Hypothesis Testing Under Communication Constraints

  • Rudolf AhlswedeEmail author
Part of the Foundations in Signal Processing, Communications and Networking book series (SIGNAL, volume 15)


A new class of statistical problems is introduced, involving the presence of communication constraints on remotely collected data. Bivariate hypothesis testing, \(H_0: P_{XY}\) against \(H_1: P_{\bar{XY}}\), is considered when the statistician has direct access to Y data but can be informed about X data only at a prescribed finite rate R. For any fixed R the smallest achievable probability of an error of type 2 with the probability of an error of type 1 being at most \(\epsilon \) is shown to go to zero with an exponential rate not depending on \(\epsilon \) as the sample size goes to infinity.


  1. 1.
    H. Chernoff, A measure of asymptotic efficiency for tests of a hypothesis based on a sum of observations. Ann. Math. Stat. 23, 493–507 (1952)MathSciNetCrossRefGoogle Scholar
  2. 2.
    S. Kullback, Information Theory and Statistics (Wiley, New York, 1959)zbMATHGoogle Scholar
  3. 3.
    S. Kullback, R.A. Leibler, On information and sufficiency. Ann. Math. Stat. 22, 79–86 (1951)MathSciNetCrossRefGoogle Scholar
  4. 4.
    I. Csiszár, J. Körner, Information Theory: Coding Theorems for Discrete Memoryless Systems (Academic, New York, 1982)zbMATHGoogle Scholar
  5. 5.
    R. Ahlswede, I. Wegener, Search Problems, Wiley Interscience Series in Discrete Mathematics and Optimization (Wiley, New York, 1987)zbMATHGoogle Scholar
  6. 6.
    A. Perez, Discrimination rate loss in simple statistical hypotheses by unfitted decision procedures, in Probability and Related Topics: Papers in Honour of Octav Onicescu (Nagard, 1983), pp. 381–390Google Scholar
  7. 7.
    H. Chernoff, The identification of an element of a large population in the presence of noise. Ann. Stat. 8, 1179–1197 (1980)MathSciNetCrossRefGoogle Scholar
  8. 8.
    R. Ahlswede, J. Körner, Source coding with side information and a converse for degraded broadcast channels. IEEE Trans. Inf. Theory 21, 629–637 (1975)MathSciNetCrossRefGoogle Scholar
  9. 9.
    R. Ahlswede, Coloring hypergraphs: a new approach to multi-user source coding, Part I. J. Comb. Inf. Syst. Sci. 1, 76–115 (1979)zbMATHGoogle Scholar
  10. 10.
    R. Ahlswede, P. Gács, J. Körner, Bounds on conditional probabilities with applications in multi-user communication. Z. Wahrscheinlichkeitsth. verw. Gebiete 34, 157–177 (1976)MathSciNetCrossRefGoogle Scholar
  11. 11.
    R. Ahlswede, Coloring hypergraphs: a new approach to multi-user source coding, Part II. J. Comb. Inf. Syst. Sci. 5, 220–268 (1980)zbMATHGoogle Scholar
  12. 12.
    W. Hoeffding, Asymptotically optimal tests for multinomial distributions. Ann. Math. Stat. 36, 369–400 (1965)MathSciNetCrossRefGoogle Scholar
  13. 13.
    I. Csiszár, G. Longo, On the error exponent for source coding and for testing simple statistical hypotheses. Stud. Sci. Math. Hung. 6, 181–191 (1971)MathSciNetzbMATHGoogle Scholar
  14. 14.
    G.D. Forney, Exponential error bounds for erasure, list, and decision feedback schemes. IEEE Trans. Inf. Theory 14, 206–220 (1968)MathSciNetCrossRefGoogle Scholar

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.BielefeldGermany

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