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

Olena Kravchuk Brownian bridge and nonparametric rank tests2 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

Olena Kravchuk Brownian bridge and nonparametric rank tests3 Definition of Brownian bridge

Olena Kravchuk Brownian bridge and nonparametric rank tests4 Construction of the BB

Olena Kravchuk Brownian bridge and nonparametric rank tests5 Varying the coefficients of the bridge

Olena Kravchuk Brownian bridge and nonparametric rank tests6 Two useful properties

Olena Kravchuk Brownian bridge and nonparametric rank tests7 Ranks and anti-ranks First sampleSecond sample Index Data Rank Anti-rank354612

Olena Kravchuk Brownian bridge and nonparametric rank tests8 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.

Olena Kravchuk Brownian bridge and nonparametric rank tests9 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.

Olena Kravchuk Brownian bridge and nonparametric rank tests10 From real data to the random bridge First sampleSecond sample Index, i Data, X Constant, c Rank, R Anti-rank, D Bridge, Z

Olena Kravchuk Brownian bridge and nonparametric rank tests11 Symmetric distributions and the BB

Olena Kravchuk Brownian bridge and nonparametric rank tests12 Random walk model: no difference in distributions

Olena Kravchuk Brownian bridge and nonparametric rank tests13 Location and scale alternatives

Olena Kravchuk Brownian bridge and nonparametric rank tests14 Random walk: location and scale alternatives Shift = 2 Scale = 2

Olena Kravchuk Brownian bridge and nonparametric rank tests15 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.

Olena Kravchuk Brownian bridge and nonparametric rank tests16 Trigonometric scores rank statistics The Cramer-von Mises statistic The first and second Fourier coefficients:

Olena Kravchuk Brownian bridge and nonparametric rank tests17 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 (S W is the Wilcoxon statistic and S A-B is the Ansari-Bradley, adopted Wilcoxon, statistic)

Olena Kravchuk Brownian bridge and nonparametric rank tests18 Percentage points for the first component (one-sample) Durbin and Knott – Components of Cramer-von Mises Statistics

Olena Kravchuk Brownian bridge and nonparametric rank tests19 Percentage points for the first component (two-sample) Kravchuk – Rank test of location optimal for HSD

Olena Kravchuk Brownian bridge and nonparametric rank tests20 Some tests of location

Olena Kravchuk Brownian bridge and nonparametric rank tests21 Trigonometric scores rank estimators Location estimator of the HSD (Vaughan) Scale estimator of the Cauchy distribution (Rublik) Trigonometric scores rank estimator (Kravchuk)

Olena Kravchuk Brownian bridge and nonparametric rank tests22 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.

Olena Kravchuk Brownian bridge and nonparametric rank tests23 Functionals on the bridge When the score function is defined and differentiable, it is easy to derive the corresponding functional.

Olena Kravchuk Brownian bridge and nonparametric rank tests24 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

Olena Kravchuk Brownian bridge and nonparametric rank tests25 Numerical examples: test of location t-testWilcoxonS1S1 p-value CI 95% (-172.4,28.6)(-185.0,25.0)(-183.0,25.0) 1.Normal, N(500,100 2 ) 2.Normal, N(580,100 2 )

Olena Kravchuk Brownian bridge and nonparametric rank tests26 Numerical examples: test of scale F-testSiegel-TukeyS2S2 p-value Normal, N(300,200 2 ) 2.Normal, N(300,100 2 )

Olena Kravchuk Brownian bridge and nonparametric rank tests27 Numerical examples: combined test F-testt-testS 1 2 +S 2 2 LepageCM p-value Normal, N(580,200 2 ) 2.Normal, N(500,100 2 )

Olena Kravchuk Brownian bridge and nonparametric rank tests28 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

Olena Kravchuk Brownian bridge and nonparametric rank tests29 Application: grey-scale images

Olena Kravchuk Brownian bridge and nonparametric rank tests30 Application: grey-scale images, histograms

Olena Kravchuk Brownian bridge and nonparametric rank tests31 Application: colour images

Olena Kravchuk Brownian bridge and nonparametric rank tests32 Useful books 1.H. Cramer. Mathematical Methods of Statistics. Princeton University Press, Princeton, 19 th edition, G. Grimmett and D. Stirzaker. Probability and Random Processes. Oxford University Press, N.Y., J. Hajek, Z. Sidak and P.K. Sen. Theory of Rank Tests. Academic Press, San Diego, California, F. Knight. Essentials of Brownian Motion and Diffusion. AMS, Providence, R.I., K. Knight. Mathematical Statistics. Chapman & Hall, Boca Raton, J. Maritz. Distribution-free Statistical Methods. Monographs on Applied Probability and Statistics. Chapman & Hall, London, 1981.

Olena Kravchuk Brownian bridge and nonparametric rank tests33 Interesting papers 1.J. Durbin and M. Knott. Components of Cramer – von Mises statistics. Part 1. Journal of the Royal Statistical Society, Series B., 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, N. Henze and Ya.Yu. Nikitin. Two-sample tests based on the integrated empirical processes. Communications in Statistics – Theory and Methods, A. Janseen. Testing nonparametric statistical functionals with application to rank tests. Journal of Statistical Planning and Inference, F.Rublik. A quantile goodness-of-fit test for the Cauchy distribution, based on extreme order statistics. Applications of Mathematics, D.C. Vaughan. The generalized secant hyperbolic distribution and its properties. Communications in Statistics – Theory and Methods, 2002.