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 - PowerPoint PPT PresentationTRANSCRIPT
Brownian Bridge and nonparametric rank tests
Olena Kravchuk School of Physical SciencesDepartment of MathematicsUQ
Olena Kravchuk Brownian bridge and nonparametric rank tests
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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 tests
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Definition of Brownian bridge
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Construction of the BB
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Varying the coefficients of the bridge
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Two useful properties
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Ranks and anti-ranks
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First sample Second sample
Index 1 2 3 4 5 6
Data 5 7 0 3 1 4
Rank 5 6 1 3 2 4
Anti-rank 3 5 4 6 1 2
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Olena Kravchuk Brownian bridge and nonparametric rank tests
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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.
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Olena Kravchuk Brownian bridge and nonparametric rank tests
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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.
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Olena Kravchuk Brownian bridge and nonparametric rank tests
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From real data to the random bridge
First sample Second sample
Index, i 1 2 3 4 5 6
Data, X 5 7 0 3 1 4
Constant, c 0.41 0.41 0.41 -0.41 -0.41 -0.41
Rank, R 5 6 1 3 2 4
Anti-rank, D 3 5 4 6 1 2
Bridge, Z 0.41 0 -0.41 -0.82 -0.41 0
41.06/1,3 cnm
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Symmetric distributions and the BB
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LHSD
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Random walk model: no difference in distributions
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Location and scale alternatives
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Random walk: location and scale alternatives
Shift = 2 Scale = 2
Olena Kravchuk Brownian bridge and nonparametric rank tests
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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.
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Olena Kravchuk Brownian bridge and nonparametric rank tests
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Trigonometric scores rank statistics
The Cramer-von Mises statistic
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Olena Kravchuk Brownian bridge and nonparametric rank tests
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Combined trigonometric scores rank statistics
The first and second coefficients are uncorrelated
Fast convergence to the asymptotic distribution
22
2121 SSS
2221 ~ S
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)
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Percentage points for the first component (one-sample)
Durbin and Knott – Components of Cramer-von Mises Statistics
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Percentage points for the first component (two-sample)
Kravchuk – Rank test of location optimal for HSD
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Some tests of location
Olena Kravchuk Brownian bridge and nonparametric rank tests
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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 tests
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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.
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Olena Kravchuk Brownian bridge and nonparametric rank tests
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Functionals on the bridge
When the score function is defined and differentiable, it is easy to derive the corresponding functional.
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Olena Kravchuk Brownian bridge and nonparametric rank tests
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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 tests
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Numerical examples: test of location
Normal 2Normal 1
750
700
650
600
550
500
450
400
350
300
Boxplots of Normal 1 and Normal 2(means are indicated by solid circles)
t-test Wilcoxon S1
p-value 0.150 0.162 0.154
CI95% (-172.4,28.6) (-185.0,25.0) (-183.0,25.0)
1. Normal, N(500,1002)2. Normal, N(580,1002)
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Normals2Normals1
600
500
400
300
200
100
Nor
mal
s1
Numerical examples: test of scale
F-test Siegel-Tukey S2
p-value 0.123 0.064 0.054
1. Normal, N(300,2002)2. Normal, N(300,1002)
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Normalc2Normalc1
1000
900
800
700
600
500
400
300
200
Nor
mal
c1Numerical examples: combined test
F-test t-test S12+S2
2 Lepage CM
p-value 0.021 0.174 0.018 0.035 0.010
1. Normal, N(580,2002)2. Normal, N(500,1002)
Olena Kravchuk Brownian bridge and nonparametric rank tests
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When two colour histograms are compared, nonparametric tests are required as a priori knowledge about the colour probability distribution isgenerally 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 tests
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Application: grey-scale images
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Application: grey-scale images, histograms
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Application: colour images
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Useful books
1. H. Cramer. Mathematical Methods of Statistics. Princeton University Press, Princeton, 19th edition, 1999.
2. G. Grimmett and D. Stirzaker. Probability and Random Processes. Oxford University Press, N.Y., 1982.
3. J. Hajek, Z. Sidak and P.K. Sen. Theory of Rank Tests. Academic Press, San Diego, California, 1999.
4. F. Knight. Essentials of Brownian Motion and Diffusion. AMS, Providence, R.I., 1981.
5. K. Knight. Mathematical Statistics. Chapman & Hall, Boca Raton, 2000.
6. J. Maritz. Distribution-free Statistical Methods. Monographs on Applied Probability and Statistics. Chapman & Hall, London, 1981.
Olena Kravchuk Brownian bridge and nonparametric rank tests
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Interesting papers
1. J. Durbin and M. Knott. Components of Cramer – von Mises statistics. Part 1. Journal of the Royal Statistical Society, Series B., 1972.
2. 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.
3. N. Henze and Ya.Yu. Nikitin. Two-sample tests based on the integrated empirical processes. Communications in Statistics – Theory and Methods, 2003.
4. A. Janseen. Testing nonparametric statistical functionals with application to rank tests. Journal of Statistical Planning and Inference, 1999.
5. F.Rublik. A quantile goodness-of-fit test for the Cauchy distribution, based on extreme order statistics. Applications of Mathematics, 2001.
6. D.C. Vaughan. The generalized secant hyperbolic distribution and its properties. Communications in Statistics – Theory and Methods, 2002.