · #Bayesian Methods and Mixture Models #Combinatorics #Cumulative distribution function #Distribution (mathematics) #Diverse Scientific and Engineering Research #Infinity #Limit (mathematics) #Mathematical analysis #Mathematics #Moment (physics) #Normal distribution #Physics #Probability density function #Random variable #Statistic #Statistical hypothesis testing #Statistics #Stochastic processes and statistical mechanics #Test statistic
paper · pdf · doi:10.1214/aoms/1177730491
published in The Annals of Mathematical Statistics 18(1), 50-60 (Institute of Mathematical Statistics)
openalex publication_date 1947/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Abstract
Let x and y be two random variables with continuous cumulative distribution functions f and g. A statistic U depending on the relative ranks of the x's and y's is proposed for testing the hypothesis f = g. Wilcoxon proposed an equivalent test in the Biometrics Bulletin, December, 1945, but gave only a few points of the distribution of his statistic. Under the hypothesis f = g the probability of obtaining a given U in a sample of n x's and m y's is the solution of a certain recurrence relation involving n and m. Using this recurrence relation tables have been computed giving the probability of U for samples up to n = m = 8. At this point the distribution is almost normal. From the recurrence relation explicit expressions for the mean, variance, and fourth moment are obtained. The 2rth moment is shown to have a certain form which enabled us to prove that the limit distribution is normal if m, n go to infinity in any arbitrary manner. The test is shown to be consistent with respect to the class of alternatives f(x) > g(x) for every x.
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- Investigating Affect Mining Techniques for Annotation Sample Selection in the Creation of Finnish Affective Speech Corpus
- Dynamic Graph Embedding Through Hub-aware Random Walks
- The Third Pillar of Causal Analysis? A Measurement Perspective on Causal Representations
- A Nonparametric Test of Dependence Based on Ensemble of Decision Trees
- Memorization or Reasoning? Exploring the Idiom Understanding of LLMs
- University of Indonesia at SemEval-2025 Task 11: Evaluating State-of-the-Art Encoders for Multi-Label Emotion Detection
- Uncertainty-Aware Crossmodal Fusion for Classification of Animal Behavior
- Smart Traffic Signals: Comparing MARL and Fixed-Time Strategies
- Finding Distributions that Differ, with False Discovery Rate Control
- Learning to Adapt to Position Bias in Vision Transformer Classifiers
- Does Differentially Private Synthetic Data Lead to Synthetic Discoveries?
- Latent Two-Sample Testing for Fair Autonomous Vehicle Road Evaluation
- Doc2CI: A Multi-Service Study of CI Configuration Generation Using Large Language Models
- Patterns of Reservoir‐Triggered Seismicity in a Low‐Seismicity Region of France
- Inactivation of Salmonella, Enterococcus faecium and natural microbiota on dry food matrices with microwave-driven plasma-processed air
- Wisdom from Diversity: Bias Mitigation Through Hybrid Human-LLM Crowds
- Exaggerated false positives by popular differential expression methods when analyzing human population samples
- Sustained software development, not number of citations or journal choice, is indicative of accurate bioinformatic software
- MUREQ: a multilayer framework for analyzing and operationalizing visualization requirements
- From Digital to Physical Reservoir Computing: Co-Optimizing Soft Robotic Reservoirs via Dynamics Matching
- GOUHFI: a novel contrast- and resolution-agnostic segmentation tool for Ultra-High Field MRI
- A Perturbation and Speciation-Based Algorithm for Dynamic Optimization Uninformed of Change
- Desirability of outcome ranking (DOOR) analysis for multivariate survival outcomes with application to ACTT-1 trial
- Benchmarking cross-project defect prediction approaches with costs metrics
- How synchronous participation affects the willingness to subscribe to social live streaming services: the role of co-Interactive behavior on Twitch
- A Representation Learning Approach to Feature Drift Detection in Wireless Networks
- Nonparametric Method for Clustered Data in Pre-Post Factorial Design
- LizAI XT -- Artificial Intelligence-Powered Platform for Healthcare Data Management: A Study on Clinical Data Mega-Structure, Semantic Search, and Insights of Sixteen Diseases
- An Approach for Clustering Subjects According to Similarities in Cell Distributions within Biopsies
- Distinct disease trajectories in frontotemporal dementia–motor neuron disease and behavioural variant frontotemporal dementia: A longitudinal study
- Support trees in vanilla agroforests of Madagascar: diversity, composition and origin
- AI-Mediated Code Comment Improvement
- Beyond Basic A/B testing: Improving Statistical Efficiency for Business Growth
- Two-step penalised logistic regression for multi-omic data with an application to cardiometabolic syndrome
- 'Congratulations, morons': Dynamics of Toxicity and Interaction Polarization in the Covid Vaccination and Ukraine War Twitter Debates
- Real Faults in Model Context Protocol (MCP) Software: a Comprehensive Taxonomy
- A RANKING VARIABLE ANALOGUE OF BISERIAL CORRELATION: IMPLICATIONS FOR SHORT‐CUT ITEM ANALYSIS
- Political Homophily in Independence Movements: Analysing and Classifying Social Media Users by National Identity
- Unified and Simple Sample Size Calculations for Individual or Cluster Randomized Trials with Skewed or Ordinal Outcomes
- Bayesian Model for Multiple Change-points Detection in Multivariate Time Series
- SYSML: StYlometry with Structure and Multitask Learning: Implications\n for Darknet Forum Migrant Analysis
- Unblind the charts: Towards Making Interactive Charts Accessible in Android Applications
- Reduction in hydraulic conductivity for high burn severity soils begins after the first rainfall event: Results from laboratory-scale rainfall simulation experiments
- Body size of orthoconic cephalopods from the late Silurian and Devonian of the Anti‐Atlas (Morocco)
- Persistent interaction patterns across social media platforms and over time
- Widespread Occurrence of Trenching Patterns in the Granulation Field: Evidence for Roll Convection?
- Studying Developer Perceptions on the Potential of CI Recommendation Systems
- Profile-Likelihood and Baseline-Sensitivity Diagnostics for Digitized Radiation-Sensor Decay Datasets
- A Penny for Your Prompts: Experiments Detecting and Mitigating LLM Usage by Survey Respondents
- Perceptual Implications of Automatic Anonymization in Pathological Speech
- Spatial Pattern of Groundwater Arsenic Occurrence and Association with Bedrock Geology in Greater Augusta, Maine
- Survival in Young-Onset Metastatic Colorectal Cancer: Findings From Cancer and Leukemia Group B (Alliance)/SWOG 80405
- Hyperbolic comets as an indicator of a hypothetical planet 9 in the solar system
- Physics-Informed Neural Networks for Cardiac Activation Mapping
- Spider Monkey Optimization algorithm for numerical optimization
- Identifying differences in physical activity and autonomic function patterns between psychotic patients and controls over a long period of continuous monitoring using wearable sensors
- Effects of summer mesoscale hydrographic features on epipelagic siphonophore assemblages in the surrounding waters of Taiwan, western North Pacific Ocean
- Exploring Statistical Change Point Detection Techniques for Performance Anomaly Detection at Mozilla
- Polymorphism of apolipoprotein E
- Visual and tactile learning of ground structures in desert ants
- Link prediction based on local random walk
- Sub-residue sharpness of protein helix-coil transitions reveals a spatial-spectral uncertainty limit
- Learning Only from Relevant Keywords and Unlabeled Documents
- A systematic study of variability among OB-stars based on HIPPARCOS photometry
- Press-pulse: a general theory of mass extinction?
- Some Extensions of the Idea of Bias
- Locating Faulty Methods with a Mixed RNN and Attention Model
- When Does Critique Improve AI-Assisted Theoretical Physics? SCALAR: Structured Critic--Actor Loop for Agentic Reasoning
- Mapping the Italian Telegram Ecosystem: Communities, Toxicity, and Hate Speech
- Conflict, migration and land‐cover changes in Indochina: a hydrological assessment
- An examination of range effects when evaluating discomfort due to glare in Singaporean buildings
- Regole, Consuetudini, Statuti nella storia degli Ordini religiosi.
- Reasoning-Based Refinement of Unsupervised Text Clusters with LLMs
- Do Automatic Comment Generation Techniques Fall Short? Exploring the Influence of Method Dependencies on Code Understanding
- Gamifying Testing in IntelliJ: A Replicability Study
- Regional variation in the terminal Pleistocene and early Holocene radiocarbon record of eastern North America
- Sojourner under Sabotage: A Serious Testing and Debugging Game
- Neuroevolution of Self-Attention Over Proto-Objects
- Investigating Co-Constructive Behavior of Large Language Models in Explanation Dialogues
- The Willingness to Pay–Willingness to Accept Gap, the “Endowment Effect,” Subject Misconceptions, and Experimental Procedures for Eliciting Valuations
- Statistical Significance of Minimum of the Order Parameter Fluctuations of Seismicity Before Major Earthquakes in Japan
- Android Instrumentation Testing in Continuous Integration: Practices, Patterns, and Performance
- Multi-class Probabilistic Bounds for Self-learning
- Dominance statistics: Ordinal analyses to answer ordinal questions.
- Prosodic Markers of Saliency in Humorous Narratives
- Students’ epistemologies about experimental physics: Validating the Colorado Learning Attitudes about Science Survey for experimental physics
- How Closely Do LLM Reviews Align with Human Peer Review?
- Success-Odds: An improved Win-Ratio
- The Simple Difference Formula: An Approach to Teaching Nonparametric Correlation
- Dynamic adaptation of software-defined networks for IoT systems
- Improving the open cluster census
- What is a cool-core cluster? a detailed analysis of the cores of the X-ray flux-limitedHIFLUGCScluster sample
- Correlation between gamma-ray bursts and gravitational waves
- Bridging Econometrics and AI: VaR Estimation via Reinforcement Learning and GARCH Models
- Automated patch assessment for program repair at scale
- Developing skills versus reinforcing concepts in physics labs: Insight from a survey of students’ beliefs about experimental physics
- Neutron-Induced, Single-Event Effects on Neuromorphic Event-Based Vision Sensor: A First Step and Tools to Space Applications
- What's the Difference? Supporting Users in Identifying the Effects of Prompt and Model Changes Through Token Patterns
- Foundations of Descriptive and Inferential Statistics
- Changing thunderstorms environments in Saudi Arabia: Frequency, variability and drivers of change
- Model-Based Design of Energy-Efficient Applications for IoT Systems
- PUMiner
- SWFC-ART: A cost-effective approach for Fixed-Size-Candidate-Set Adaptive Random Testing through small world graphs
- Influence of sociodemographic characteristics on human mobility
- The ‘as code’ activities: development anti-patterns for infrastructure as code
- Theory of Pixel Lensing toward M31. II. The Velocity Anisotropy and Flattening of the MACHO Distribution
- On the Estimation of Relative Potency in Dilution (-Direct) Assays by Distribution-Free Methods
- Thatcherite Ideology, Housing Tenure and Crime: The Socio-spatial Consequences of the Right to Buy for Domestic Property Crime
- The Code Barrier: What LLMs Actually Understand?
- Who Speaks for Ethics? How Demographics Shape Ethical Advocacy in Software Development
- Uniformity of some recent color metrics tested with an accurate color-difference tolerance dataset
- Surrogate-assisted analysis of weighted functional brain networks
- Mental model shifts in human-LLM interactions
- NF-kappa B genes have a major role in inflammatory breast cancer. [europepmc]
- Rank-based inverse normal transformations are increasingly used, but are they merited? [europepmc]
- miRNAs in lung cancer - studying complex fingerprints in patient's blood cells by microarray experiments. [europepmc]
- Motif Enrichment Analysis: a unified framework and an evaluation on ChIP data. [europepmc]
- Estimation of allele frequency and association mapping using next-generation sequencing data. [europepmc]
- Metagenomic biomarker discovery and explanation. [europepmc]
- Improved statistical methods enable greater sensitivity in rhythm detection for genome-wide data. [europepmc]
- Notch promotes recurrence of dormant tumor cells following HER2/neu-targeted therapy. [europepmc]
- Calculating an optimal box size for ligand docking and virtual screening against experimental and predicted binding pockets. [europepmc]
- Therapeutic CD94/NKG2A blockade improves natural killer cell dysfunction in chronic lymphocytic leukemia. [europepmc]
- AURKA, DLGAP5, TPX2, KIF11 and CKAP5: Five specific mitosis-associated genes correlate with poor prognosis for non-small cell lung cancer patients. [europepmc]
- pKWmEB: integration of Kruskal-Wallis test with empirical Bayes under polygenic background control for multi-locus genome-wide association study. [europepmc]
- Drosophila melanogaster White Mutant w 1118 Undergo Retinal Degeneration. [europepmc]
- Effects of tobacco smoke and electronic cigarette vapor exposure on the oral and gut microbiota in humans: a pilot study. [europepmc]
- Spatially and functionally distinct subclasses of breast cancer-associated fibroblasts revealed by single cell RNA sequencing. [europepmc]
- Bacteroides ovatus ATCC 8483 monotherapy is superior to traditional fecal transplant and multi-strain bacteriotherapy in a murine colitis model. [europepmc]
- Species-specific enhancement of enterohemorrhagic E. coli pathogenesis mediated by microbiome metabolites. [europepmc]
- CNNs for automatic glaucoma assessment using fundus images: an extensive validation. [europepmc]
- Stop codon context influences genome-wide stimulation of termination codon readthrough by aminoglycosides. [europepmc]
- Open Development and Clinical Validation of Multiple 3D-Printed Nasopharyngeal Collection Swabs: Rapid Resolution of a Critical COVID-19 Testing Bottleneck. [europepmc]
- Estimating infectiousness throughout SARS-CoV-2 infection course. [europepmc]
- UCell: Robust and scalable single-cell gene signature scoring. [europepmc]
- Exaggerated false positives by popular differential expression methods when analyzing human population samples. [europepmc]
- Evidence for embracing normative modeling. [europepmc]
- Single-cell and spatial transcriptomics analysis of non-small cell lung cancer. [europepmc]
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