Improved least squares topology testing and estimation.

Generalized least squares (GLS) methods provide a relatively fast means of constructing a confidence set of topologies. Because they utilize information about the covariances between distances, it is reasonable to expect additional efficiency in estimation and confidence set construction relative to other least squares (LS) methods. Difficulties have been found to arise in a number of practical settings due to estimates of covariance matrices being ill conditioned or even noninvertible. We present here new ways of estimating the covariance matrices for distances that are much more likely to be positive definite, as the actual covariance matrices are. A thorough investigation of performance is also conducted. An alternative to GLS that has been proposed for constructing confidence sets of topologies is weighted least squares (WLS). As currently implemented, this approach is equivalent to the use of GLS but with covariances set to zero rather than being estimated. In effect, this approach assumes normality of the estimated distances and zero covariances. As the results here illustrate, this assumption leads to poor performance. A 95% confidence set is almost certain to contain the true topology but will contain many more topologies than are needed. On the other hand, the results here also indicate that, among LS methods, WLS performs quite well at estimating the correct topology. It turns out to be possible to improve the performance of WLS for confidence set construction through a relatively inexpensive normal parametric bootstrap that utilizes the same variances and covariances of GLS. The resulting procedure is shown to perform at least as well as GLS and thus provides a reasonable alternative in cases where covariance matrices are ill conditioned.