Electronic Journal of Statistics

Asymptotic hypotheses testing for the colour blind problem

Laura Dumitrescu and Estate V. Khmaladze

Full-text: Open access

Abstract

Within a nonparametric framework, we consider the problem of testing the equality of marginal distributions for a sequence of independent and identically distributed bivariate data, with unobservable order in each pair. In this case, it is not possible to construct the corresponding empirical distributions functions and yet this article shows that a systematic approach to hypothesis testing is possible and provides an empirical process on which inference can be based. Furthermore, we identify the linear statistics that are asymptotically optimal for testing the hypothesis of equal marginal distributions against contiguous alternatives. Finally, we exhibit an interesting property of the proposed stochastic process: local alternatives of dependence can also be detected.

Article information

Source
Electron. J. Statist., Volume 13, Number 2 (2019), 4573-4595.

Dates
Received: March 2019
First available in Project Euclid: 12 November 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1573527691

Digital Object Identifier
doi:10.1214/19-EJS1634

Mathematical Reviews number (MathSciNet)
MR4029803

Zentralblatt MATH identifier
07136625

Subjects
Primary: 62G10: Hypothesis testing
Secondary: 62G20: Asymptotic properties

Keywords
Asymptotically optimal test contiguous alternatives dependence alternatives empirical process goodness of fit Kolmogorov–Smirnov statistics unordered pairs

Rights
Creative Commons Attribution 4.0 International License.

Citation

Dumitrescu, Laura; Khmaladze, Estate V. Asymptotic hypotheses testing for the colour blind problem. Electron. J. Statist. 13 (2019), no. 2, 4573--4595. doi:10.1214/19-EJS1634. https://projecteuclid.org/euclid.ejs/1573527691


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References

  • [1] Banerjee, T., Chattopadhyay, G. and Banerjee, K. (2017). Two stages test of means of unordered pairs., Statistics in Medicine, 36, 2466–2480.
  • [2] Bernstein, A. V. and Sidorov, A. A. (1972). Estimates of the set of expectations for a normal population., Theory of Probability and Its Applications, 17, 723–726.
  • [3] Blum, J. R., Kiefer, J. and Rosenblatt, M. (1961). Distribution free tests of independence based on the sample distribution function., Annals of Mathematical Statistics, 32, 485–496.
  • [4] Davies, P. and Phillips, A. J. (1988). Nonparametric tests of population differences and estimation of the probability of misidentification with unidentified paired data., Biometrika, 75, 753–760.
  • [5] Day, S. J. and Altman, D. G. (2000). Blinding in clinical trials and other studies., BMJ, 321, 504.
  • [6] Gindilis, V. M. (1966). Mitotic chromosome spiralization and karyogrammic analysis in man., Citologia, 8, 144–157.
  • [7] Hájek, J., Šidák, Z. and Sen, P. K. (1999)., Theory of rank tests, 2nd edition. Academic Press.
  • [8] Hinkley, D. V. (1973). Two-sample tests with unordered pairs., Journal of the Royal Statistical Society, Series B, 35, 337–346.
  • [9] Janssen, A. and Rahnenführer, J. (2002). A hazard-based approach to dependence tests for bivariate censored models., Mathematical Methods of Statistics, 11, 297–322.
  • [10] Kuo, H. -H. (1975)., Gaussian measures in Banach spaces, Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York.
  • [11] Lauder, I. J. (1977). Tracing quantitative measurements on human chromosomes in family studies., Annals of Human Genetics, 41, 77–82.
  • [12] Li, P. and Qin, J. (2011). A new nuisance-parameter elimination method with application to the unordered homologous chromosome pairs problem., Journal of the American Statistical Association, 106, 1476–1484.
  • [13] Miller, F., Friede, T. and Kieser, M. (2009). Blinded assessment of treatment effects utilizing information about the randomization block length., Statistics in Medicine, 28, 1690–1706.
  • [14] Oosterhoff J. and van Zwet W. R. (2012). A note on contiguity and hellinger distance. In: van de Geer S., Wegkamp M. (eds) Selected Works of Willem van Zwet. Selected Works in Probability and Statistics. Springer, New York, NY.
  • [15] Parsadanishvili, E. G. and Khmaladze, E. V. (1982). The testing of statistical hypotheses on unidentifiable objects., Theory of Probability and Its Applications, 27, 175–182.
  • [16] van der Vaart, A. W. and Wellner, J. A. (1996)., Weak convergence and empirical processes: With applications to statistics. Springer Series in Statistics. Springer-Verlag, New York.