Trigonometric scores rank
statistics
Olena Kravchuk
(supervisor: Phil Pollett)
Department of Mathematics, UQ
Ranks and anti-ranks
N

ci a ( ri ) 
i 1
ri ( y ) # y ' s  yi
N

d ri  rd i  i
ai c( d i )
Ri  ri (Y )
i 1
First sample
Di  d i (Y )
Second sample
Index
1
2
3
4
5
6
Data
5
7
0
3
1
4
Rank
5
6
1
3
2
4
Antirank 3
5
4
6
1
2
Olena Kravchuk
Trigonometric scores rank statistics
2
Simple linear rank statistic
Let us consider the two-sample location problem. Assume that the distributions
are continuous of the same location family, f, and may differ in location, μ, only.
The inference is made from two random samples of size m and n, N=m+n,
drawn from the distributions.
1
H 0 :   0, H A :   0; P( R  r | H 0 ) 
N!
N

1
S  ci a ( Ri ). E ( S | H 0 ) 
N
i 1
N
N
c a ;
i
i 1
i
i 1
N
1 N
var( S | H 0 ) 
( ci  c ) 2 ( ai  a ) 2
N  1 i 1
i 1

Olena Kravchuk

Trigonometric scores rank statistics
3
Random walk model
Let us start a random walk at the origin and walk on the pooled data sample
moving up every time we see an observation from the first sample and down
every time we see an observation from the second sample. Let us pin the walk
T down by assigning the appropriate up/down steps, c’s.
  1
mn
, 1  i  m,

c

m mn

.
ci  
c    1 mn , m  i  N

n mn
mn

ci  0,

ci2  1.
i 1
i 1
Ti 
mn
i
 c( D ).
i
j 1
T  Tc (), Tc  B.
Olena Kravchuk
Trigonometric scores rank statistics
4
Brownian Bridge


iid
Zi
B (t ) 
sin( it ), Z i ~ N (0,1)
 j 1 i
2
Cramer-von Mises statistic
One of the common form of the statistic is given below. There di is the
difference between the sample distribution functions at the ith point in the
pooled sample.
W 2  mn ( m  n ) 2
N

d i2 , d i 
i 1
1
W2 
N
Olena Kravchuk
N

i 1
ni mi
 .
n m
1

Ti 2 , W 2  B 2 (t )dt 
0
Trigonometric scores rank statistics

Z i2
i 
i 1
2
2
.
5
First components of CM
Durbin and Knott – Components of Cramer-von mises Statistics
Zi 
2
N
N
cos( jx ),
r
xr  F ( yr ) ~ Uniform(0,1)
j 1
The random variable cos(jπx) is the projection of a unit vector on a fixed
vector where the angle between the two vectors is distributed uniformly
between 0 and jπ. Evidently for testing the significance of individual
components we only need significance points for the first component.
Olena Kravchuk
Trigonometric scores rank statistics
6
Percentage points for the first component
(one-sample)
Durbin and Knott – Components of Cramer-von mises Statistics
Olena Kravchuk
Trigonometric scores rank statistics
7
Percentage points for the first component
(two-sample)
Kravchuk – Rank test of location optimal for HSD
Olena Kravchuk
Trigonometric scores rank statistics
8
Hyperbolic secant distribution
f ( y) 
1

sech( y)
X 1 ~ N (0,1), X 2 ~ N (0,1), log
Y
X1
~ HSD(0,1)
X2
X1
~ Cauchy(0,1), log Y ~ HSD(0,1)
X2
Olena Kravchuk
Trigonometric scores rank statistics
9
Some tests of location
Olena Kravchuk
Trigonometric scores rank statistics
10
Random walks under the alternative
Olena Kravchuk
Trigonometric scores rank statistics
11
Small-sample power
Olena Kravchuk
Trigonometric scores rank statistics
12
Small-sample power
Olena Kravchuk
Trigonometric scores rank statistics
13
Trigonometric scores rank estimators
Location estimator of the HSD
Scale estimator of the Cauchy
Trigonometric scores rank estimator
Olena Kravchuk
Trigonometric scores rank statistics
14