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  1. Brownian Bridge and nonparametric rank tests Olena Kravchuk School of Physical Sciences Department of Mathematics UQ

  2. Lecture outline • Definition and important characteristics of the Brownian bridge (BB) • Interesting measurable events on the BB • Asymptotic behaviour of rank statistics • Cramer-von Mises statistic • Small and large sample properties of rank statistics • Some applications of rank procedures • Useful references Brownian bridge and nonparametric rank tests

  3. Definition of Brownian bridge Brownian bridge and nonparametric rank tests

  4. Construction of the BB Brownian bridge and nonparametric rank tests

  5. Varying the coefficients of the bridge Brownian bridge and nonparametric rank tests

  6. Two useful properties Brownian bridge and nonparametric rank tests

  7. Ranks and anti-ranks Brownian bridge and nonparametric rank tests

  8. Simple linear rank statistic • Any simple linear rank statistic is a linear combination of the scores, a’s, and the constants, c’s. • When the constants are standardised, the first moment is zero and the second moment is expressed in terms of the scores. • The limiting distribution is normal because of a CLT. Brownian bridge and nonparametric rank tests

  9. Constrained random walk on pooled data • Combine all the observations from two samples into the pooled sample, N=m+n. • Permute the vector of the constants according to the anti-ranks of the observations and walk on the permuted constants, linearly interpolating the walk Z between the steps. • Pin down the walk by normalizing the constants. • This random bridge Z converges in distribution to the Brownian Bridge as the smaller sample increases. Brownian bridge and nonparametric rank tests

  10. From real data to the random bridge Brownian bridge and nonparametric rank tests

  11. Symmetric distributions and the BB Brownian bridge and nonparametric rank tests

  12. Random walk model: no difference in distributions Brownian bridge and nonparametric rank tests

  13. Location and scale alternatives Brownian bridge and nonparametric rank tests

  14. Random walk: location and scale alternatives Scale = 2 Shift = 2 Brownian bridge and nonparametric rank tests

  15. Simple linear rank statistic again • The simple linear rank statistic is expressed in terms of the random bridge. • Although the small sample properties are investigated in the usual manner, the large sample properties are governed by the properties of the Brownian Bridge. • It is easy to visualise a linear rank statistic in such a way that the shape of the bridge suggests a particular type of statistic. Brownian bridge and nonparametric rank tests

  16. Trigonometric scores rank statistics • The Cramer-von Mises statistic • The first and second Fourier coefficients: Brownian bridge and nonparametric rank tests

  17. Combined trigonometric scores rank statistics • The first and second coefficients are uncorrelated • Fast convergence to the asymptotic distribution • The Lepage test is a common test of the combined alternative (SW is the Wilcoxon statistic and SA-B is the Ansari-Bradley, adopted Wilcoxon, statistic) Brownian bridge and nonparametric rank tests

  18. Percentage points for the first component (one-sample) Durbin and Knott – Components of Cramer-von Mises Statistics Brownian bridge and nonparametric rank tests

  19. Percentage points for the first component (two-sample) Kravchuk – Rank test of location optimal for HSD Brownian bridge and nonparametric rank tests

  20. Some tests of location Brownian bridge and nonparametric rank tests

  21. Trigonometric scores rank estimators Location estimator of the HSD (Vaughan) Scale estimator of the Cauchy distribution (Rublik) Trigonometric scores rank estimator (Kravchuk) Brownian bridge and nonparametric rank tests

  22. Optimal linear rank test • An optimal test of location may be found in the class of simple linear rank tests by an appropriate choice of the score function, a. • Assume that the score function is differentiable. • An optimal test statistic may be constructed by selecting the coefficients, b’s. Brownian bridge and nonparametric rank tests

  23. Functionals on the bridge • When the score function is defined and differentiable, it is easy to derive the corresponding functional. Brownian bridge and nonparametric rank tests

  24. Result 4: trigonometric scores estimators • Efficient location estimator for the HSD • Efficient scale estimator for the Cauchy distribution • Easy to establish exact confidence level • Easy to encode into automatic procedures Brownian bridge and nonparametric rank tests

  25. Numerical examples: test of location • Normal, N(500,1002) • Normal, N(580,1002) Brownian bridge and nonparametric rank tests

  26. Numerical examples: test of scale • Normal, N(300,2002) • Normal, N(300,1002) Brownian bridge and nonparametric rank tests

  27. Numerical examples: combined test • Normal, N(580,2002) • Normal, N(500,1002) Brownian bridge and nonparametric rank tests

  28. When two colour histograms are compared, nonparametric tests are required as a priori knowledge about the colour probability distribution is generally not available. The difficulty arises when statistical tests are applied to colour images: whether one should treat colour distributions as continuous, discrete or categorical. Application: palette-based images Brownian bridge and nonparametric rank tests

  29. Application: grey-scale images Brownian bridge and nonparametric rank tests

  30. Application: grey-scale images, histograms Brownian bridge and nonparametric rank tests

  31. Application: colour images Brownian bridge and nonparametric rank tests

  32. Useful books • H. Cramer. Mathematical Methods of Statistics. Princeton University Press, Princeton, 19th edition, 1999. • G. Grimmett and D. Stirzaker. Probability and Random Processes. Oxford University Press, N.Y., 1982. • J. Hajek, Z. Sidak and P.K. Sen. Theory of Rank Tests. Academic Press, San Diego, California, 1999. • F. Knight. Essentials of Brownian Motion and Diffusion. AMS, Providence, R.I., 1981. • K. Knight. Mathematical Statistics. Chapman & Hall, Boca Raton, 2000. • J. Maritz. Distribution-free Statistical Methods. Monographs on Applied Probability and Statistics. Chapman & Hall, London, 1981. Brownian bridge and nonparametric rank tests

  33. Interesting papers • J. Durbin and M. Knott. Components of Cramer – von Mises statistics. Part 1. Journal of the Royal Statistical Society, Series B., 1972. • K.M. Hanson and D.R. Wolf. Estimators for the Cauchy distribution. In G.R. Heidbreder, editor, Maximum entropy and Bayesian methods, Kluwer Academic Publisher, Netherlands, 1996. • N. Henze and Ya.Yu. Nikitin. Two-sample tests based on the integrated empirical processes. Communications in Statistics – Theory and Methods, 2003. • A. Janseen. Testing nonparametric statistical functionals with application to rank tests. Journal of Statistical Planning and Inference, 1999. • F.Rublik. A quantile goodness-of-fit test for the Cauchy distribution, based on extreme order statistics. Applications of Mathematics, 2001. • D.C. Vaughan. The generalized secant hyperbolic distribution and its properties. Communications in Statistics – Theory and Methods, 2002. Brownian bridge and nonparametric rank tests