brownian bridge and nonparametric rank tests

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

)()(

'#)(

YdDYrR

irdysyyr

iiii

dr

ii

ii

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

)()(

)()(

1

11

i

jj

i

N

iii

N

ii

DciT

dcarac

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.

)1,0()var(

.1

1)var(,0)(

,1,1,0

).()(

1

2

11

2

1

11

NS

SaaN

SSE

aaN

cc

DcaRacS

N

ii

N

ii

N

ii

N

ii

N

iii

N

iii

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.

.1,)1()(,

),(

.1,0)(;

.,1

,,1

1

2

1

tNitNTtNiTtNitNiT

tiZ

ccDcT

miNmn

n

miNmn

mc

ii

i

i iii

i

jjii

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

2exp

21)(,

241)(),(1)(

2yyfysechyfysechyf N2

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.

1

0

111

)()(

).(

dttBtbS

RacNb

cZNbS

N

iii

N

i

N

Rj

ji

N

ii

i

i

Olena Kravchuk Brownian bridge and nonparametric rank tests

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Trigonometric scores rank statistics

The Cramer-von Mises statistic

)1,0(~.)(,1

.,)(

122

21

0

22

1

22

1

222

NXiXdttBWT

NW

mm

nnnmmnW

i

iN

ii

iii

N

ii

The first and second Fourier coefficients:

dttBtSTNi

NS

dttBtSTNi

NS

N

ii

N

ii

)()2sin(22,2sin22

2

)()sin(2,sin22

1

02

12

1

01

11

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.

.),(])[1(lim

]1,0[,))(())(('),( 1

1

dudbuuNa

uuFfuFffu

N

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.

)4/1,0(~)5.0()5.0()(

)()(),,0(~)()(,)()(

1

0

1

0

1

0

21

0

222

0

NBdtttB

dttdttNdttBtbdxxbtt

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

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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.