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

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

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

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

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

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

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

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

Hyperbolic secant distribution Olena Kravchuk Trigonometric scores rank statistics

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

Random walks under the alternative Olena Kravchuk Trigonometric scores rank statistics

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

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

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