Calibrating Silverman's Test for Multimodality


Principal Investigator

Peter Hall

Centre for Mathematics and Its Applications

The project was concerned with testing for the number of modes in a population. We were interested in a popular test (proposed by B.W. Silverman) for multimodality which is widely believed to be conservative. Our aim was to simulate the behaviour of the test in order to measure its conservativeness and to correct for it, at least, for large sample sizes. Theoretical results derived by us show that for large sample sizes the behaviour of the test does not depend on the distribution from which our samples were drawn.    



Matthew York

Centre for Mathematics and Its Applications





v65, w12 - VPP


What are the results to date and the future of the work?

The test was calibrated to remove its conservativeness. The calculation for the amount of correction required for the test took 9.5 hours on the VPP. It is estimated that the same calculations would have taken over six months on a departmental work station. Once the correction was developed we performed a large scale simulation study of our new test to investigate its performance. We plan to use this corrected test in work related to the estimation of the components in a mixture of smooth regression curves.

What computational techniques are used?

Our algorithm consisted of drawing a random sample from a distribution and constructing several kernel density estimates of the distribution's probability density function from the sample, using different smoothing parameters. These density estimates were efficiently calculated using fast Fourier transforms. The aim was to search for the smallest smoothing parameter that yielded a density estimate with one mode. (The number of modes of a density estimate is a decreasing function of the smoothing parameter.) A large number (3000-7000) of resamples were then generated and this critical smoothing parameter was computed similarly for each of these resamples.

This whole procedure was then repeated a large number (again 3000-7000) of times.

Once the critical smoothing parameters had been determined for all the samples and resamples they could be used to approximate the distribution of the test statistic for the test for multimodality. This allowed us to determine the level error of the existing test and to correct for this error. Similar simulations permitted us to assess the performance of the new test in a wide range of situations.

- Appendix A

This procedure could be applied to all of the samples (and resamples) simultaneously as none of the calculations from one data set were required in the calculations for another. This structure of the problem allowed for a high degree of vectorisation to be employed. Vectorisation rates of over 90% were generally recorded.
Appendix A -