Principles of Hypothesis Testing

  • M. M. Rao
Part of the Mathematics and Its Applications book series (MAIA, volume 508)


This chapter is devoted to some serious aspects of the hypothesis testing problems, including both the simple and composite cases. These consist of the fundamental lemma of Neyman-Pearson, in its abstract version due to Grenander, and a few of its applications as well as a technique in reducing composite hypotheses by means of weights. The latter contains a detailed Bayes methodology with iterated priors and some uniformity conditions that admit extensions to stochastic processes. Some of these considerations are classical, but they are seen to allow sharper analysis, in contrast with a use of the general (decision) theory, and these are examined carefully in the first four sections which also contain vector analysis approaches. It may be noted that this work demands an employment of deeper mathematical tools in solving some fundamental questions such as the Behrens-Fisher problem, and this is detailed in the fifth section. The last one is devoted to complementing the the preceding results, as exercises often with hints.


Critical Region Vector Measure Similar Test Nuisance Parameter Powerful Region 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Bibliographical notes

  1. [1]
    Grenander, U. “Stochastic processes and statistical inference,” Arkiv fur Mat., 1 (1950), 195–277.MathSciNetGoogle Scholar
  2. [1]
    On the theory of estimation with convex loss functions,“ Proc. Symp. in honor of J. Neyman,PWN publishers, Warszawa, (1977), 177–202. Google Scholar
  3. [1]
    Chernoff, H., and Scheffé, H. “A generalization of the Neyman-Pearson fundamental lemma,” Ann. Math. Statist. 23 (1952), 213–225. Google Scholar
  4. [1]
    Chernoff, H. “Large sample theory: parametric case,” Ann. Math. Statist. 27 (1956), 1–22. Google Scholar
  5. [1]
    Chan, N. H., and Wei, C.-Z. “Limiting distributions of least squares estimates of unstable autoregressive processes,” Ann. Statist.,16 (1988), 367–401. Chang, D. K., and Rao, M. M.Google Scholar
  6. [9]
    Remarks on a Radon-Nikodÿm theorem for vector measures,“ Proc. Conf. Vector and Operator valued Measures and Applications,Academic Press, (1973), 303–317.Google Scholar
  7. [1]
    Rényi, A. On a new axiomatic theory of probability,“ Acta Math. Hung.,6 (1955), 285–333.Google Scholar
  8. [3]
    Grenander, U., and Miller, M. I. “Representations of knowledge in complex systems,” J. R. Statist. Soc. Ser B, 56 (1994), 549–603.MathSciNetzbMATHGoogle Scholar
  9. [1]
    Hwang, C.- R. Conditioning by (EQUAL, LINEAR),“ Trans. Am. Math. Soc.,274 (1983), 69–83.Google Scholar
  10. [21]
    Rao, M. M., and Ren, Z. D. Theory of Orlicz Spaces,Marcel Dekker, New York, 1991. Rao, M. M., and Sazonov, V. V.Google Scholar
  11. [2]
    Doob, J. L. Stochastic Processes,Wiley, New York, 1953. Google Scholar
  12. [1]
    Grenander, U. “Stochastic processes and statistical inference,” Arkiv fur Mat., 1 (1950), 195–277.MathSciNetGoogle Scholar
  13. [1]
    Shiryayev, A. N. Statistics of Random Processes,I, II„ Springer, New York, 1977. Lloyd, S. P.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2000

Authors and Affiliations

  • M. M. Rao
    • 1
  1. 1.University of CaliforniaRiversideUSA

Personalised recommendations