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

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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

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Definition of Brownian bridge
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Construction of the BB
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Varying the coefficients of the bridge
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Two useful properties
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Ranks and anti-ranks
First sample First sample First sample Second sample Second 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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Simple linear rank statistic

• Any simple linear rank statistic is a linear
  combination of the scores, as, and the
  constants, cs.
• 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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Constrained random walk on pooled data

• Combine all the observations from two samples
  into the pooled sample, Nmn.
• 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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From real data to the random bridge
First sample First sample First sample Second sample Second 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
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Symmetric distributions and the BB
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Random walk model no difference in distributions
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Location and scale alternatives
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Random walk location and scale alternatives
Scale 2
Shift 2
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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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Trigonometric scores rank statistics

• The Cramer-von Mises statistic

• The first and second Fourier coefficients

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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)

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Percentage points for the first component
(one-sample)

• Durbin and Knott Components of Cramer-von Mises
  Statistics

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Percentage points for the first component
(two-sample)

• Kravchuk Rank test of location optimal for HSD

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Some tests of location
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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)
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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, bs.

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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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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

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Numerical examples test of location

1. Normal, N(500,1002)
2. Normal, N(580,1002)

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)
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Numerical examples test of scale

1. Normal, N(300,2002)
2. Normal, N(300,1002)

F-test Siegel-Tukey S2
p-value 0.123 0.064 0.054
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Numerical examples combined test

1. Normal, N(580,2002)
2. Normal, N(500,1002)

F-test t-test S12S22 Lepage CM
p-value 0.021 0.174 0.018 0.035 0.010
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Application palette-based images
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Application grey-scale images
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Application grey-scale images, histograms
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Application colour images
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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.

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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.