ASYMPTOTIC OPTIMALITY OF RANK TESTS OF LOCATION AND THE BROWNIAN BRIDGE

1. ASYMPTOTIC OPTIMALITY OF RANK TESTS OF LOCATION AND THE BROWNIAN BRIDGE Kravchuk, O.Y. Department of Mathematics, The University of Queensland

2. Foreword A two-sample rank test of location can be simply constructed by manipulating the score function. There are many well-known two-sample rank tests like the Wilcoxon, median, Fisher, van der Waerden, etc. Recently, we have introduced a new test with trigonometric scores. What is the practical advantage of having so many rank tests of location? Given a particular data sample, how can one select the most appropriate rank test? This talk demonstrates how interpreting a simple linear rank statistic as a functional on the Brownian Bridge helps visualise a two-sample rank test procedure and decide upon an efficient test. Visualizing rank tests of location

3. Two-sample rank test of location • We are given two random samples of size m and n correspondingly. • Null hypothesis: all the observations are independent random variables identically distributed according to some one-dimensional absolutely continuous density f. • Alternative hypothesis: the distributions of the observations of the first and second samples differ in the location parameter only. Visualizing rank tests of location

4. 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. Visualizing rank tests of location

5. Constrained random walk • Combine all the observations into the pooled sample. • 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 B converges in distribution to the Brownian Bridge as the smaller sample increases. Visualizing rank tests of location

6. From the real data to the random bridge Visualizing rank tests of location

7. 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. Visualizing rank tests of location

8. Data distributions, hypotheses and random bridges • We illustrate the bridges under the null and alternative hypotheses on large samples from three bell-shaped distributions: normal, hyperbolic secant and logistic. Visualizing rank tests of location

9. 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. Visualizing rank tests of location

10. Functionals on the bridge • When the score function is defined and differentiable, it is easy to derive the corresponding functional. • The tools developed for the Brownian Bridge are used to investigate on the properties of the rank statistics. Visualizing rank tests of location

11. After word The behaviour of the real data is reflected by the shape of the random bridge constructed on the sorted data sample. Interpreting rank statistics as functionals on the random bridge allows one to make a connection between the rank statistic and the data distribution in order to select an efficient test of the location alternative. The visualization approach to rank tests discussed in this talk may be useful in practical applications as well as in teaching Nonparametric Statistics. Visualizing rank tests of location

12. Acknowledgements • Dr Phil Pollett (UQ) for his excellent supervision of my PhD on nonparametric tests, numerous valuable suggestions on and discussions of the stochastic properties of the statistics and general support and understanding during this research • Prof Kaye Basford (UQ) for her support and encouragement of the research and for her helpful suggestions for the presentations and applications of the tests • Prof Jerzy Filar and A/Prof Bruce Brown (UniSA) for their initial discussion of the trigonometric scores statistics Visualizing rank tests of location