1. Trigonometric scores rank statistics
Olena Kravchuk
(supervisor: Phil Pollett)
Department of Mathematics, UQ

2. Trigonometric scores rank statistics
Ranks and anti-ranks
First sample
Second sample
Index 1 2 3 4 5 6
Data 7
Rank
Antirank
Olena Kravchuk
Trigonometric scores rank statistics

3. 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.
Olena Kravchuk
Trigonometric scores rank statistics

4. Trigonometric scores rank statistics
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.
Olena Kravchuk
Trigonometric scores rank statistics

5. Cramer-von Mises statistic
Brownian Bridge
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.
Olena Kravchuk
Trigonometric scores rank statistics

6. Trigonometric scores rank statistics
First components of CM
Durbin and Knott – Components of Cramer-von mises Statistics
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

7. Percentage points for the first component (one-sample)
Durbin and Knott – Components of Cramer-von mises Statistics
Olena Kravchuk
Trigonometric scores rank statistics

8. Percentage points for the first component (two-sample)
Kravchuk – Rank test of location optimal for HSD
Olena Kravchuk
Trigonometric scores rank statistics

9. Hyperbolic secant distribution
Olena Kravchuk
Trigonometric scores rank statistics

10. Trigonometric scores rank statistics
Some tests of location
Olena Kravchuk
Trigonometric scores rank statistics

11. Random walks under the alternative
Olena Kravchuk
Trigonometric scores rank statistics

12. Trigonometric scores rank statistics
Small-sample power
Olena Kravchuk
Trigonometric scores rank statistics

13. Trigonometric scores rank statistics
Small-sample power
Olena Kravchuk
Trigonometric scores rank statistics

14. Trigonometric scores rank estimators
Location estimator of the HSD
Scale estimator of the Cauchy
Trigonometric scores rank estimator
Olena Kravchuk
Trigonometric scores rank statistics