Bahadur Efficiency of Tests for a Shift in Location of Normal Populations

  • Jaap Praagman

Summary

A sequence of independent random variables X 1,...,X N is said to have a change point n, if X 1,...,X N have a common distribution F and X n+1,..., X N have a common distribution G, GF. Consider the problem of testing the null hypothesis of no change, against the alternative of a one-sided change at an unknown change point n, when both F and G are normal with equal variance σ2. Most of the test statistics for this problem can be interpreted as generalizations of two-sample statistics (n known). In this chapter we derive the Bahadur efficiencies for two classes of statistics that are generalizations of the two-sample likelihood ratio statistics. The asymptotic results are compared with some small-sample power estimates based on Monte Carlo experiments.

Keywords

ogeN Prool 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bahadur, R.R. (1971), Some Limit Theorems in Statistics. Philadelphia, PA: SIAM.CrossRefGoogle Scholar
  2. Bhattacharyya, G.K. and Johnson, R.A. (1968), Nonparametric tests for shift at an unknown time point. Annals of Mathematical Statistics, 39, 1731–1743.Google Scholar
  3. Billingsley, P. (1968), Convergence of Probability Measures. New York: John Wiley.Google Scholar
  4. Chernoff, H. and Zacks, S. (1964), Estimating the current mean of a normal distribution which is subjected to changes in time. Annals of Mathematical Statistics, 35, 999–1018.CrossRefGoogle Scholar
  5. Chow, Y.S. and Lai, T.L. (1973), Limiting behaviour of weighted sums of random variables. Annals of Probability, 1, 810–824.CrossRefGoogle Scholar
  6. Deshayes, J. and Picard, D. (1982), Tests of disorder of regression: Asymptotic comparison. Theory of Probability and its Applications, 27, 100–115.CrossRefGoogle Scholar
  7. Feller, W. (1957), An Introduction to Probability Theory and Its Applications, vol. 1, second edn. New York: John Wiley.Google Scholar
  8. Gardner, L.A. (1969), On detecting changes in the mean of normal variates. Annals of Mathematical Statistics, 40, 116–126.CrossRefGoogle Scholar
  9. Groeneboom, P. and Oosterhoff, J. (1977), Bahadur efficiency and probabilities of large deviations. Statistica Neerlandica., 31, 1–24.Google Scholar
  10. Haccou, P., Meelis, E., and van de Geer, S. (1985), On the likelihood ratio test for a change point in a sequence of independent exponentially distributed random variables. Report MS-R8507. Amsterdam: Centre for Mathematics and Computer Science.Google Scholar
  11. Hawkins, D.M. (1977), Testing a sequence of observations for a shift in location. Journal of the American Statistical Association, 72, 180–186.CrossRefGoogle Scholar
  12. Killeen, T.J. and Hettmansperger, T.P. (1972), Bivariate tests for location and their Bahadur efficiencies. Annals of Mathematical Statistics, 43, 1507–1516.CrossRefGoogle Scholar
  13. Killeen, T.J., Hettmansperger, T.P., and Sievers, G.L. (1972), An elementary theorem on the probability of large deviations. Annals of Mathematical Statistics, 43, 181–192.CrossRefGoogle Scholar
  14. Law, A.M. and Kelton, W.D. (1982), Simulation Modeling and Analysis. New York: McGraw-Hill.Google Scholar
  15. Praagman, J. (1986), Efficiency of Change-Point Tests, Unpublished Ph.D. thesis, Eindhoven University of Technology.Google Scholar
  16. Raghavachari, M. (1970), On a theorem of Bahadur on the rate of convergence of test statistics. Annals of Mathematical Statistics, 41, 1695–1699.CrossRefGoogle Scholar
  17. Sen, A. and Srivastava, M.S. (1975a), Some one-sided tests for change in level. Technometrics, 17, 61–64.CrossRefGoogle Scholar
  18. Sen, A. and Srivastava, M.S. (1975b), On tests for detecting change in mean. Annals of Statistics, 3, 98–108.CrossRefGoogle Scholar
  19. Worsley, K.J. (1979), On the likelihood ratio test for a shift in location of normal populations. Journal of the American Statistical Association, 74, 365–367.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Jaap Praagman

There are no affiliations available

Personalised recommendations