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## Brownian Bridge and nonparametric rank tests

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**Brownian Bridge and nonparametric rank tests**Olena Kravchuk School of Physical Sciences Department of Mathematics UQ**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**Definition of Brownian bridge**Brownian bridge and nonparametric rank tests**Construction of the BB**Brownian bridge and nonparametric rank tests**Varying the coefficients of the bridge**Brownian bridge and nonparametric rank tests**Two useful properties**Brownian bridge and nonparametric rank tests**Ranks and anti-ranks**Brownian bridge and nonparametric rank tests**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**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**From real data to the random bridge**Brownian bridge and nonparametric rank tests**Symmetric distributions and the BB**Brownian bridge and nonparametric rank tests**Random walk model: no difference in distributions**Brownian bridge and nonparametric rank tests**Location and scale alternatives**Brownian bridge and nonparametric rank tests**Random walk: location and scale alternatives**Scale = 2 Shift = 2 Brownian bridge and nonparametric rank tests**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**Trigonometric scores rank statistics**• The Cramer-von Mises statistic • The first and second Fourier coefficients: Brownian bridge and nonparametric rank tests**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**Percentage points for the first component (one-sample)**Durbin and Knott – Components of Cramer-von Mises Statistics Brownian bridge and nonparametric rank tests**Percentage points for the first component (two-sample)**Kravchuk – Rank test of location optimal for HSD Brownian bridge and nonparametric rank tests**Some tests of location**Brownian bridge and nonparametric rank tests**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**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**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**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**Numerical examples: test of location**• Normal, N(500,1002) • Normal, N(580,1002) Brownian bridge and nonparametric rank tests**Numerical examples: test of scale**• Normal, N(300,2002) • Normal, N(300,1002) Brownian bridge and nonparametric rank tests**Numerical examples: combined test**• Normal, N(580,2002) • Normal, N(500,1002) Brownian bridge and nonparametric rank tests**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**Application: grey-scale images**Brownian bridge and nonparametric rank tests**Application: grey-scale images, histograms**Brownian bridge and nonparametric rank tests**Application: colour images**Brownian bridge and nonparametric rank tests**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**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