pp 1–26 | Cite as

Testing equality of a large number of densities under mixing conditions

  • Marta Cousido-RochaEmail author
  • Jacobo de Uña-Álvarez
  • Jeffrey D. Hart
Original Paper


In certain settings, such as microarray data, the sampling information is formed by a large number of possibly dependent small data sets. In special applications, for example in order to perform clustering, the researcher aims to verify whether all data sets have a common distribution. For this reason we propose a formal test for the null hypothesis that all data sets come from a single distribution. The asymptotic setting is that in which the number of small data sets goes to infinity, while the sample size remains fixed. The asymptotic null distribution of the proposed test is derived under mixing conditions on the sequence of small data sets, and the power properties of our test under two reasonable fixed alternatives are investigated. A simulation study is conducted, showing that the test respects the nominal level, and that it has a power which tends to 1 when the number of data sets tends to infinity. An illustration involving microarray data is provided.


Dependent data Kernel density estimation k-Sample problem Smooth tests U-statistics 

Mathematics Subject Classification




This work has received financial support of the Call 2015 Grants for Ph.D. contracts for training of doctors of the Ministry of Economy and Competitiveness, cofinanced by the European Social Fund (Ref. BES-2015-074958). We acknowledge support from MTM2014-55966-P project, Ministry of Economy and Competitiveness, and MTM2017-89422-P project, Ministry of Economy, Industry and Competitiveness, State Research Agency, and Regional Development Fund, UE. We also acknowledge the financial support provided by the SiDOR research group through the grant Competitive Reference Group, 2016–2019 (ED431C 2016/040), funded by the “Consellería de Cultura, Educación e Ordenación Universitaria. Xunta de Galicia.” To finish, the first author would like to thank the University of Vigo, and its Escola Internacional de Doutoramento (EIDO) by the financial support provided through mobility doctorate grants. The authors also thank Professors Raymond J. Carroll and Robert Chapkin for allowing use of their data.

Supplementary material

11749_2018_625_MOESM1_ESM.pdf (399 kb)
Supplementary Materials: Supplementary Material includes formal definitions of mixing dependence, stationarity and regularity conditions needed for the technical results, a remark about Theorem 5, the proof of Theorem 6, an additional real data analysis, and additional simulation results. (pdf 394KB)


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Copyright information

© Sociedad de Estadística e Investigación Operativa 2019

Authors and Affiliations

  1. 1.Department of Statistics and Operations Research and SiDOR Research Group, Faculty of EconomicsUniversity of VigoVigoSpain
  2. 2.Centro de Investigaciones Biomédicas (CINBIO)University of VigoVigoSpain
  3. 3.Department of StatisticsTexas A&M UniversityCollege StationUSA

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